Thermal Energy and Calorimetry
3. Heating and Cooling Curves
Learning outcomes
- I can interpret heating and cooling curves.
- I can identify regions where temperature changes and where phase changes occur.
- I can explain how particle motion changes throughout a heating or cooling process.
- I can relate energy input to changes in temperature and state.
- I can predict the behavior of substances during heating and cooling.
Heating and Cooling Curves
What Is a Heating Curve?
A heating curve is a graph showing how the temperature of a substance changes as thermal energy is added.
A typical heating curve follows a substance as it changes through:
solid → liquid → gas
The graph usually contains two types of regions:
- sloping regions, where temperature changes
- flat regions, where a phase change occurs
Heating curves are useful because they show that adding energy does not always cause the temperature to increase.
Reading the Axes
A heating curve normally has:
Vertical axis: temperature
Horizontal axis: energy added or heating time
If the heater supplies energy at a constant rate, time can be used as an indication of the amount of energy transferred.
Moving from left to right means that more energy is being supplied to the substance.
The Five Main Regions
A complete heating curve often contains five regions:
solid warming → melting → liquid warming → boiling → gas warming
These regions alternate between sloping and flat sections.
| Region | State | Temperature | What happens to energy? |
|---|---|---|---|
| Solid warming | Solid | Increases | Particle kinetic energy increases |
| Melting | Solid + liquid | Constant | Potential energy increases |
| Liquid warming | Liquid | Increases | Particle kinetic energy increases |
| Boiling | Liquid + gas | Constant | Potential energy increases |
| Gas warming | Gas | Increases | Particle kinetic energy increases |
Understanding this pattern makes heating curves much easier to interpret.
Solid Warming
At the beginning of a typical heating curve, the substance is completely solid.
Particles in a solid:
- are closely packed
- remain in relatively fixed positions
- vibrate around those positions
As thermal energy is supplied, the particles vibrate more vigorously.
Their average kinetic energy increases.
Therefore:
energy added → kinetic energy increases → temperature increases
The graph slopes upward.
No phase change occurs during this section.
Reaching the Melting Point
Eventually, the solid reaches its melting point.
At this temperature, additional energy begins changing the substance from solid to liquid.
The important observation is:
energy continues to enter the substance, but its temperature remains approximately constant.
The heating curve becomes flat.
The Melting Plateau
During melting:
solid → liquid
Both solid and liquid are present.
The transferred energy is used mainly to overcome some of the attractive interactions maintaining the solid structure.
Instead of substantially increasing average particle kinetic energy, the energy increases the potential energy associated with particle arrangement.
Therefore:
average kinetic energy remains approximately constant
so:
temperature remains constant
This flat section is called a plateau.
Latent Heat During Melting
The energy transferred during melting is called latent heat.
The energy required can be calculated using:
Q = mL_f
where:
- Q = energy transferred, in J
- m = mass, in kg
- L_f = specific latent heat of fusion, in J/kg
No ΔT appears in this equation because the temperature does not change during the ideal phase change.
Liquid Warming
Once all the solid has melted, the substance is completely liquid.
Additional energy now increases the average kinetic energy of the liquid particles.
The particles:
- move faster
- collide more frequently and energetically
- have greater average kinetic energy
Therefore the temperature begins increasing again.
The heating curve slopes upward.
Energy transferred during this region can be calculated using:
Q = mcΔT
Reaching the Boiling Point
Eventually, the liquid reaches its boiling point.
Additional energy now begins converting the liquid into gas.
The temperature again stops increasing while the phase change takes place.
The graph enters another flat region.
The Boiling Plateau
During boiling:
liquid → gas
Both liquid and gas are present.
Energy continues to enter the system, but it is used mainly to overcome attractive interactions between particles.
The particles become much more widely separated as they enter the gas phase.
Therefore:
energy added → potential energy increases
while:
average kinetic energy remains approximately constant → temperature remains constant
Latent Heat During Boiling
The energy required to vaporize a substance can be calculated using:
Q = mL_v
where:
- Q = energy transferred
- m = mass
- L_v = specific latent heat of vaporization
For water, the specific latent heat of vaporization is much larger than its specific latent heat of fusion.
This means considerably more energy is required to vaporize 1 kg of water than to melt 1 kg of ice at their respective phase-change temperatures.
Gas Warming
Once all the liquid has become gas, additional energy again increases particle kinetic energy.
Gas particles:
- move faster
- have greater average kinetic energy
- collide more energetically
The gas temperature therefore increases.
The heating curve once again slopes upward.
Slopes and Plateaus
A simple rule helps when interpreting heating curves:
Sloping line
Temperature is changing.
The substance remains primarily in one state.
Use:
Q = mcΔT
Flat line
A phase change is occurring.
Temperature remains approximately constant.
Use:
Q = mL
So:
slope = temperature change
plateau = phase change
Particle Motion Along a Heating Curve
Heating curves can also be interpreted using the particle model.
Solid warming
Particles vibrate faster.
Kinetic energy increases.
Melting
Particles begin leaving their fixed arrangement.
Potential energy increases.
Liquid warming
Particles move faster.
Kinetic energy increases.
Boiling
Particles separate substantially and enter the gas phase.
Potential energy increases.
Gas warming
Particles move increasingly rapidly.
Kinetic energy increases.
This gives an important relationship:
temperature change → mainly kinetic energy change
phase change → mainly potential energy change
A Heating Curve for Water
Consider water being heated from ice below 0°C to steam above 100°C at standard atmospheric pressure.
The sequence is:
ice below 0°C
Temperature rises.
↓
ice + water at approximately 0°C
Melting occurs while temperature remains approximately constant.
↓
liquid water from approximately 0°C to 100°C
Temperature rises.
↓
water + steam at approximately 100°C
Boiling occurs while temperature remains approximately constant.
↓
steam above 100°C
Temperature rises.
This assumes approximately standard atmospheric pressure.
Energy Calculations Across a Heating Curve
A complete heating problem may require several calculations.
Suppose ice begins at −20°C and ends as steam above 100°C.
The total energy cannot be found using one single Q = mcΔT calculation.
Instead, divide the process into stages.
Heating the Ice
Q₁ = mc_iceΔT
Melting the Ice
Q₂ = mL_f
Heating the Liquid Water
Q₃ = mc_waterΔT
Vaporizing the Water
Q₄ = mL_v
Heating the Steam
Q₅ = mc_steamΔT
Then:
Q_total = Q₁ + Q₂ + Q₃ + Q₄ + Q₅
Each section of the graph represents a different physical process.
What Does the Length of a Plateau Mean?
If energy is being supplied at a constant rate, a longer plateau means more energy is required for that phase change.
For water, the boiling plateau is typically much longer than the melting plateau under comparable conditions because:
L_v > L_f
Therefore, vaporizing water requires considerably more energy per kilogram than melting ice.
The horizontal length of a heating curve must be interpreted carefully, however. It represents energy directly only if the horizontal axis is energy, or if energy is being supplied at a constant rate when the axis is time.
What Does the Slope Tell Us?
During a region without a phase change:
Q = mcΔT
Rearranging:
ΔT = Q/(mc)
For the same mass and the same rate of energy input, a substance with a lower specific heat capacity experiences a greater temperature change.
Its heating curve will therefore rise more steeply.
A substance with a higher specific heat capacity experiences a smaller temperature change for the same energy input.
Its curve will be less steep.
Therefore, the slope can provide information about the thermal properties of a substance.
What Is a Cooling Curve?
A cooling curve shows what happens as thermal energy is transferred from a substance to its surroundings.
The processes on a cooling curve are essentially the reverse of those on a heating curve.
For a substance beginning as a gas:
gas → liquid → solid
The temperature decreases during single-state regions and remains approximately constant during phase changes.
Gas Cooling
Initially, the substance may be entirely gaseous.
As energy leaves the gas:
- particles move more slowly
- average kinetic energy decreases
- temperature decreases
The graph slopes downward.
Eventually, the gas reaches its condensation temperature.
Condensation Plateau
During condensation:
gas → liquid
Energy continues to leave the substance.
However, the temperature remains approximately constant while the phase change occurs.
Particles move closer together and form stronger intermolecular attractions.
Energy is released to the surroundings.
This is the reverse of vaporization.
Vaporization absorbs latent heat.
Condensation releases latent heat.
Liquid Cooling
Once condensation is complete, the substance is entirely liquid.
As more energy leaves:
- particles move more slowly
- average kinetic energy decreases
- temperature decreases
The cooling curve slopes downward again.
Eventually, the freezing point is reached.
Freezing Plateau
During freezing:
liquid → solid
Energy continues to leave the substance, but its temperature remains approximately constant.
Particles become arranged into the more ordered structure of the solid.
Latent heat is released to the surroundings.
The process is the reverse of melting.
Melting absorbs latent heat.
Freezing releases latent heat.
Solid Cooling
Once freezing is complete, the substance is entirely solid.
Further energy loss reduces the vibrational motion of the particles.
Average kinetic energy decreases.
Therefore:
temperature decreases
The graph slopes downward again.
Heating vs Cooling
| Heating | Cooling |
|---|---|
| Energy enters substance | Energy leaves substance |
| Solid warms | Gas cools |
| Melting absorbs energy | Condensation releases energy |
| Liquid warms | Liquid cools |
| Boiling absorbs energy | Freezing releases energy |
| Gas warms | Solid cools |
The same phase-change temperatures appear on ideal heating and cooling curves for a pure substance at the same pressure.
Predicting What Happens
Heating and cooling curves allow us to predict the state of a substance.
Suppose a substance has:
melting point = 40°C
boiling point = 120°C
Then, under the specified pressure:
Below 40°C:
solid
At 40°C while melting:
solid + liquid
Between 40°C and 120°C:
liquid
At 120°C while boiling:
liquid + gas
Above 120°C:
gas
This information can often be determined directly from a heating curve.
Example: Identifying a Phase Change
Suppose a substance is heated continuously.
Its temperature increases from 20°C to 65°C.
It then remains at 65°C for several minutes even though energy continues to be supplied.
Afterward, the temperature begins increasing again.
The flat section indicates a phase change.
If the substance began as a solid, then 65°C is its approximate melting point under those conditions.
During the plateau:
solid + liquid are present
Example: Predicting Particle Behaviour
Suppose a liquid is heated but has not yet reached its boiling point.
As energy enters:
- particle average kinetic energy increases
- particles move faster
- temperature increases
Once boiling begins:
- the average kinetic energy remains approximately constant
- particles continue separating into the gas phase
- potential energy increases
- temperature remains approximately constant
This explains why a temperature graph changes from a slope to a plateau.
Evaporation vs a Heating-Curve Plateau
Evaporation should not automatically be confused with the boiling plateau.
Evaporation can occur from the surface of a liquid at temperatures below its boiling point.
Boiling occurs throughout the liquid at its boiling point for the given pressure.
A standard heating curve usually shows the boiling phase change, not ordinary surface evaporation.
Heating Curves and Pressure
Melting and boiling points can depend on pressure.
This is especially important for boiling.
At lower atmospheric pressure, liquids generally boil at lower temperatures.
For example, water boils below 100°C at sufficiently high altitudes.
At higher pressure, its boiling temperature increases.
Therefore, a heating curve should be interpreted with the pressure conditions in mind.
Heating Curves in Everyday Life
Heating and cooling curves help explain many familiar processes.
Melting Ice
Ice can absorb considerable energy while remaining close to 0°C as it melts.
Boiling Water
Water can continue absorbing energy at its boiling point without becoming significantly hotter while liquid water remains.
Freezing Food
Energy must be removed not only to cool food to its freezing point but also to complete the phase change.
Refrigeration
Refrigerants repeatedly evaporate and condense, absorbing and releasing latent heat.
Industrial Processing
Engineers use heating and cooling curves when controlling:
- metal processing
- food production
- chemical manufacturing
- refrigeration
- material testing
Common Mistakes
Mistake: "Every part of a heating curve slopes upward."
During phase changes, temperature remains approximately constant and the curve forms a plateau.
Mistake: "No energy is being transferred during a flat section."
Energy is still being transferred. It is changing the state rather than the temperature.
Mistake: "Particles stop moving during a plateau."
Particles continue moving. Their average kinetic energy remains approximately constant during the ideal phase change.
Mistake: "Temperature measures all the internal energy of a substance."
Temperature is related to average kinetic energy. Internal energy also includes potential energy associated with particle interactions.
Mistake: "Use Q = mcΔT everywhere on a heating curve."
Use Q = mcΔT when temperature changes and Q = mL during a phase change.
Mistake: "A flat line always means melting."
It could represent melting, boiling, freezing or condensation depending on whether the substance is being heated or cooled and which states are present.
Key Terms
Heating curve — graph showing how temperature and state change as energy is supplied to a substance.
Cooling curve — graph showing how temperature and state change as energy is removed.
Plateau — flat region of a heating or cooling curve where a phase change occurs at approximately constant temperature.
Melting point — temperature at which solid and liquid phases coexist during melting or freezing under specified pressure.
Boiling point — temperature at which boiling occurs at a specified external pressure.
Kinetic energy — energy associated with particle motion.
Potential energy — energy associated with particle positions and interactions.
Specific heat capacity — energy required to raise the temperature of 1 kg of a substance by 1°C or 1 K.
Specific latent heat — energy transferred per kilogram during a phase change without a temperature change.
Latent heat of fusion — energy associated with the solid–liquid phase change.
Latent heat of vaporization — energy associated with the liquid–gas phase change.
Condensation — change from gas to liquid.
Freezing — change from liquid to solid.
Key Takeaways
- Heating and cooling curves show how temperature and state change as energy is transferred.
- Sloping sections represent temperature changes within a single state.
- Flat sections represent phase changes.
- During a sloping section, average particle kinetic energy changes.
- During an ideal phase change, temperature and average kinetic energy remain approximately constant while particle potential energy changes.
- Melting and boiling absorb latent heat.
- Freezing and condensation release latent heat.
- Use Q = mcΔT when temperature changes without a phase change.
- Use Q = mL during a phase change.
- A longer plateau represents a larger energy transfer if the horizontal scale represents energy or energy is supplied at a constant rate.
- The slope of a heating curve can be related to specific heat capacity when mass and heating rate are controlled.
- Heating and cooling processes are essentially reverses of one another.
- Heating curves can be used to identify melting points, boiling points and the physical state of a substance.
- A complete heating or cooling problem may require several separate energy calculations.
Check Your Understanding
- What does a sloping section of a heating curve represent?
- What does a flat section represent?
- Why does temperature remain approximately constant while a solid melts?
- Describe how particle motion changes while a solid is heated toward its melting point.
- What happens to particle potential energy during melting?
- Why does the temperature begin rising again after melting is complete?
- What two states are present during the boiling plateau?
- Explain the difference between what happens to energy during a sloping region and during a plateau.
- Which equation should be used during a temperature-changing region: Q = mcΔT or Q = mL?
- Which equation should be used during a phase change?
- Describe what happens to particles as a gas cools and condenses.
- A substance is heated at a constant rate and remains at 80°C for several minutes before its temperature begins rising again. Explain what is happening during the constant-temperature region.