Gas Laws and Kinetic Theory
| Safle: | Young Education |
| Cwrs: | Thermal Energy and Gas Laws |
| Llyfrau: | Gas Laws and Kinetic Theory |
| Argraffwyd gan: | Guest user |
| Dyddiad: | Dydd Llun, 9 Tachwedd 2026, 4:42 PM |
1. Kinetic Molecular Theory
Learning outcomes
- I can describe the main assumptions of the kinetic molecular theory of gases.
- I can explain how gas pressure results from particle collisions.
- I can relate temperature to the average kinetic energy of gas particles.
- I can explain why gases are compressible and expand to fill their containers.
- I can use the particle model to explain the behavior of gases.
Kinetic Molecular Theory
What Is the Kinetic Molecular Theory?
The kinetic molecular theory (KMT) is a particle model used to explain the behavior of gases.
Instead of thinking of a gas as a continuous substance, KMT describes it as an enormous number of tiny particles moving constantly and randomly.
The model helps explain:
- gas pressure
- temperature
- expansion
- compression
- diffusion
- changes in pressure when gases are heated or cooled
The Main Assumptions of Kinetic Molecular Theory
For an ideal gas, the kinetic molecular theory makes several simplifying assumptions.
Gas particles are extremely small
Individual particles occupy very little volume compared with the total volume of the gas.
Most of the space in a gas is therefore empty space.
Gas particles move continuously
Particles are in constant, rapid and random motion.
They travel in approximately straight lines between collisions.
Particles collide
Gas particles collide with:
- other gas particles
- the walls of their container
Collisions are elastic
In an idealized elastic collision, the total kinetic energy of the colliding particles is conserved.
Particles may exchange kinetic energy during a collision, but the total is unchanged.
Attractive and repulsive forces are negligible
Ideal gas particles are assumed not to attract or repel each other significantly except during collisions.
Temperature is related to average kinetic energy
The average translational kinetic energy of gas particles depends on the gas's absolute temperature.
These assumptions form the basis of the ideal-gas particle model.
The Particle Model of a Gas
Gas particles are much farther apart than particles in solids or liquids.
They move freely throughout their container.
This particle arrangement explains many familiar gas properties.
| Property | Particle Explanation |
|---|---|
| No fixed shape | Particles move freely |
| No fixed volume | Particles spread throughout container |
| Compressible | Large spaces exist between particles |
| Low density | Particles are widely separated |
| Produces pressure | Particles collide with container walls |
| Diffuses | Particles move randomly and mix |
Gas Particles Are in Constant Motion
Gas particles never simply sit still.
They move:
- rapidly
- randomly
- in many different directions
Between collisions, an ideal gas particle travels approximately in a straight line.
After a collision, its direction may change.
A sample of gas therefore contains enormous numbers of particles continually moving and colliding.
What Is Gas Pressure?
Pressure is force acting per unit area.
p = F/A
Gas pressure is produced by collisions between gas particles and the walls of their container.
Every time a particle strikes a wall and rebounds, its momentum changes.
The wall exerts a force on the particle.
By Newton's Third Law, the particle exerts a force on the wall.
One collision produces an extremely small force.
But a container may experience billions upon billions of particle collisions.
Together, these collisions produce measurable gas pressure.
Pressure at the Particle Level
Imagine gas trapped inside a balloon.
Particles constantly strike the inner surface of the balloon.
Each collision exerts a tiny force.
The combined effect of these collisions creates an outward pressure on the balloon.
The surrounding atmosphere also produces pressure on the outside.
The balloon's size depends partly on the balance of these forces and the tension in the rubber.
Increasing the Number of Particles
Suppose more gas particles are added to a rigid container without changing its volume significantly.
There are now more particles available to strike the walls.
Therefore:
more particles → more wall collisions → greater pressure
if temperature and volume are otherwise unchanged.
This explains why pumping additional air into a fixed-volume container increases its pressure.
Temperature and Particle Motion
Temperature has a precise meaning in kinetic molecular theory.
The absolute temperature of a gas is directly related to the average translational kinetic energy of its particles.
For an individual particle:
KE = ½mv²
where:
- KE = kinetic energy
- m = particle mass
- v = particle speed
Increasing particle speed has a particularly strong effect because speed is squared.
Temperature and Average Kinetic Energy
For an ideal gas:
average kinetic energy ∝ absolute temperature
Therefore:
higher temperature → greater average kinetic energy
and:
lower temperature → lower average kinetic energy
Temperature must be measured using the kelvin scale when using this proportional relationship.
What Happens When a Gas Is Heated?
Suppose gas is trapped in a rigid sealed container.
When the gas is heated:
temperature increases
↓
average kinetic energy increases
↓
particles move faster on average
↓
particles collide with the walls more frequently
and
collisions involve greater momentum changes
↓
pressure increases
This provides a particle-level explanation for why heating a gas at constant volume increases its pressure.
What Happens When a Gas Is Cooled?
Cooling produces the opposite effect.
temperature decreases
↓
average kinetic energy decreases
↓
particles move more slowly on average
↓
wall collisions become less frequent and less forceful
↓
pressure decreases
provided the container's volume remains constant.
Particles Do Not All Move at the Same Speed
Temperature describes the average kinetic energy of the particles.
It does not mean every particle has exactly the same speed or kinetic energy.
At any moment:
- some particles move relatively slowly
- many have intermediate speeds
- some move very rapidly
Collisions continually redistribute kinetic energy between particles.
Therefore, a gas contains a distribution of particle speeds.
Why Gases Are Compressible
One of the most important properties of gases is that they can be compressed significantly.
Why?
Because gas particles are separated by relatively large distances.
Most of a gas's volume is empty space between particles.
When a gas is compressed:
container volume decreases
but the particles themselves do not need to become much smaller.
Instead:
the spaces between the particles decrease
This is very different from most solids and liquids, where particles are already packed relatively close together.
Compressing a Gas
Imagine air trapped inside a sealed syringe.
When the plunger is pushed inward:
- the number of particles remains approximately constant
- the available volume decreases
- particles travel shorter distances before hitting the walls
- collisions with the walls become more frequent
If temperature remains constant:
volume decreases → pressure increases
This is the particle explanation behind Boyle's law.
Why Gases Fill Their Containers
A gas has no fixed shape and no fixed volume.
Gas particles move freely in all directions.
If gas is released into a larger container, particles continue moving randomly until they are distributed throughout the available space.
Therefore:
a gas expands to fill its entire container
The particles are not deliberately moving toward empty regions.
Their random motion naturally spreads them throughout the available volume.
Gas Expansion
Suppose a partition separates two halves of a container.
One side contains gas.
The other side is empty.
When the partition is removed, gas particles move into the empty region.
Eventually, particles become distributed throughout the entire container.
This demonstrates why gases expand spontaneously into available space.
Diffusion
Diffusion is the net spreading of particles from regions of higher concentration toward regions of lower concentration as a result of random particle motion.
A familiar example is perfume released in one part of a room.
Eventually, the smell can be detected farther away.
The perfume particles:
- move randomly
- collide with air molecules
- gradually spread throughout the room
Diffusion is therefore strong evidence for the constant random motion of particles.
Temperature and Diffusion
Increasing temperature generally increases the average kinetic energy of gas particles.
Particles therefore move faster on average.
This tends to increase the rate of diffusion.
Therefore:
higher temperature → faster particle motion → generally faster diffusion
Gas Density
Gas density is:
ρ = m/V
Gases usually have much lower densities than solids and liquids because their particles are widely separated.
If a fixed mass of gas expands:
volume increases
so:
density decreases
If the same gas is compressed:
volume decreases
so:
density increases
Heating a Gas in a Flexible Container
Now consider gas inside a flexible balloon rather than a rigid container.
When heated:
- particles gain average kinetic energy
- particles move faster
- collisions initially exert greater pressure
- the balloon can expand
As the balloon expands, the particles have more space.
A new balance may eventually be established between internal pressure, external pressure and the balloon's elastic forces.
This demonstrates why the result of heating a gas depends on whether its volume is allowed to change.
Heating Air and Convection
Kinetic molecular theory also helps explain convection.
Suppose air is heated and allowed to expand.
The particles move faster and the gas expands.
If approximately the same mass occupies a larger volume:
ρ = m/V
so density decreases.
The warmer, less-dense air may rise through surrounding cooler, denser air.
This contributes to convection currents.
Hot-Air Balloons
A hot-air balloon provides a practical example.
Air inside the balloon is heated.
The air:
- gains average kinetic energy
- expands
- becomes less dense than the surrounding cooler air
The surrounding denser air produces a buoyant force on the balloon.
If the buoyant force is sufficiently large, the balloon rises.
Kinetic theory therefore connects microscopic particle behavior with a large-scale observable effect.
Gas Pressure in a Tire
Air inside a tire contains particles moving rapidly and colliding with the tire walls.
These collisions create pressure.
If the tire becomes hotter:
temperature increases → average kinetic energy increases → pressure tends to increase
if the tire's volume does not change much.
This is why tire pressure can change with temperature.
Aerosol Cans
Aerosol cans contain gases under pressure.
If an aerosol can is strongly heated:
- gas temperature increases
- particle kinetic energy increases
- collisions with the container become more frequent and forceful
- internal pressure can increase substantially
This is why pressurized containers carry warnings about exposure to excessive heat.
Gas in a Piston
Consider gas trapped beneath a movable piston.
The particles collide with the piston and exert pressure.
If the gas is heated:
- average kinetic energy increases
- particles move faster
- the force on the piston increases
If the piston can move, it may rise.
The gas then expands.
This is an important principle in:
- engines
- turbines
- compressors
- heat engines
Microscopic particle motion can therefore produce macroscopic mechanical motion.
Atmospheric Pressure
The atmosphere consists of enormous numbers of moving gas particles.
These particles collide with:
- Earth's surface
- buildings
- people
- other objects
These collisions contribute to atmospheric pressure.
At sea level, atmospheric pressure is roughly:
101 kPa
We normally do not notice this enormous pressure because forces from atmospheric pressure act in many directions and pressures inside our bodies help balance external pressure.
Pressure Acts in All Directions
Gas particles move randomly in all directions.
They therefore collide with all surfaces of a container.
Gas pressure consequently acts:
- upward
- downward
- sideways
- in every other direction
This is why an inflated balloon pushes outward over its entire surface.
Ideal Gases and Real Gases
The kinetic molecular theory described here is an idealized model.
Real gas particles:
- have finite size
- can attract and repel each other
- do not behave perfectly ideally under all conditions
The ideal-gas approximation works especially well when:
- pressure is relatively low
- temperature is relatively high
Under these conditions, particles are generally far apart and intermolecular forces have less effect.
At very high pressures or low temperatures, real gases can deviate significantly from ideal behavior.
Connecting Microscopic and Macroscopic Properties
One of the strengths of kinetic molecular theory is that it connects invisible particle behavior with properties that can be measured.
| Macroscopic Observation | Particle Explanation |
|---|---|
| Gas has pressure | Particles collide with container walls |
| Heating increases pressure at constant volume | Particles move faster and collisions become more frequent/forceful |
| Gas can be compressed | Large spaces exist between particles |
| Gas fills container | Particles move randomly throughout available space |
| Gas diffuses | Constant random particle motion spreads particles |
| Gas expands when heated | Faster particles can produce expansion if boundaries can move |
| Warm gas can become less dense | Expansion increases volume for the same mass |
This connection between the microscopic and macroscopic worlds is central to thermal physics.
A Particle Explanation Strategy
When asked to explain gas behavior using kinetic molecular theory, think about four questions:
How are the particles arranged?
Are they close together or far apart?
How are the particles moving?
Has their average speed changed?
How often are they colliding?
Has the collision rate changed?
What happens during collisions with the walls?
Has the momentum change or force changed?
This approach can explain many gas-law situations without simply memorizing formulas.
Worked Example: Heating a Sealed Container
A rigid sealed container contains gas at 300 K.
The gas is heated.
Explain why the pressure increases.
Step 1: Temperature
Temperature increases.
Step 2: Kinetic Energy
The average kinetic energy of the particles increases.
Step 3: Particle Motion
Particles move faster on average.
Step 4: Collisions
Particles collide with the container walls more frequently and with greater momentum changes.
Step 5: Pressure
The average force exerted on the walls increases.
Therefore:
gas pressure increases.
Worked Example: Compressing a Gas
A student pushes the plunger of a sealed syringe inward slowly while keeping temperature approximately constant.
Explain why pressure increases.
The number of particles remains constant, but the volume decreases.
The particles have less distance to travel between wall collisions.
Therefore, collisions with the walls occur more frequently.
The average force exerted on the walls increases.
Therefore:
pressure increases as volume decreases.
Common Mistakes
Mistake: "Gas particles expand when a gas expands."
The particles themselves do not significantly increase in size. The average spacing between particles increases.
Mistake: "Gas pressure is caused by particles pushing continuously on the walls."
Gas pressure results from enormous numbers of particle collisions with the walls.
Mistake: "Higher temperature means there are more particles."
Temperature is related to the average kinetic energy of the particles, not the number of particles.
Mistake: "All particles at one temperature move at the same speed."
Gas particles have a range of speeds. Temperature relates to their average kinetic energy.
Mistake: "There is no empty space in a gas."
Most of the volume of an ideal gas consists of space between particles.
Mistake: "Gas particles stop moving when pressure decreases."
Particles continue moving as long as the temperature is above absolute zero.
Mistake: "Heating always increases pressure."
Heating increases pressure strongly when volume is fixed. If the gas can expand, some of the effect appears as increased volume instead.
Key Terms
Kinetic molecular theory (KMT) — model explaining gas behavior in terms of constantly moving microscopic particles.
Ideal gas — theoretical gas whose particles follow the assumptions of kinetic molecular theory.
Kinetic energy — energy an object or particle possesses because of its motion.
Random motion — particle movement with constantly changing directions and no preferred overall direction.
Elastic collision — collision in which total kinetic energy is conserved.
Gas pressure — force per unit area produced by particle collisions with surfaces.
Absolute temperature — temperature measured from absolute zero using the kelvin scale.
Average kinetic energy — average energy of motion of the particles in a sample.
Compression — reduction in the volume occupied by a gas.
Expansion — increase in the volume occupied by a gas.
Diffusion — net spreading of particles resulting from their random motion.
Density — mass per unit volume, calculated using ρ = m/V.
Idealization — simplified representation used to make a physical system easier to understand and analyze.
Key Takeaways
- Kinetic molecular theory explains gases using the motion and collisions of microscopic particles.
- Gas particles are extremely small compared with the distances between them.
- Most of the volume of a gas is empty space between particles.
- Gas particles move continuously and randomly.
- Ideal gas particles travel approximately in straight lines between collisions.
- Collisions are treated as elastic in the ideal-gas model.
- Gas pressure results from particles colliding with container walls.
- Increasing the collision frequency or momentum change can increase pressure.
- Absolute temperature is directly related to the average translational kinetic energy of gas particles.
- Heating a gas increases its particles' average kinetic energy.
- Gas particles do not all move at exactly the same speed.
- Gases are highly compressible because there are large spaces between their particles.
- Gases expand to fill containers because particles move freely and randomly through the available space.
- Diffusion results from constant random particle motion.
- Heating a gas at constant volume generally increases its pressure.
- Compressing a gas at constant temperature increases its pressure.
- Kinetic molecular theory connects microscopic particle behavior to observable properties such as pressure, volume, temperature and density.
Check Your Understanding
- State four assumptions of the kinetic molecular theory of gases.
- How does the spacing between particles in a gas compare with that in a solid or liquid?
- Explain how gas particles produce pressure on the walls of a container.
- What happens to the average kinetic energy of gas particles when temperature increases?
- Why must kelvin be used when relating temperature directly to average kinetic energy?
- Do all gas particles at the same temperature move at exactly the same speed? Explain.
- Why are gases much more compressible than most liquids and solids?
- Explain why a gas expands to fill its container.
- Use kinetic molecular theory to explain why heating a gas in a rigid container increases its pressure.
- Explain why compressing a gas in a sealed syringe increases its pressure.
- Why does diffusion generally occur faster at higher temperatures?
- Explain how heating the air inside a hot-air balloon can cause the air to become less dense.
- A sealed tire becomes warmer during a long journey. Predict what happens to its pressure and explain your answer using particle theory.
- Explain why real gases may behave differently from the ideal-gas model at high pressures or low temperatures.
2. Translational Speed and Kinetic Energy
Learning outcomes
- I can explain that gas particles are in constant random motion.
- I can describe translational motion as the movement of particles from one location to another.
- I can relate particle speed to temperature.
- I can explain the relationship between average translational kinetic energy and absolute temperature.
- I can predict how changes in temperature affect the motion and energy of gas particles.
Translational Speed and Kinetic Energy
Constant Random Motion
Gas particles are in constant random motion.
At any instant, particles are moving:
- at different speeds
- in different directions
- through different parts of the container
They repeatedly collide with other particles and with the walls of their container. These collisions continually change their directions of motion.
This constant motion is one of the central ideas of the kinetic molecular theory of gases.
Translational Motion
Translational motion means movement from one location to another.
For example, a gas particle travelling from one side of a container to the other is undergoing translational motion.
This is different from other possible types of molecular motion, such as:
- rotation
- vibration
For studying ideal gases, translational motion is especially important because it is directly related to gas temperature and pressure.
Three-Dimensional Motion
Gas particles do not simply move left and right.
They move randomly in three dimensions.
A particle might move:
- upward
- downward
- forward
- backward
- sideways
- in any combination of these directions
Between collisions, the ideal gas model treats a particle as moving approximately in a straight line.
After a collision, its direction may change.
Translational Speed
Speed describes how quickly a particle changes position.
For an individual particle:
speed = distance/time
Gas particles typically move extremely quickly.
Even at ordinary temperatures, individual molecules may travel hundreds of metres per second.
However, because they continually collide and change direction, a molecule does not simply travel in a straight line across a room at that speed.
Its overall path is extremely complicated.
Speed Is Not the Same as Velocity
Speed tells us how fast a particle is moving.
Velocity includes both:
- speed
- direction
Because gas particles constantly change direction, their velocities constantly change.
In a gas that is stationary overall, there is no preferred direction of motion.
Particles may have large individual velocities even though the average velocity of the whole sample is approximately zero.
This is an important distinction.
Gas Particles Have Different Speeds
At a particular temperature, gas particles do not all move at exactly the same speed.
Instead, there is a distribution of speeds.
Some particles are:
- relatively slow
- near the average
- relatively fast
Collisions continually transfer kinetic energy between particles.
A particle that is fast at one moment may lose energy in a collision and become slower.
Another particle may gain energy and become faster.
The distribution therefore continually changes at the microscopic level.
Translational Kinetic Energy
Because gas particles are moving, they possess translational kinetic energy.
For one particle:
KE = ½mv²
where:
- KE = translational kinetic energy
- m = mass of the particle
- v = speed of the particle
The equation shows that kinetic energy depends on both mass and speed.
Why Speed Has a Large Effect
Notice that speed is squared:
KE ∝ v²
If the speed of a particle doubles:
KE_new = ½m(2v)²
KE_new = ½m(4v²)
Therefore:
KE_new = 4KE_original
Doubling speed produces four times the kinetic energy for the same particle.
Worked Example
A particle has mass:
m = 4.0 × 10⁻²⁶ kg
and speed:
v = 500 m/s
Calculate its translational kinetic energy.
Use:
KE = ½mv²
KE = ½(4.0 × 10⁻²⁶)(500)²
KE = ½(4.0 × 10⁻²⁶)(250,000)
KE = 5.0 × 10⁻²¹ J
The energy of a single molecule is extremely small because the particle's mass is extremely small.
A macroscopic gas, however, contains an enormous number of particles.
Temperature and Particle Motion
Temperature is directly connected to particle motion.
When a gas is heated:
temperature increases
↓
average translational kinetic energy increases
↓
particles generally move faster
When a gas is cooled:
temperature decreases
↓
average translational kinetic energy decreases
↓
particles generally move more slowly
Average Translational Kinetic Energy
Temperature does not tell us the kinetic energy of one particular particle.
Instead, it is related to the average translational kinetic energy of the enormous collection of particles.
For a monatomic ideal gas particle:
average KE = 3/2 k_B T
where:
- average KE = average translational kinetic energy per particle
- k_B = Boltzmann constant
- T = absolute temperature in K
The Boltzmann constant is:
k_B = 1.38 × 10⁻²³ J/K
The key relationship is:
average KE ∝ T
Absolute Temperature
The relationship between temperature and average kinetic energy requires the kelvin scale.
Kelvin temperature begins at absolute zero.
0 K = −273.15°C
Convert Celsius to kelvin using:
T(K) = T(°C) + 273.15
For example:
27°C ≈ 300 K
127°C ≈ 400 K
The kelvin scale is essential because average translational kinetic energy is proportional to absolute temperature, not Celsius temperature.
Temperature Is Not Particle Speed
A common shortcut is to say:
"temperature measures how fast particles move."
This is useful as a starting idea, but it is not completely precise.
More accurately:
temperature is related to the average translational kinetic energy of particles.
Because:
KE = ½mv²
temperature is related to particle speeds, but particle mass also matters.
Different gases at the same temperature can have different typical molecular speeds.
Same Temperature, Same Average Translational Kinetic Energy
Consider helium and oxygen gases at the same temperature.
The particles have very different masses.
However, if both gases behave ideally:
their particles have the same average translational kinetic energy.
This follows from:
average KE = 3/2 k_B T
The equation contains temperature but not particle mass.
Therefore, at the same temperature:
average KE_He = average KE_O₂
But this does not mean they have the same average speed.
Lighter Particles Move Faster
Since:
KE = ½mv²
two particles with the same kinetic energy but different masses must have different speeds.
A lighter particle must move faster to have the same kinetic energy as a heavier particle.
Therefore, at the same temperature:
lighter gas particles generally have higher characteristic speeds
than heavier gas particles.
Example: Helium and Oxygen
Helium atoms have much less mass than oxygen molecules.
Suppose both gases are at:
300 K
Their average translational kinetic energies are the same because their temperatures are the same.
However:
helium particles move faster on average than oxygen molecules
because helium particles have less mass.
This is an important application of the kinetic-energy equation.
What Happens When Temperature Doubles?
Because:
average KE ∝ T
if absolute temperature doubles:
average KE doubles
For example:
300 K → 600 K
means:
average translational KE doubles
However, particle speed does not double.
Temperature and Speed
Since:
KE ∝ v²
and:
average KE ∝ T
we can combine these ideas to obtain, for particles of the same gas:
typical speed ∝ √T
This means speed increases with the square root of absolute temperature.
Therefore:
300 K → 600 K
does not make typical particle speed twice as large.
Instead:
v_new/v_old = √(600/300)
v_new/v_old = √2
v_new ≈ 1.41v_old
So doubling absolute temperature increases a characteristic particle speed by about 41%, not 100%.
Worked Example: Changing Temperature
A gas is heated from:
200 K to 800 K
Compare the average translational kinetic energies.
Since:
average KE ∝ T
KE₂/KE₁ = 800/200
KE₂/KE₁ = 4
The average translational kinetic energy becomes four times as large.
Now compare characteristic particle speeds:
v₂/v₁ = √(800/200)
v₂/v₁ = √4
v₂/v₁ = 2
The characteristic speed doubles.
Celsius Can Be Misleading
Suppose a gas is heated from:
20°C to 40°C
It may look as though the temperature has doubled.
But convert to kelvin:
20°C ≈ 293 K
40°C ≈ 313 K
The absolute temperature has only increased by about:
313/293 ≈ 1.07
So the average translational kinetic energy increases by only about 7%, not 100%.
Always use kelvin when comparing temperature ratios.
Speed Distribution and Temperature
Heating a gas changes the entire distribution of particle speeds.
At higher temperature:
- the average kinetic energy is greater
- characteristic speeds are greater
- more particles have very high speeds
- the speed distribution becomes broader
The particles still do not all move at the same speed.
Instead, the distribution shifts toward higher speeds.
Cooling a Gas
Cooling produces the opposite effect.
As temperature decreases:
- average translational kinetic energy decreases
- characteristic particle speeds decrease
- fewer particles have very high speeds
The particles continue moving randomly as long as the gas remains above absolute zero.
Absolute Zero
Absolute zero is:
0 K
or approximately:
−273.15°C
In the simplified classical kinetic model:
T → 0 K → average translational kinetic energy → 0
Absolute zero therefore represents the lower limit of the thermodynamic temperature scale.
Modern quantum physics makes the behavior of matter near absolute zero more complicated than the simple classical particle model suggests, but the classical relationship remains extremely useful for ordinary gases.
Particle Speed and Gas Pressure
Particle speed also helps explain gas pressure.
Gas pressure results from particles colliding with container walls.
If a gas in a rigid container is heated:
- temperature increases
- average kinetic energy increases
- particles move faster
- wall collisions occur more frequently
- momentum changes during collisions become greater
Therefore:
pressure increases
if volume and number of particles remain constant.
Particle Speed and Diffusion
Gas particles spread through random motion.
This process is called diffusion.
Higher temperatures produce greater characteristic particle speeds.
Therefore, diffusion generally occurs faster at higher temperatures.
For example, gases and vapors tend to spread more rapidly when their particles have greater kinetic energy.
Particle Speed and Evaporation
The same general particle-energy idea helps explain evaporation.
Particles in a liquid have a range of kinetic energies.
Some surface particles have enough energy to escape from the liquid.
At higher temperatures:
- average kinetic energy is greater
- more particles have sufficiently high energy
- evaporation generally occurs more rapidly
This connects the distribution of particle energies to observable thermal behavior.
Brownian Motion
Evidence for microscopic random motion can be seen through Brownian motion.
Small visible particles suspended in a fluid may appear to move randomly.
The visible particles are being struck unevenly by much smaller, rapidly moving molecules.
Brownian motion provided important evidence for the particle model of matter.
From Microscopic Motion to Macroscopic Temperature
Temperature is a macroscopic quantity—we measure it for an entire sample.
Translational kinetic energy describes microscopic particles.
Kinetic theory connects the two:
microscopic particle motion
↓
average translational kinetic energy
↓
macroscopic temperature
This connection is one of the fundamental ideas of thermal physics.
Predicting the Effect of Temperature Changes
A useful reasoning chain is:
When temperature increases:
T ↑
↓
average KE ↑
↓
characteristic particle speed ↑
↓
particles move more rapidly
This can lead to:
- more frequent wall collisions
- increased pressure at constant volume
- faster diffusion
- faster evaporation
- expansion if the gas is free to expand
When temperature decreases:
T ↓
↓
average KE ↓
↓
characteristic particle speed ↓
↓
particle motion becomes slower on average
Worked Example: Comparing Two Gases
Gas A is at:
250 K
Gas B is the same gas at:
500 K
Average Kinetic Energy
KE_B/KE_A = 500/250
KE_B/KE_A = 2
Gas B's particles have twice the average translational kinetic energy.
Characteristic Speed
v_B/v_A = √(500/250)
v_B/v_A = √2
v_B/v_A ≈ 1.41
The characteristic particle speed is about 1.41 times greater, not twice as great.
Common Mistakes
Mistake: "Gas particles move only when the gas is heated."
Gas particles are already in constant motion. Heating increases their average kinetic energy.
Mistake: "All gas particles at the same temperature have the same speed."
Particles have a distribution of speeds.
Mistake: "Temperature is directly proportional to particle speed."
Temperature is directly proportional to average translational kinetic energy. Characteristic speed is proportional to √T for a given gas.
Mistake: "Doubling temperature doubles particle speed."
Doubling absolute temperature doubles average kinetic energy but increases characteristic speed by √2.
Mistake: "Different gases at the same temperature have the same average speed."
They have the same average translational kinetic energy in the ideal-gas model, but lighter particles generally move faster.
Mistake: "40°C is twice as hot as 20°C."
Temperature ratios involving kinetic energy must use kelvin.
Mistake: "Average velocity and average speed mean the same thing."
In a stationary gas, average velocity can be approximately zero because directions cancel, while average speed is definitely greater than zero.
Key Terms
Translational motion — movement of a particle from one location to another.
Random motion — particle movement without a preferred overall direction.
Speed — rate at which distance is travelled.
Velocity — rate of displacement, including direction.
Translational kinetic energy — energy associated with a particle's movement from one location to another.
Average kinetic energy — mean translational kinetic energy of particles in a sample.
Absolute temperature — temperature measured from absolute zero using the kelvin scale.
Absolute zero — lowest limit of thermodynamic temperature, 0 K.
Boltzmann constant — constant connecting the microscopic energy scale of particles to absolute temperature.
Speed distribution — range of different particle speeds present within a gas.
Diffusion — spreading of particles due to random motion.
Brownian motion — random movement of microscopic visible particles caused by collisions with molecules in a surrounding fluid.
Key Takeaways
- Gas particles are in constant random translational motion.
- Translational motion means movement from one location to another.
- Gas particles travel approximately in straight lines between collisions in the ideal-gas model.
- Particles continually change direction because of collisions.
- Gas particles at one temperature do not all have the same speed.
- Translational kinetic energy is calculated using KE = ½mv².
- Absolute temperature is directly proportional to the average translational kinetic energy of ideal gas particles.
- For a monatomic ideal gas particle, average KE = 3/2 k_B T.
- Temperature comparisons involving kinetic energy must use kelvin.
- Increasing temperature increases average translational kinetic energy.
- Decreasing temperature decreases average translational kinetic energy.
- Doubling absolute temperature doubles average translational kinetic energy.
- Characteristic particle speed is proportional to √T for the same gas.
- Doubling absolute temperature therefore does not double particle speed.
- Different ideal gases at the same temperature have the same average translational kinetic energy.
- Lighter particles generally move faster than heavier particles at the same temperature.
- Increased particle speed can affect pressure, diffusion and other observable gas behavior.
- Kinetic theory connects microscopic particle motion with the macroscopic quantity we call temperature.
Check Your Understanding
- What is meant by translational motion?
- Why is the motion of gas particles described as random?
- Do all particles in a gas move at the same speed? Explain.
- Write the equation for the translational kinetic energy of one particle.
- How does kinetic energy change if a particle's speed doubles?
- What happens to the average translational kinetic energy of a gas when its absolute temperature increases?
- A gas is heated from 300 K to 600 K. By what factor does its average translational kinetic energy change?
- For the same change from 300 K to 600 K, by approximately what factor does a characteristic particle speed change?
- Why must kelvin rather than Celsius be used when comparing temperature ratios?
- Helium and oxygen are both at 300 K. Compare their average translational kinetic energies.
- Which particles generally move faster at the same temperature: helium atoms or oxygen molecules? Explain.
- Explain why heating a gas in a rigid container can increase its pressure.
- How does increasing temperature affect the distribution of particle speeds?
- Explain how kinetic molecular theory connects the microscopic motion of particles with the macroscopic quantity of temperature.
3. Gas Laws
Learning outcomes
- I can describe Boyle's Law, Charles' Law, and Gay-Lussac's Law.
- I can identify which variables remain constant in each gas law.
- I can predict how pressure, volume, and temperature change in different situations.
- I can solve problems involving the basic gas laws.
- I can explain gas laws using the kinetic molecular theory.
Gas Laws
Describing the Behavior of Gases
The gas laws describe relationships between three important properties of a gas:
- pressure (P)
- volume (V)
- temperature (T)
For a fixed amount of gas, changing one of these variables often causes another to change.
Three fundamental gas laws are:
- Boyle's Law — pressure and volume
- Charles' Law — volume and temperature
- Gay-Lussac's Law — pressure and temperature
Each law keeps one important variable constant.
| Gas Law | Variables Compared | Held Constant | Relationship |
|---|---|---|---|
| Boyle's Law | P and V | Temperature | Inverse |
| Charles' Law | V and T | Pressure | Direct |
| Gay-Lussac's Law | P and T | Volume | Direct |
The amount of gas is also assumed to remain constant.
Pressure, Volume and Temperature
Before examining the laws, it is useful to review the three variables.
Pressure
Gas pressure results from particles colliding with the walls of their container.
Common pressure units include:
- pascal (Pa)
- kilopascal (kPa)
- atmosphere (atm)
Volume
Volume is the amount of space occupied by the gas.
Common units include:
- litres (L)
- millilitres (mL)
- cubic metres (m³)
Temperature
Temperature is related to the average translational kinetic energy of gas particles.
For gas-law calculations involving temperature ratios, temperature must be measured in:
kelvin (K)
Convert using:
T(K) = T(°C) + 273.15
Boyle's Law
Boyle's Law describes the relationship between pressure and volume when:
- temperature remains constant
- amount of gas remains constant
The relationship is:
pressure ∝ 1/volume
Therefore:
volume decreases → pressure increases
and:
volume increases → pressure decreases
This is an inverse relationship.
Boyle's Law Equation
For two states of the same gas:
P₁V₁ = P₂V₂
where:
- P₁ = initial pressure
- V₁ = initial volume
- P₂ = final pressure
- V₂ = final volume
Temperature must remain constant.
Boyle's Law and Particle Theory
Imagine gas trapped inside a sealed syringe.
When the plunger is pushed inward:
- volume decreases
- particles have less space
- particles travel shorter distances between wall collisions
- collisions with the walls occur more frequently
Therefore:
pressure increases
The particles themselves do not become smaller.
The space between the particles decreases.
Boyle's Law Prediction
Suppose a gas occupies:
4.0 L at 100 kPa
If the volume is reduced to:
2.0 L
at constant temperature, the volume has been halved.
Therefore, the pressure doubles:
100 kPa → 200 kPa
So:
V ÷ 2 → P × 2
Boyle's Law Worked Example
A gas occupies 6.0 L at a pressure of 120 kPa. It is compressed to 4.0 L at constant temperature.
Find the new pressure.
Use:
P₁V₁ = P₂V₂
Substitute:
120 × 6.0 = P₂ × 4.0
720 = 4.0P₂
P₂ = 180 kPa
Answer: 180 kPa
The answer makes sense because volume decreased, so pressure should increase.
Boyle's Law Graph
If pressure is plotted against volume, the graph is a curve.
As volume increases, pressure decreases.
The graph is not a straight line because:
P ∝ 1/V
However, a graph of pressure against 1/V would be linear for an ideal gas at constant temperature.
Boyle's Law in Everyday Life
Breathing
During inhalation, the chest cavity expands.
The volume available to the lungs increases, reducing the pressure inside relative to atmospheric pressure.
Air then moves into the lungs.
During exhalation, the volume decreases and pressure increases, helping push air outward.
Syringes
Pulling a syringe plunger outward increases volume and lowers pressure inside.
Pushing it inward decreases volume and raises pressure.
Bicycle Pumps
Compressing air inside a bicycle pump decreases its volume and increases its pressure.
Charles' Law
Charles' Law describes the relationship between volume and temperature when:
- pressure remains constant
- amount of gas remains constant
The relationship is:
volume ∝ absolute temperature
Therefore:
temperature increases → volume increases
and:
temperature decreases → volume decreases
This is a direct relationship.
Charles' Law Equation
For two states:
V₁/T₁ = V₂/T₂
where:
- V₁ = initial volume
- T₁ = initial temperature
- V₂ = final volume
- T₂ = final temperature
Temperatures must be in kelvin.
Charles' Law and Particle Theory
Suppose gas is trapped beneath a movable piston.
When the gas is heated:
- temperature increases
- average kinetic energy increases
- particles move faster
- collisions with the piston become more energetic
The piston moves outward.
The gas expands until its internal pressure again balances the external pressure.
Therefore:
temperature increases → volume increases
when pressure remains constant.
Charles' Law Prediction
Suppose a gas has:
V₁ = 2.0 L
at:
T₁ = 300 K
If the temperature increases to:
T₂ = 600 K
the absolute temperature doubles.
At constant pressure:
volume also doubles
Therefore:
V₂ = 4.0 L
Charles' Law Worked Example
A balloon has a volume of 3.0 L at 300 K.
It is heated to 400 K while pressure remains approximately constant.
Find its new volume.
Use:
V₁/T₁ = V₂/T₂
Substitute:
3.0/300 = V₂/400
V₂ = (3.0 × 400)/300
V₂ = 4.0 L
Answer: 4.0 L
The answer makes sense because temperature increased, so volume increased.
Why Celsius Cannot Be Used
Suppose a gas is heated from:
20°C to 40°C
It would be incorrect to say the temperature doubled.
Convert to kelvin:
20°C = 293 K
40°C = 313 K
The absolute temperature only increased slightly.
Therefore, the gas volume would increase only slightly at constant pressure.
Gas laws involving temperature ratios require kelvin.
Charles' Law Graph
A graph of volume against absolute temperature is approximately a straight line for an ideal gas.
Since:
V ∝ T
doubling absolute temperature doubles volume if pressure remains constant.
An idealized line extrapolates toward zero volume at:
T = 0 K
This historical relationship helped scientists develop the concept of absolute zero.
Charles' Law in Everyday Life
Balloons
A balloon placed in a cold environment may shrink.
Cooling reduces the particles' average kinetic energy, and the flexible balloon contracts until its internal and external pressures rebalance.
Hot-Air Balloons
Heating the air causes it to expand.
The expanded air has lower density than the surrounding cooler air, helping produce buoyancy.
Sports Balls
Flexible balls can change slightly in size when their temperature changes, although pressure may also change.
Gay-Lussac's Law
Gay-Lussac's Law describes the relationship between pressure and temperature when:
- volume remains constant
- amount of gas remains constant
The relationship is:
pressure ∝ absolute temperature
Therefore:
temperature increases → pressure increases
and:
temperature decreases → pressure decreases
This is another direct relationship.
Gay-Lussac's Law Equation
For two states:
P₁/T₁ = P₂/T₂
where:
- P₁ = initial pressure
- T₁ = initial temperature
- P₂ = final pressure
- T₂ = final temperature
Again:
temperature must be measured in kelvin.
Gay-Lussac's Law and Particle Theory
Imagine gas trapped in a rigid sealed container.
The volume cannot change.
When the gas is heated:
- temperature increases
- average kinetic energy increases
- particles move faster
- particles strike the walls more frequently
- collisions involve greater momentum changes
Therefore:
pressure increases
Because the container is rigid, the gas cannot expand enough to relieve the increased pressure.
Gay-Lussac's Law Prediction
Suppose gas in a rigid container is at:
300 K and 100 kPa
The temperature is increased to:
600 K
Absolute temperature doubles.
Therefore, at constant volume:
pressure doubles
So:
P₂ = 200 kPa
Gay-Lussac's Law Worked Example
A sealed rigid container contains gas at:
P₁ = 150 kPa
T₁ = 300 K
The gas is heated to:
T₂ = 400 K
Find the final pressure.
Use:
P₁/T₁ = P₂/T₂
Substitute:
150/300 = P₂/400
P₂ = (150 × 400)/300
P₂ = 200 kPa
Answer: 200 kPa
The answer makes sense because the temperature increased while volume remained constant.
Gay-Lussac's Law in Everyday Life
Tire Pressure
When a tire becomes warmer, the air particles move faster.
If the tire volume changes relatively little:
temperature increases → pressure increases
This is why tire pressure can vary with temperature.
Pressurized Containers
Heating a sealed pressurized container can increase its internal pressure.
This is why pressurized cans should not be exposed to excessive heat.
Pressure Cookers
Heating gases and vapor in a mostly sealed container increases pressure, although pressure-cooker behavior also involves boiling and phase changes and is therefore more complicated than a simple ideal-gas example.
Comparing the Three Gas Laws
The easiest way to distinguish the laws is to identify what stays constant.
| Law | Constant | Relationship | Equation |
|---|---|---|---|
| Boyle's | Temperature | P ↑ when V ↓ | P₁V₁ = P₂V₂ |
| Charles' | Pressure | V ↑ when T ↑ | V₁/T₁ = V₂/T₂ |
| Gay-Lussac's | Volume | P ↑ when T ↑ | P₁/T₁ = P₂/T₂ |
A useful memory pattern is:
Boyle: P–V
Charles: V–T
Gay-Lussac: P–T
Direct and Inverse Relationships
Direct Relationships
Charles' Law:
V ∝ T
Gay-Lussac's Law:
P ∝ T
If one variable increases, the other increases.
If one decreases, the other decreases.
Inverse Relationship
Boyle's Law:
P ∝ 1/V
If pressure increases, volume decreases.
If pressure decreases, volume increases.
Predict Before Calculating
Before solving a gas-law problem, predict the direction of change.
For example:
Gas is compressed at constant temperature
Volume ↓
Therefore:
Pressure ↑
Gas is heated at constant pressure
Temperature ↑
Therefore:
Volume ↑
Gas is cooled in a rigid container
Temperature ↓
Therefore:
Pressure ↓
Making a prediction first is a powerful way to check whether a calculated answer is reasonable.
Choosing the Correct Gas Law
Ask:
Which variable stays constant?
If temperature stays constant:
Boyle's Law
If pressure stays constant:
Charles' Law
If volume stays constant:
Gay-Lussac's Law
Then identify the initial and final values before substituting numbers.
Problem-Solving Strategy
For most gas-law calculations:
Identify the variables
Write down:
P₁, V₁, T₁
P₂, V₂, T₂
Identify the constant
Which variable does not change?
Select the correct law
Choose Boyle, Charles or Gay-Lussac.
Convert temperature
If temperature is involved:
°C → K
Rearrange if necessary
Solve the equation for the unknown.
Substitute values
Keep units consistent.
Check your answer
Ask whether the direction of change makes physical sense.
Worked Example: Identifying the Law
A sealed syringe contains 50 mL of air.
The plunger is pushed inward until the volume becomes 25 mL. Temperature remains constant.
Which gas law applies?
Temperature is constant.
Therefore:
Boyle's Law
Volume decreased by half.
Therefore, pressure should:
double
Worked Example: Temperature in Celsius
A balloon occupies 5.0 L at 27°C.
It is heated to 127°C at constant pressure.
Find the new volume.
First convert temperatures:
T₁ = 27 + 273 = 300 K
T₂ = 127 + 273 = 400 K
Use Charles' Law:
V₁/T₁ = V₂/T₂
5.0/300 = V₂/400
V₂ = 6.67 L
Answer: approximately 6.7 L
Notice that 127°C is not used directly.
Worked Example: Cooling a Rigid Container
Gas in a rigid tank has:
P₁ = 240 kPa
T₁ = 360 K
It cools to:
T₂ = 270 K
Find P₂.
Volume is constant, so use Gay-Lussac's Law:
P₁/T₁ = P₂/T₂
240/360 = P₂/270
P₂ = 180 kPa
Answer: 180 kPa
The pressure decreased because the gas cooled.
Gas Laws and Kinetic Molecular Theory
The gas laws are not simply mathematical rules.
They can be explained using particle behavior.
Boyle's Law
Compression reduces the distance particles travel between wall collisions.
V ↓ → collision frequency ↑ → P ↑
Charles' Law
Heating increases particle kinetic energy. If pressure is kept constant, the gas expands until collision effects per unit area rebalance.
T ↑ → particle speed ↑ → V ↑
Gay-Lussac's Law
Heating increases particle kinetic energy while volume remains fixed.
T ↑ → faster particles → stronger/more frequent wall collisions → P ↑
The same particle model therefore explains all three laws.
What If Nothing Is Held Constant?
Real situations often involve changes in pressure, volume and temperature simultaneously.
For a fixed amount of gas, these relationships can be combined into the combined gas law:
P₁V₁/T₁ = P₂V₂/T₂
This equation combines Boyle's, Charles' and Gay-Lussac's relationships.
For the current topic, the three individual gas laws are the main focus, but recognizing the combined relationship helps show how they fit together.
The Ideal Gas Law
A broader relationship is the ideal gas law:
PV = nRT
where:
- P = pressure
- V = volume
- n = amount of gas in moles
- R = gas constant
- T = absolute temperature
The three basic gas laws can be understood as special cases of this relationship when certain variables remain constant.
Real Gases
The gas laws describe ideal gases most accurately.
Real gases behave approximately ideally under many ordinary conditions.
However, differences become more important at:
- very high pressures
- very low temperatures
Under these conditions:
- particle volume becomes more significant
- intermolecular attractions become more important
The simple gas laws are nevertheless excellent models for many everyday and scientific situations.
Common Mistakes
Mistake: "Boyle's Law says pressure and volume increase together."
They have an inverse relationship. If one increases, the other decreases at constant temperature.
Mistake: "Charles' Law keeps temperature constant."
Charles' Law keeps pressure constant and compares volume with temperature.
Mistake: "Gay-Lussac's Law keeps pressure constant."
Gay-Lussac's Law keeps volume constant and compares pressure with temperature.
Mistake: "Celsius can be used in temperature ratios."
Gas-law calculations involving temperature ratios require kelvin.
Mistake: "Gas particles get larger when a gas expands."
The particles themselves remain essentially the same size. Their average separation increases.
Mistake: "Heating always increases gas pressure."
If volume is fixed, pressure increases. If the gas can expand at constant pressure, volume increases instead.
Mistake: "You need to memorize a formula before deciding what is happening."
First identify what remains constant and predict the physical change.
Key Terms
Gas law — mathematical relationship describing the behavior of gases under specified conditions.
Pressure — force per unit area resulting from particle collisions with surfaces.
Volume — amount of space occupied by a gas.
Absolute temperature — temperature measured from absolute zero using kelvin.
Boyle's Law — inverse relationship between pressure and volume at constant temperature.
Charles' Law — direct relationship between volume and absolute temperature at constant pressure.
Gay-Lussac's Law — direct relationship between pressure and absolute temperature at constant volume.
Direct relationship — relationship in which both variables increase or decrease together.
Inverse relationship — relationship in which one variable increases while the other decreases.
Kelvin — SI unit of thermodynamic temperature.
Kinetic molecular theory — particle model explaining gas behavior through particle motion and collisions.
Combined gas law — equation connecting pressure, volume and temperature changes for a fixed amount of gas.
Ideal gas — theoretical gas that follows the assumptions of kinetic molecular theory and gas-law relationships exactly.
Key Takeaways
- Gas behavior can be described using pressure, volume and temperature.
- Boyle's Law relates pressure and volume while temperature remains constant.
- Boyle's Law is an inverse relationship.
- Boyle's Law can be written as P₁V₁ = P₂V₂.
- Charles' Law relates volume and absolute temperature while pressure remains constant.
- Charles' Law is a direct relationship.
- Charles' Law can be written as V₁/T₁ = V₂/T₂.
- Gay-Lussac's Law relates pressure and absolute temperature while volume remains constant.
- Gay-Lussac's Law is a direct relationship.
- Gay-Lussac's Law can be written as P₁/T₁ = P₂/T₂.
- Temperature must be measured in kelvin for gas-law temperature ratios.
- Boyle's Law can be explained by changes in the frequency of particle collisions as volume changes.
- Charles' Law can be explained by increased particle kinetic energy causing expansion when pressure remains constant.
- Gay-Lussac's Law can be explained by faster particles producing greater pressure when volume remains constant.
- Predicting whether a variable should increase or decrease is an important way to check calculations.
- The three basic gas laws can be connected through the combined gas law and ideal gas law.
Check Your Understanding
- What three variables are described by the basic gas laws?
- Which variable remains constant in Boyle's Law?
- Describe the relationship between pressure and volume in Boyle's Law.
- A gas's volume is halved at constant temperature. Predict what happens to its pressure.
- A gas occupies 4.0 L at 100 kPa and is compressed to 2.0 L at constant temperature. Calculate its new pressure.
- Which variable remains constant in Charles' Law?
- A gas is heated from 300 K to 450 K at constant pressure. Predict what happens to its volume.
- A gas occupies 2.0 L at 250 K. Calculate its volume at 500 K if pressure remains constant.
- Why must Celsius temperatures be converted to kelvin in gas-law calculations?
- Which variable remains constant in Gay-Lussac's Law?
- A rigid container is heated from 300 K to 600 K. What happens to its gas pressure?
- A gas has a pressure of 120 kPa at 300 K. Calculate its pressure at 450 K if volume remains constant.
- Use kinetic molecular theory to explain Boyle's Law.
- Use kinetic molecular theory to explain why pressure increases when a gas is heated in a rigid container.
- A balloon becomes smaller when placed in a cold environment. Which gas law best explains this behavior?
- Explain how you would decide which of the three basic gas laws to use in a calculation.
4. Ideal Gas Law and Molar Volume
Learning outcomes
- I can identify the variables in the Ideal Gas Law.
- I can calculate pressure, volume, temperature, or amount of gas using the Ideal Gas Law.
- I can explain the concept of molar volume.
- I can use molar volume to determine the volume or amount of a gas at standard conditions.
- I can compare calculations using the Ideal Gas Law and molar volume.
Ideal Gas Law and Molar Volume
Bringing the Gas Variables Together
The basic gas laws describe relationships between pressure, volume and temperature. The Ideal Gas Law combines these relationships and also includes the amount of gas present.
The four important variables are:
- pressure (P)
- volume (V)
- amount of gas
- absolute temperature (T)
This makes the Ideal Gas Law useful when the amount of gas is known or needs to be calculated.
The Ideal Gas Law
The relationship between the variables is:
PV = nRT
where:
- P = pressure
- V = volume
- n = amount of gas in moles
- R = ideal gas constant
- T = absolute temperature in kelvin
The Ideal Gas Law describes the behavior of an ideal gas and provides a good approximation for many real gases under ordinary conditions.
Understanding the Variables
Pressure
Pressure results from gas particles colliding with the walls of their container.
Common units include:
- Pa
- kPa
- atm
Volume
Volume is the space occupied by the gas.
Common units include:
- m³
- L
- mL
Amount of Gas
The amount of gas is measured in moles.
Symbol:
n
One mole contains approximately:
6.022 × 10²³ particles
This number is called the Avogadro constant.
Temperature
Temperature must be measured in kelvin.
Convert using:
T(K) = T(°C) + 273.15
The Gas Constant
The value of R depends on the units used for pressure and volume.
Two commonly used forms are:
R = 8.31 J/(mol·K)
when using SI units:
- pressure in Pa
- volume in m³
or:
R = 8.31 kPa·L/(mol·K)
when using:
- pressure in kPa
- volume in L
These work because:
1 kPa·L = 1 J
Another common chemistry form is:
R = 0.0821 L·atm/(mol·K)
when pressure is measured in atmospheres.
The important rule is:
choose an R value that matches your units.
Rearranging the Ideal Gas Law
Starting with:
PV = nRT
To calculate pressure:
P = nRT/V
To calculate volume:
V = nRT/P
To calculate amount of gas:
n = PV/RT
To calculate temperature:
T = PV/nR
These four forms allow us to solve many gas problems.
Solving an Ideal Gas Problem
A reliable strategy is:
- Identify P, V, n and T.
- Convert temperature to kelvin.
- Check that pressure and volume units match R.
- Rearrange PV = nRT for the unknown.
- Substitute the values.
- Include the correct unit.
- Check whether the answer is physically reasonable.
Worked Example: Finding Volume
Calculate the volume occupied by 2.00 mol of an ideal gas at:
T = 300 K
P = 100 kPa
Use:
V = nRT/P
Using:
R = 8.31 kPa·L/(mol·K)
Substitute:
V = (2.00 × 8.31 × 300)/100
V = 49.86 L
V ≈ 49.9 L
Worked Example: Finding Pressure
A 10.0 L container contains 0.50 mol of gas at 300 K.
Calculate the pressure.
Use:
P = nRT/V
P = (0.50 × 8.31 × 300)/10.0
P = 124.65 kPa
P ≈ 125 kPa
Worked Example: Finding the Amount of Gas
A 12.0 L container holds gas at:
P = 200 kPa
T = 300 K
Calculate the amount of gas.
Use:
n = PV/RT
n = (200 × 12.0)/(8.31 × 300)
n = 2400/2493
n ≈ 0.963 mol
Worked Example: Finding Temperature
A 5.0 L container holds 0.20 mol of gas at a pressure of 150 kPa.
Find the temperature.
Use:
T = PV/nR
T = (150 × 5.0)/(0.20 × 8.31)
T ≈ 451 K
T ≈ 451 K
In Celsius:
451 − 273 ≈ 178°C
Why Temperature Must Be in Kelvin
The Ideal Gas Law uses absolute temperature.
Suppose a gas is at:
27°C
Convert:
T = 27 + 273
T ≈ 300 K
Using 27 directly in PV = nRT would give a completely incorrect result.
This is because 0°C does not represent zero molecular kinetic energy.
The kelvin scale begins at absolute zero.
Unit Consistency
One of the most common sources of errors is mixing incompatible units.
For example, if:
R = 8.31 kPa·L/(mol·K)
then use:
- P in kPa
- V in L
- n in mol
- T in K
If:
R = 8.31 Pa·m³/(mol·K)
then use:
- P in Pa
- V in m³
A useful conversion is:
1 L = 0.001 m³
and:
1 kPa = 1000 Pa
Connecting the Ideal Gas Law to Particle Theory
The Ideal Gas Law can also be understood using kinetic molecular theory.
If more particles are placed into the same volume:
n ↑ → more wall collisions → P ↑
If temperature increases:
T ↑ → average kinetic energy ↑ → P tends to ↑
If volume increases:
V ↑ → wall collisions become less frequent per unit area → P tends to ↓
The mathematical relationships in PV = nRT therefore reflect microscopic particle behavior.
What Is a Mole?
Gas samples contain enormous numbers of particles.
Instead of counting individual atoms or molecules, chemists use the mole.
One mole contains:
6.022 × 10²³ particles
Therefore:
2 mol contains approximately:
1.204 × 10²⁴ particles
and:
0.5 mol contains approximately:
3.011 × 10²³ particles
The mole allows microscopic particle numbers to be connected to measurable quantities of gas.
What Is Molar Volume?
Molar volume is the volume occupied by one mole of a substance under specified conditions.
For gases:
molar volume = volume/amount
or:
V_m = V/n
where:
- V_m = molar volume
- V = gas volume
- n = amount of gas
Typical units are:
L/mol
or:
m³/mol
Why Conditions Matter
A gas does not have one fixed molar volume under all conditions.
Gas volume depends strongly on:
- temperature
- pressure
Therefore, a molar volume must always be associated with specified conditions.
For example, one mole of gas occupies a larger volume at:
- higher temperature
- lower pressure
and a smaller volume at:
- lower temperature
- higher pressure
Standard Temperature and Pressure
A commonly used reference condition is standard temperature and pressure, or STP.
A widely used chemistry definition is:
T = 273.15 K (0°C)
P = 100 kPa
At these conditions, one mole of an ideal gas occupies approximately:
22.7 L/mol
However, some textbooks use the older convention:
T = 273.15 K
P = 1 atm = 101.325 kPa
Under those conditions:
molar volume ≈ 22.4 L/mol
This difference is small but important.
Always check which definition your course or question uses.
Why You May See 22.4 L and 22.7 L
Both values can be correct because they refer to slightly different standard pressures.
| Conditions | Molar Volume |
|---|---|
| 0°C and 100 kPa | ≈ 22.7 L/mol |
| 0°C and 1 atm | ≈ 22.4 L/mol |
For calculations, use the convention specified in the question.
If none is given, state which standard you are using.
Using Molar Volume
If molar volume is known:
V = nV_m
where:
- V = gas volume
- n = amount in moles
- V_m = molar volume
To find moles:
n = V/V_m
This can make calculations at standard conditions much quicker than using the complete Ideal Gas Law.
Worked Example: Volume from Moles
Using:
V_m = 22.7 L/mol
calculate the volume of 3.0 mol of ideal gas at 0°C and 100 kPa.
Use:
V = nV_m
V = 3.0 × 22.7
V = 68.1 L
Worked Example: Moles from Volume
A gas occupies 45.4 L at 0°C and 100 kPa.
Calculate the amount of gas.
Use:
n = V/V_m
n = 45.4/22.7
n = 2.00 mol
Worked Example: A Smaller Gas Sample
What volume would 0.25 mol of ideal gas occupy at 0°C and 100 kPa?
V = nV_m
V = 0.25 × 22.7
V = 5.68 L
Why Different Gases Have Similar Molar Volumes
At the same temperature and pressure, equal amounts of ideal gases occupy equal volumes.
This is related to Avogadro's law.
For example, at the same temperature and pressure:
1 mol helium
1 mol oxygen
1 mol nitrogen
1 mol hydrogen
all occupy approximately the same volume if they behave ideally.
The particles themselves may have very different masses, but the relationship between particle number, pressure, temperature and volume produces the same ideal-gas molar volume.
Molar Volume Does Not Mean Equal Mass
Suppose we have:
1 mol He
and:
1 mol O₂
At the same temperature and pressure, they occupy approximately the same volume.
But their masses are very different.
Approximately:
1 mol He ≈ 4 g
1 mol O₂ ≈ 32 g
Therefore:
equal molar volume does not mean equal mass.
It means equal amounts in moles occupy equal volumes at the same temperature and pressure.
Finding Molar Volume from the Ideal Gas Law
The molar volume relationship can actually be derived from the Ideal Gas Law.
Start with:
PV = nRT
Rearrange:
V/n = RT/P
But:
V/n = V_m
Therefore:
V_m = RT/P
This shows clearly that molar volume depends on:
- temperature
- pressure
Calculating Molar Volume at 100 kPa
At:
T = 273.15 K
P = 100 kPa
Use:
V_m = RT/P
V_m = (8.31 × 273.15)/100
V_m ≈ 22.7 L/mol
So one mole occupies approximately:
22.7 L
at 0°C and 100 kPa.
Molar Volume at Room Temperature
Suppose the temperature is approximately:
25°C = 298 K
and pressure is:
100 kPa
Calculate the molar volume.
V_m = RT/P
V_m = (8.31 × 298)/100
V_m ≈ 24.8 L/mol
Therefore, at approximately room temperature and 100 kPa:
1 mol ideal gas occupies about 24.8 L
This is larger than its volume at 0°C because the temperature is higher.
STP vs Room Conditions
This comparison is useful:
| Conditions | Approximate Molar Volume |
|---|---|
| 0°C, 100 kPa | 22.7 L/mol |
| 25°C, 100 kPa | 24.8 L/mol |
Heating the gas while keeping pressure constant causes it to expand.
This agrees with Charles' Law.
Using the Ideal Gas Law or Molar Volume?
Both methods can solve some of the same problems.
Use molar volume when:
- conditions match a known standard
- the molar volume is given
- only volume and moles are needed
Use the Ideal Gas Law when:
- temperature is not a standard value
- pressure is not a standard value
- pressure must be calculated
- temperature must be calculated
- conditions are changing
- a more general solution is needed
Comparing the Two Methods
Suppose 2.0 mol of gas is at:
0°C and 100 kPa.
Molar Volume Method
V = nV_m
V = 2.0 × 22.7
V = 45.4 L
Ideal Gas Law Method
V = nRT/P
V = (2.0 × 8.31 × 273.15)/100
V ≈ 45.4 L
Both methods give approximately the same answer.
The molar-volume method is simply faster when the appropriate standard conditions apply.
Molar Volume and Particle Number
Since one mole contains:
6.022 × 10²³ particles
approximately 22.7 L of an ideal gas at 0°C and 100 kPa contains:
6.022 × 10²³ particles
This is true whether the gas is:
- helium
- hydrogen
- nitrogen
- oxygen
provided ideal behavior is a reasonable approximation.
Connecting Mass, Moles and Gas Volume
Gas problems may also provide mass instead of moles.
Use:
n = m/M
where:
- n = amount in mol
- m = mass
- M = molar mass
Once moles are known, use either:
PV = nRT
or:
V = nV_m
depending on the conditions.
Worked Example: Mass to Gas Volume
What volume would 32 g of oxygen gas occupy at 0°C and 100 kPa?
O₂ has molar mass:
M = 32 g/mol
First calculate moles:
n = m/M
n = 32/32
n = 1.00 mol
Now use molar volume:
V = nV_m
V = 1.00 × 22.7
V = 22.7 L
Worked Example: Gas Volume to Mass
A sample of oxygen occupies 11.35 L at 0°C and 100 kPa.
First find moles:
n = V/V_m
n = 11.35/22.7
n = 0.500 mol
O₂ has molar mass:
32 g/mol
Therefore:
m = nM
m = 0.500 × 32
m = 16.0 g
Gas Density
The Ideal Gas Law can also help explain gas density.
Density is:
ρ = m/V
For one mole:
m = M
and:
V = V_m
Therefore:
ρ = M/V_m
A gas with a larger molar mass generally has a greater density under the same temperature and pressure conditions.
For example, oxygen gas is denser than helium at the same temperature and pressure.
Ideal Gases and Real Gases
The Ideal Gas Law assumes ideal behavior.
Real gases are made of particles that:
- have finite volume
- experience intermolecular forces
Real gases behave most like ideal gases when:
- pressure is relatively low
- temperature is relatively high
At high pressures, particles are crowded closer together and their actual volume becomes important.
At low temperatures, attractive forces between particles become more significant.
Why the Ideal Gas Law Is Useful
Even though no real gas is perfectly ideal, PV = nRT provides a very useful approximation for:
- laboratory gases
- atmospheric calculations
- balloons
- gas storage
- chemistry calculations
- engineering
- environmental science
It provides a bridge between microscopic particle behavior and measurable macroscopic quantities.
Common Mistakes
Mistake: "Temperature can be entered in Celsius."
The Ideal Gas Law requires kelvin.
Mistake: "R is always 8.31 regardless of units."
The numerical value and units of R must match the pressure and volume units being used.
Mistake: "One mole of gas always occupies 22.4 L."
Molar volume depends on temperature and pressure.
Mistake: "22.4 L/mol and 22.7 L/mol cannot both be correct."
They correspond to different standard-pressure conventions.
Mistake: "All gases have the same density because they have the same molar volume."
Different gases have different molar masses.
Mistake: "Molar volume can be used at any temperature and pressure."
A quoted molar volume applies only to its specified conditions.
Mistake: "Moles and molecules are the same thing."
A mole is an amount containing 6.022 × 10²³ particles.
Key Terms
Ideal Gas Law — relationship connecting pressure, volume, amount of gas and absolute temperature.
Ideal gas — theoretical gas whose particles follow the assumptions of the ideal-gas model.
Pressure (P) — force per unit area produced by gas-particle collisions.
Volume (V) — amount of space occupied by a gas.
Amount of gas — quantity of gas measured in moles.
Absolute temperature (T) — temperature measured using the kelvin scale.
Ideal gas constant (R) — proportionality constant connecting P, V, n and T.
Mole — amount of substance containing approximately 6.022 × 10²³ particles.
Avogadro constant — number of particles per mole, approximately 6.022 × 10²³ mol⁻¹.
Molar volume (V_m) — volume occupied by one mole of a substance under specified conditions.
Standard temperature and pressure (STP) — defined reference conditions used for gas calculations; the exact pressure convention should always be checked.
Avogadro's law — equal amounts of ideal gases occupy equal volumes at the same temperature and pressure.
Molar mass (M) — mass of one mole of a substance.
Key Takeaways
- The Ideal Gas Law combines pressure, volume, amount of gas and temperature.
- The equation is PV = nRT.
- P represents pressure, V volume, n moles and T absolute temperature.
- Temperature must always be expressed in kelvin when using the Ideal Gas Law.
- The units used for pressure and volume must match the chosen value of R.
- The Ideal Gas Law can be rearranged to calculate P, V, n or T.
- One mole contains approximately 6.022 × 10²³ particles.
- Molar volume is the volume occupied by one mole at specified temperature and pressure.
- Molar volume is calculated using V_m = V/n.
- At 0°C and 100 kPa, an ideal gas has a molar volume of approximately 22.7 L/mol.
- At 0°C and 1 atm, the corresponding value is approximately 22.4 L/mol.
- At about 25°C and 100 kPa, the ideal-gas molar volume is approximately 24.8 L/mol.
- Molar volume changes when temperature or pressure changes.
- Different ideal gases have approximately the same molar volume when measured at the same temperature and pressure.
- Equal molar volumes do not mean equal masses.
- Molar volume is convenient when conditions match a known reference value.
- The Ideal Gas Law is more general and can be used at many different temperatures and pressures.
- Molar-volume calculations and Ideal Gas Law calculations should agree when the same conditions are used.
Check Your Understanding
- Write the Ideal Gas Law.
- What does each symbol in PV = nRT represent?
- Why must temperature be expressed in kelvin?
- Calculate the volume of 1.5 mol of ideal gas at 300 K and 100 kPa.
- A 20.0 L container holds 0.80 mol of gas at 300 K. Calculate its pressure.
- A gas occupies 15.0 L at 150 kPa and 300 K. Calculate the number of moles present.
- Define molar volume.
- What is the approximate molar volume of an ideal gas at 0°C and 100 kPa?
- Why might a textbook give 22.4 L/mol instead of 22.7 L/mol?
- Calculate the volume occupied by 2.5 mol of ideal gas at 0°C and 100 kPa using molar volume.
- A gas occupies 56.75 L at 0°C and 100 kPa. Calculate the number of moles.
- Explain why one mole of helium and one mole of oxygen occupy approximately the same volume at the same temperature and pressure.
- Explain why those two samples do not have the same mass.
- When is using molar volume easier than using PV = nRT?
- When would the Ideal Gas Law be preferable to using a standard molar volume?
- Calculate the molar volume of an ideal gas at 298 K and 100 kPa and compare it with the molar volume at 273 K.
5. Applications of Gas Laws
Learning outcomes
- I can explain how gas laws are applied in science and engineering.
- I can describe the behavior of gases in weather, aviation, and diving.
- I can explain how gas expansion and compression are used in technology.
- I can evaluate real-world situations using gas law principles.
- I can connect gas laws, kinetic theory, and thermal energy to practical applications.
Applications of Gas Laws
Gas Laws in the Real World
Gas laws are not just mathematical relationships. They explain the behavior of gases in many natural systems and technologies.
Applications include:
- weather and the atmosphere
- aviation
- scuba diving
- engines
- refrigerators and air conditioners
- compressors and pumps
- aerosol containers
- balloons
- medical equipment
- industrial gas storage
These applications connect three major ideas:
gas laws + kinetic molecular theory + thermal energy
Reviewing the Gas Laws
The most important relationships are:
Boyle's Law
At constant temperature:
P₁V₁ = P₂V₂
Pressure and volume have an inverse relationship.
V ↓ → P ↑
V ↑ → P ↓
Charles' Law
At constant pressure:
V₁/T₁ = V₂/T₂
Volume and absolute temperature have a direct relationship.
T ↑ → V ↑
T ↓ → V ↓
Gay-Lussac's Law
At constant volume:
P₁/T₁ = P₂/T₂
Pressure and absolute temperature have a direct relationship.
T ↑ → P ↑
T ↓ → P ↓
For more complex situations where pressure, volume and temperature all change, the combined gas law can be used.
Connecting Gas Laws to Kinetic Theory
The gas laws can be explained by considering particles.
Gas particles:
- move constantly and randomly
- have kinetic energy
- collide with each other
- collide with container walls
- are separated by relatively large distances
Gas pressure results from particle collisions with surfaces.
Temperature is related to the particles' average translational kinetic energy.
Therefore, changing temperature, volume or the number of particles changes how particles move and collide.
This microscopic behavior produces the macroscopic gas laws.
Weather and the Atmosphere
Earth's atmosphere is a mixture of gases.
Atmospheric gases constantly experience changes in:
- temperature
- pressure
- density
- volume
These changes are fundamental to weather.
Heating Air in the Atmosphere
Solar radiation heats Earth's surface.
The surface then transfers thermal energy to the air above it.
As air warms:
- particle kinetic energy increases
- particles move faster
- the air tends to expand
- density decreases
Remember:
ρ = m/V
If the same mass occupies a larger volume, its density decreases.
Warm, less-dense air can therefore rise through surrounding cooler, denser air.
Rising Air and Weather
As warm air rises, the atmospheric pressure around it decreases.
The rising air expands.
Expansion requires energy, so the air generally cools.
If the air contains water vapor, sufficient cooling can cause condensation.
This can produce:
- clouds
- fog
- rain
- thunderstorms
Gas behavior therefore plays an important role in atmospheric processes.
Sinking Air
Cooler air is often denser than warmer surrounding air.
It may sink.
As an air mass descends into regions of higher atmospheric pressure:
- the gas is compressed
- its volume decreases
- its temperature can increase
This helps explain why descending air is often associated with warmer and drier atmospheric conditions.
Atmospheric Pressure and Altitude
Atmospheric pressure decreases with increasing altitude.
Near Earth's surface, there is a large column of air above us.
At higher altitudes, there is less atmosphere above.
Therefore:
altitude ↑ → atmospheric pressure ↓
This change in pressure affects:
- aircraft
- weather balloons
- human breathing
- boiling points
- sealed containers
Weather Balloons
Weather balloons provide an excellent gas-law example.
A balloon is partially inflated near Earth's surface.
As it rises:
external atmospheric pressure decreases
The gas inside expands.
Therefore:
pressure ↓ → volume ↑
Eventually, the balloon may become so large that its material reaches its limit and bursts.
Both pressure and temperature change during ascent, so the combined gas law gives a better model than Boyle's Law alone.
Aviation
Aircraft operate across large changes in atmospheric pressure and temperature.
At cruising altitude:
- atmospheric pressure is much lower
- air density is lower
- temperature is often much lower than at ground level
These conditions affect:
- aircraft cabins
- engines
- lift
- instruments
- tires
- fuel systems
Aircraft Cabin Pressurization
Humans require sufficient oxygen pressure to function normally.
At high altitude, atmospheric pressure is too low for comfortable normal breathing.
Aircraft therefore maintain a cabin pressure substantially above the outside atmospheric pressure.
The aircraft structure must withstand the pressure difference between:
- higher-pressure cabin air
- lower-pressure outside air
The fuselage effectively acts as a pressure vessel.
Why Aircraft Cabins Are Not Usually Pressurized to Sea Level
Maintaining full sea-level pressure at high altitude would create a larger pressure difference across the aircraft fuselage.
A larger pressure difference would require:
- stronger structures
- greater structural mass
- potentially greater engineering demands
Instead, passenger aircraft generally maintain a cabin pressure equivalent to a moderate altitude.
This represents an engineering compromise between:
- passenger comfort
- structural strength
- aircraft mass
- safety
Aircraft Tires
Aircraft tires can experience large changes in:
- temperature
- pressure
- load
A tire may be cold at altitude and become much warmer during landing because of braking and contact with the runway.
If the gas temperature inside a nearly fixed-volume tire increases:
T ↑ → P ↑
This behavior is related to Gay-Lussac's Law.
Engineers must design tires to tolerate these changing conditions.
Scuba Diving
Gas laws are especially important in diving because water pressure increases rapidly with depth.
Approximately every 10 m of seawater adds about another atmosphere of pressure.
This strongly affects gases in:
- lungs
- masks
- buoyancy compensators
- bubbles
- diving equipment
Diving and Boyle's Law
As a diver descends:
pressure increases
For a flexible trapped gas space at roughly constant temperature:
volume decreases
As the diver ascends:
pressure decreases
and:
gas volume increases
This is an application of Boyle's Law.
Expanding Bubbles
Consider an air bubble released by a diver.
As the bubble rises:
- surrounding water pressure decreases
- the gas expands
- bubble volume increases
This is why bubbles generally become larger as they approach the surface.
Diving and the Lungs
Gas expansion during ascent is extremely important.
If a diver breathes compressed gas at depth and then ascends while improperly holding their breath, decreasing external pressure can cause gas in the lungs to expand.
This can cause serious injury.
Divers are trained in appropriate ascent and breathing procedures specifically because pressure changes affect gas volume.
Gas Solubility and Diving
Diving also involves gases dissolving in body tissues.
At greater pressure, more nitrogen can dissolve into tissues during breathing of ordinary compressed air.
If pressure decreases too rapidly during ascent, dissolved nitrogen can form bubbles.
This can contribute to decompression sickness.
This effect is primarily described by gas-solubility principles such as Henry's law rather than Boyle's Law alone, but it is another example of why gas pressure matters in diving.
Gas Compression
Compressing a gas means forcing it into a smaller volume.
A compressor does work on a gas.
As the gas is compressed:
- particles are forced closer together
- collisions become more frequent
- pressure increases
Compression can also increase gas temperature.
Why Compressed Gas Gets Hot
When a piston compresses a gas, mechanical work is done on the gas.
Energy is transferred into the gas.
If compression happens quickly and little energy escapes:
internal energy increases
and:
temperature increases
This is why:
- bicycle pumps can become warm
- compressors require cooling
- rapidly compressed gases can become very hot
Gas compression therefore connects mechanical work and thermal energy.
Bicycle Pumps
A bicycle pump provides a simple example.
When the handle is pushed downward:
- air volume decreases
- pressure increases
- work is done on the gas
- gas temperature may increase
This involves both:
Boyle's Law
and:
energy transfer through mechanical work
The real process is therefore more complicated than a perfectly constant-temperature Boyle's Law calculation.
Gas Expansion and Cooling
The reverse can also occur.
When a compressed gas expands and does work on its surroundings, its internal energy may decrease.
The gas may cool.
This principle is important in:
- refrigeration
- air conditioning
- gas turbines
- pneumatic equipment
Gas expansion can therefore be used as part of a cooling process.
Refrigerators and Air Conditioners
Refrigerators and air conditioners move thermal energy from a cooler region to a warmer region using a refrigeration cycle.
A typical cycle includes:
- compression
- condensation
- expansion
- evaporation
The refrigerant undergoes changes in:
- pressure
- temperature
- volume
- phase
Compression in Refrigeration
The compressor compresses refrigerant vapor.
As compression occurs:
- pressure increases
- temperature increases
The hot high-pressure refrigerant can then transfer thermal energy to the surroundings.
Expansion in Refrigeration
Later in the cycle, the refrigerant passes through an expansion device.
Its pressure drops dramatically.
The refrigerant becomes cold enough to absorb thermal energy from the refrigerated space.
The complete refrigeration process involves both gas behavior and phase changes, so it cannot be explained using a single simple gas law alone.
Internal Combustion Engines
Gas expansion is also used to produce mechanical work.
In a car engine:
- fuel and air enter a cylinder
- the mixture is compressed
- combustion greatly increases temperature and pressure
- hot gases expand
- expanding gas pushes a piston
Energy is transformed approximately as:
chemical energy → thermal energy → mechanical work
The behavior of the hot gases is central to engine operation.
Gas Turbines and Jet Engines
Jet engines and gas turbines also use compression and expansion.
A simplified sequence is:
air enters
↓
compressor increases pressure
↓
fuel adds thermal energy
↓
hot gases expand
↓
expanding gases produce useful work and thrust
Gas laws and thermodynamics are therefore fundamental to aviation propulsion and electrical power generation.
Pneumatic Systems
Pneumatic systems use compressed gases to transmit energy and produce motion.
Examples include:
- pneumatic drills
- air brakes
- factory actuators
- air-powered tools
- robotic systems
Compressed air stores energy.
When released:
- the gas expands
- pressure produces forces
- those forces can move pistons or other components
Pneumatic Pistons
For a piston:
pressure = force/area
so:
F = PA
A pressure difference across a piston can therefore produce a force.
For example, if:
P = 500 kPa
A = 0.002 m²
then:
P = 500,000 Pa
F = PA
F = 500,000 × 0.002
F = 1000 N
Compressed gas can therefore generate substantial forces.
Aerosol Containers
Aerosol cans contain material under pressure.
Heating a sealed container can be dangerous.
If its volume is approximately constant:
T ↑ → P ↑
At the particle level:
- particles gain kinetic energy
- particles move faster
- wall collisions become more frequent and energetic
- internal pressure increases
This is why pressurized containers should not be exposed to excessive heat.
Gas Cylinders
Industrial gases are often stored under high pressure.
Examples include:
- oxygen
- nitrogen
- helium
- carbon dioxide
Gas cylinders must be designed to withstand large internal pressures.
Engineers consider:
- material strength
- temperature changes
- pressure limits
- valves
- impact protection
- safe storage conditions
Heating a compressed-gas cylinder can increase its pressure and create a serious hazard.
Hot-Air Balloons
A hot-air balloon demonstrates the connection between thermal energy, gas density and buoyancy.
The burner transfers thermal energy to the air.
The air:
- becomes hotter
- expands
- becomes less dense
The surrounding cooler air is denser.
The resulting buoyant force can lift the balloon.
Heating the air therefore changes its physical behavior through particle motion and expansion.
Medical Applications
Gas laws are important in medical technology.
Examples include:
- ventilators
- oxygen delivery systems
- anesthesia equipment
- respiratory testing
- hyperbaric chambers
Ventilators control the pressure and volume of gases entering and leaving the lungs.
Understanding pressure-volume relationships is essential for delivering air safely.
Breathing and Boyle's Law
Breathing itself provides a biological gas-law application.
During inhalation:
- diaphragm contracts
- chest volume increases
- lung pressure decreases
- air moves inward
During exhalation:
- chest volume decreases
- lung pressure increases
- air moves outward
This can be approximated using Boyle's Law.
Evaluating Real Gas-Law Situations
Real situations are usually more complicated than textbook examples.
A textbook might state:
volume decreases → pressure increases
But in reality, compression may also:
- increase temperature
- transfer thermal energy
- involve friction
- involve leaks
- change the amount of gas
Therefore, the first step when evaluating a real system is to identify which variables actually remain constant.
Using the Combined Gas Law
If pressure, volume and temperature all change for the same amount of gas, the combined gas law is useful:
P₁V₁/T₁ = P₂V₂/T₂
This combines the relationships represented by Boyle's, Charles' and Gay-Lussac's laws.
For example, it can model a weather balloon whose:
- pressure decreases
- temperature changes
- volume increases
as it rises through the atmosphere.
Worked Example: Weather Balloon
A balloon has:
P₁ = 100 kPa
V₁ = 5.0 L
T₁ = 300 K
At higher altitude:
P₂ = 60 kPa
T₂ = 270 K
Find the new volume.
Use:
P₁V₁/T₁ = P₂V₂/T₂
Substitute:
(100 × 5.0)/300 = (60 × V₂)/270
Rearrange:
V₂ = (100 × 5.0 × 270)/(300 × 60)
V₂ = 7.5 L
Answer: 7.5 L
The balloon expands because the decrease in external pressure has a strong effect.
Worked Example: Diving Bubble
An air bubble has a volume of:
2.0 cm³
at a pressure of:
300 kPa
It rises to a region where pressure is:
100 kPa
Assume temperature remains constant.
Use Boyle's Law:
P₁V₁ = P₂V₂
300 × 2.0 = 100 × V₂
V₂ = 6.0 cm³
Answer: 6.0 cm³
The bubble becomes three times larger.
Worked Example: Heated Gas Cylinder
A rigid gas cylinder has pressure:
P₁ = 800 kPa
at:
T₁ = 300 K
The cylinder is heated to:
T₂ = 360 K
Assume its volume remains constant.
Use:
P₁/T₁ = P₂/T₂
800/300 = P₂/360
P₂ = 960 kPa
Answer: 960 kPa
Heating increased the pressure by:
160 kPa
This demonstrates why temperature control is important for compressed-gas storage.
Gas Laws and Energy Transfer
Gas-law applications often involve energy transfers.
Compression
Work is done on the gas.
This can increase its internal energy and temperature.
Expansion
The gas may do work on its surroundings.
Its internal energy may decrease.
Heating
Thermal energy is transferred into the gas.
Average particle kinetic energy generally increases.
Cooling
Thermal energy leaves the gas.
Average particle kinetic energy generally decreases.
This connects gas laws directly with thermodynamics.
From Particles to Technology
A useful chain of reasoning is:
thermal energy changes
↓
particle kinetic energy changes
↓
particle speed changes
↓
collision behavior changes
↓
pressure, volume or temperature changes
↓
macroscopic effects
↓
useful technological application
This allows the same particle model to explain systems ranging from weather balloons to jet engines.
Choosing the Correct Model
When analyzing a real gas situation, ask:
Is temperature constant?
Consider Boyle's Law.
Is pressure constant?
Consider Charles' Law.
Is volume constant?
Consider Gay-Lussac's Law.
Are pressure, volume and temperature all changing?
Consider the combined gas law.
Does the amount of gas matter?
Consider the Ideal Gas Law:
PV = nRT
The physical situation should determine the equation—not the other way around.
Evaluating Engineering Designs
Gas-law knowledge allows engineers to predict whether a design will operate safely and efficiently.
Important questions include:
- How much will pressure increase when temperature rises?
- How much will a gas expand when pressure decreases?
- Can the container withstand the maximum pressure?
- How much work is required to compress the gas?
- How much cooling is required after compression?
- How will altitude affect the system?
- Will gas expansion produce useful mechanical work?
- Are ideal-gas assumptions reasonable?
These questions are important in designing safe technological systems.
Common Mistakes
Mistake: "One gas law perfectly describes every real situation."
Real systems often involve simultaneous changes in several variables.
Mistake: "Weather balloons expand only because they become hotter."
The large decrease in atmospheric pressure with altitude is extremely important.
Mistake: "A diver's bubbles stay the same size while rising."
Decreasing external pressure causes the bubbles to expand.
Mistake: "Compressed gases are always at the same temperature."
Rapid compression can significantly increase gas temperature.
Mistake: "Expansion always happens at constant temperature."
Expanding gases can cool when they do work.
Mistake: "Aircraft cabin pressure equals outside pressure."
At cruising altitude, cabin pressure is deliberately maintained above the much lower external atmospheric pressure.
Mistake: "Refrigerators work using only Boyle's Law."
Refrigeration involves pressure changes, energy transfer and phase changes.
Mistake: "Gas-law equations are separate from kinetic theory."
The gas laws describe macroscopic effects produced by microscopic particle motion and collisions.
Key Terms
Gas law — relationship describing how pressure, volume and temperature of a gas are connected.
Compression — reduction in gas volume, usually accompanied by increased pressure.
Expansion — increase in the volume occupied by a gas.
Atmospheric pressure — pressure produced by Earth's atmosphere.
Cabin pressurization — maintaining aircraft cabin pressure above the low external pressure at altitude.
Pneumatic system — system that uses compressed gas to transmit energy or produce motion.
Compressor — device that increases gas pressure by doing mechanical work on a gas.
Refrigerant — substance used to transfer thermal energy in refrigeration and air-conditioning systems.
Buoyancy — upward force exerted by a fluid on an immersed object.
Combined gas law — relationship connecting pressure, volume and absolute temperature for a fixed amount of gas.
Ideal Gas Law — PV = nRT, relating pressure, volume, amount of gas and absolute temperature.
Kinetic molecular theory — model explaining gas behavior through particle motion and collisions.
Thermal energy — energy associated with the microscopic behavior of particles in a system.
Key Takeaways
- Gas laws have important applications in science, engineering, medicine and everyday technology.
- Atmospheric gases expand, compress, heat and cool as weather systems develop.
- Warm air can expand and become less dense, contributing to atmospheric convection.
- Weather balloons expand as atmospheric pressure decreases with altitude.
- Aircraft must manage large pressure differences between their cabins and the outside atmosphere.
- Gas pressure in aircraft tires and other sealed systems changes with temperature.
- Water pressure increases with diving depth, compressing gas spaces.
- Gas bubbles expand as divers ascend because external pressure decreases.
- Rapid changes in gas pressure are important to diving safety.
- Compressors reduce gas volume and increase pressure.
- Doing work on a gas during compression can increase its temperature.
- Expanding gases can do mechanical work and may cool.
- Refrigerators and air conditioners use compression, expansion, thermal energy transfer and phase changes.
- Engines use hot expanding gases to perform mechanical work.
- Pneumatic systems use compressed gas to produce forces and motion.
- Heating a rigid gas container can significantly increase its pressure.
- Hot-air balloons connect gas expansion, density, thermal energy and buoyancy.
- Real applications often involve several changing variables, so the combined or Ideal Gas Law may be more appropriate than a single basic gas law.
- Kinetic molecular theory explains gas-law behavior through changes in particle speed, spacing and collisions.
- Gas-law principles allow engineers to predict performance, improve efficiency and design safer systems.
Check Your Understanding
- Explain why a weather balloon expands as it rises through the atmosphere.
- Why does atmospheric pressure decrease with altitude?
- Explain why aircraft cabins must be pressurized at high altitude.
- Why can aircraft tire pressure change as tire temperature changes?
- What happens to the volume of a trapped gas as a diver descends?
- Explain why an air bubble becomes larger as it rises toward the surface.
- Why can rapidly ascending while holding one's breath be dangerous for a scuba diver?
- Explain why a bicycle pump may become warm when air is compressed.
- How does gas compression connect mechanical work with thermal energy?
- Describe how compression and expansion are used in a refrigerator or air conditioner.
- Explain how expanding gases can perform mechanical work in an engine.
- How does a pneumatic piston convert gas pressure into force?
- Why can heating a sealed gas cylinder be dangerous?
- Explain how heating air helps a hot-air balloon rise.
- A 2.0 cm³ bubble at 300 kPa rises to a region at 100 kPa. Assuming constant temperature, calculate its new volume.
- A rigid cylinder has a pressure of 500 kPa at 300 K. What is its pressure at 360 K?
- Explain why a real engineering situation may require the combined gas law rather than Boyle's Law alone.
- Describe how gas laws, kinetic molecular theory and thermal energy are connected in one practical application of your choice.