Internal Energy

Every object contains energy at the microscopic level, even when the object is not moving as a whole. The total energy stored by the particles inside a substance is called its internal energy.

Internal energy is especially important when studying temperature, heating, changes of state, gases, and thermodynamics.

For a substance, internal energy is mainly associated with:

• the kinetic energy of its particles
• the potential energy associated with interactions between its particles

This means that heating a substance can change its internal energy even when the entire object remains stationary.

What Is Internal Energy?

Internal energy is the total microscopic kinetic and potential energy of the particles in a system.

We commonly represent internal energy using the symbol:

U

For a simple particle model:

Internal energy = microscopic kinetic energy + microscopic potential energy

The word internal is important. We are considering energy associated with particles inside the system, rather than the motion or position of the whole system.

For example, imagine a sealed container of gas sitting on a table. The container is not moving, so its macroscopic kinetic energy is zero relative to the table.

However, the gas molecules inside are moving rapidly in random directions.

The gas therefore has internal energy.

Microscopic and Macroscopic Energy

Energy can be considered on different scales.

Macroscopic energy describes the energy of an object or system as a whole.

Examples include:

• a moving car having kinetic energy
• a book raised above the floor having gravitational potential energy
• a rotating wheel having kinetic energy

Microscopic energy describes energy associated with the enormous number of particles inside matter.

Examples include:

• molecules moving randomly
• atoms vibrating in a solid
• particles rotating or vibrating
• potential energy associated with forces between particles

Internal energy concerns these microscopic forms of energy.

Particle Kinetic Energy

Particles in matter are constantly moving.

Their motion depends partly on the state of matter.

Solids

Particles mainly vibrate around fixed positions.

Liquids

Particles move around and slide past one another while remaining relatively close together.

Gases

Particles move rapidly and randomly through the available space.

The motion of these particles gives them microscopic kinetic energy.

In general, faster-moving particles have greater kinetic energy.

Particle Potential Energy

Particles can also store potential energy because forces act between them.

The potential energy depends on factors such as:

• the distance between particles
• their arrangement
• the attractive or repulsive forces between them

This becomes particularly important during changes of state.

For example, when a solid melts, energy is transferred into the substance. During the phase change, some of this energy changes the arrangement and separation of particles rather than increasing their average speed.

The potential-energy part of the internal energy therefore changes.

Internal Energy and Temperature

Temperature is closely related to the microscopic motion of particles.

For many introductory particle models:

higher temperature → greater average kinetic energy of the particles

If a gas is heated, its particles generally move faster.

If it is cooled, they generally move more slowly.

However, temperature and internal energy are not the same thing.

Temperature is related to the average kinetic energy of the particles.

Internal energy depends on the total microscopic energy of all the particles in the system.

This distinction is extremely important.

Temperature Is Not Total Energy

Consider two containers of water:

• Cup A contains 100 mL of water at 50°C.
• Container B contains 10 L of water at 50°C.

Both have the same temperature, so the average kinetic energy per particle is approximately the same.

However, Container B contains far more particles.

It therefore has much more total internal energy.

So:

same temperature does not necessarily mean same internal energy.

How Internal Energy Changes

The internal energy of a system can change when energy is transferred into or out of it.

Two particularly important transfer mechanisms are:

• heating
• doing work

We can represent the change as:

ΔU = energy transferred by heating + energy transferred by work done on the system

Here:

ΔU means "change in internal energy."

If more energy enters the system than leaves it, internal energy increases.

If more energy leaves than enters, internal energy decreases.

Increasing Internal Energy by Heating

Suppose a metal block is placed on a hot plate.

Energy is transferred from the hot plate to the block.

The particles in the block vibrate more vigorously.

Their average kinetic energy increases.

As a result:

• internal energy increases
• temperature usually increases

The energy has been transferred into the microscopic energy stores of the material.

Decreasing Internal Energy by Cooling

Now suppose the hot metal block is placed in cooler surroundings.

Energy is transferred from the block to the surroundings.

Its particles gradually vibrate less vigorously.

Therefore:

• microscopic kinetic energy decreases
• internal energy decreases
• temperature decreases

Eventually, the block and its surroundings may reach the same temperature.

Internal Energy Can Change Without a Temperature Change

Heating does not always cause the temperature to rise.

During a change of state, energy can increase the internal energy of a substance without increasing its temperature.

Consider melting ice.

As energy is transferred to ice at its melting point, the temperature can remain constant while the ice changes into liquid water.

The transferred energy changes the potential energy associated with the particle arrangement.

Therefore:

internal energy increases even though temperature remains constant during the phase change.

This is one reason temperature cannot be used as a direct measure of total internal energy.

Internal Energy in Gases

Gases provide an especially useful way to study internal energy.

Gas particles:

• move continuously
• travel in random directions
• collide with each other
• collide with the walls of their container

In an idealized gas model, much of our attention is placed on the microscopic kinetic energy of the particles.

Higher temperature means greater average kinetic energy.

Therefore, increasing the temperature of a fixed amount of an ideal gas increases its internal energy.

Heating a Gas in a Rigid Container

Imagine gas inside a strong, sealed container whose volume cannot change.

If energy is transferred to the gas by heating:

1. the gas particles gain kinetic energy
2. they move faster
3. collisions with the container walls become more frequent and/or more forceful
4. the temperature increases
5. the pressure increases

The gas cannot expand because the container is rigid.

The added energy therefore increases the gas's internal energy.

Gas Expansion

Now imagine gas inside a cylinder with a movable piston.

When the gas is heated, it may expand and push the piston outward.

In this situation, some transferred energy may:

• increase the internal energy of the gas
• be transferred as the gas does work on the piston

This is an important thermodynamic idea:

Energy supplied to a gas does not necessarily remain entirely as internal energy.

Some can be transferred from the gas by mechanical work.

Compressing a Gas

Internal energy can also be increased by doing work on a gas.

Imagine pushing down on a piston and compressing the gas inside.

Work is done on the gas.

This transfers energy into the gas.

If the compression happens rapidly enough that little energy escapes to the surroundings, the temperature of the gas rises.

This occurs because the internal energy of the gas increases.

A familiar example is a bicycle pump. When air is compressed repeatedly, the pump can become warm.

Expansion Can Cool a Gas

The reverse can also occur.

If a gas expands and does work on its surroundings without receiving enough energy by heating to replace the transferred energy, its internal energy can decrease.

Its temperature may then decrease.

So:

compression can increase internal energy

while:

expansion can decrease internal energy

depending on the energy transfers taking place.

Internal Energy and Amount of Matter

Internal energy depends partly on how much matter is present.

Imagine two samples of the same substance at the same temperature:

• Sample A has a mass of 1 kg.
• Sample B has a mass of 5 kg.

Sample B contains more particles.

Although the particles have similar average kinetic energies, Sample B has a greater total microscopic energy.

Therefore, it generally has greater internal energy.

Internal energy is therefore an extensive property: it depends on the amount of matter in the system.

Worked Example

A gas is heated while contained in a rigid container.

What happens to its internal energy?

Energy is transferred into the gas by heating.

The particles move faster, increasing their average kinetic energy.

Because the container is rigid, the gas cannot expand and do significant boundary work.

Therefore:

the internal energy of the gas increases.

Its temperature also rises.

Worked Example: Compression

A gas is compressed quickly using a piston.

What happens?

Work is done on the gas.

Energy is transferred into the gas.

Its internal energy increases.

The average kinetic energy of its particles can increase.

Therefore:

the temperature of the gas can rise.

Energy Transfer vs. Energy Storage

A useful distinction is between energy stores and energy transfers.

Internal energy is an energy store within a system.

Heating is an energy-transfer process.

Mechanical work is also an energy-transfer process.

So it is better to say:

Energy is transferred to the object's internal energy store by heating.

rather than:

The object contains heat.

This distinction helps describe energy changes more accurately.

Connecting Internal Energy to the Particle Model

The particle model gives us a microscopic explanation for thermal behaviour.

When a substance is heated:

• particles may move faster
• particles may vibrate more strongly
• particle spacing may change
• particle arrangement may change
• microscopic kinetic energy may increase
• microscopic potential energy may increase

All of these effects can contribute to changes in internal energy.

This explains why the concept of internal energy connects macroscopic observations, such as temperature and state, with microscopic particle behaviour.

Real-World Connections

Car Engines

Hot gases inside an engine have internal energy. Expanding gases push against pistons and transfer energy mechanically.

Refrigerators

Refrigerators use compression and expansion of a refrigerant to transfer energy from inside the refrigerator to the surroundings.

Bicycle Pumps

Rapidly compressing air can raise its temperature because work transfers energy into the gas.

Aerosol Cans

Gas escaping from a pressurized container can expand and cool.

Steam Turbines

High-temperature steam expands and does work on turbine blades, transferring some of its internal energy into mechanical energy.

Common Mistakes

Thinking temperature and internal energy are the same

Temperature is related to the average kinetic energy of particles. Internal energy is the total microscopic kinetic and potential energy.

Thinking a stationary object has no energy

An object can have zero macroscopic kinetic energy while its particles still possess microscopic kinetic and potential energy.

Thinking heating always raises temperature

During a phase change, internal energy can increase while temperature remains constant.

Thinking gases only store kinetic energy in all situations

The ideal gas model emphasizes molecular kinetic energy, but real gases can also involve intermolecular potential energy.

Calling internal energy "heat"

Heat is not best treated as something stored inside an object. Heating describes an energy-transfer process.

Ignoring the amount of matter

Two objects at the same temperature can have very different internal energies if they contain different amounts or types of matter.

Key Terms

Internal energy — the total microscopic kinetic and potential energy of the particles in a system.

Microscopic energy — energy associated with individual particles and their interactions.

Macroscopic energy — energy associated with the motion, position, or behaviour of a system as a whole.

Temperature — a measure related to the average kinetic energy of particles in a substance.

Microscopic kinetic energy — energy associated with the random motion or vibration of particles.

Microscopic potential energy — energy associated with particle positions, arrangements, and interactions.

System — the object or collection of matter being studied.

Heating — transfer of energy resulting from a temperature difference.

Work — transfer of energy when a force causes displacement or through processes such as gas compression and expansion.

Compression — reducing the volume of a gas, often by doing work on it.

Expansion — an increase in gas volume, during which the gas may do work on its surroundings.

Ideal gas — a simplified model of a gas in which particles are treated as having negligible volume and intermolecular interactions are neglected except during collisions.

Key Takeaways

• Internal energy is the total microscopic energy stored within a system.
• It includes microscopic kinetic energy and potential energy.
• Macroscopic motion is different from microscopic particle motion.
• Temperature is related to the average kinetic energy of particles.
• Temperature and internal energy are not the same quantity.
• Objects at the same temperature can have different internal energies.
• Internal energy can change through heating or mechanical work.
• Internal energy can increase without temperature increasing during a change of state.
• Heating a gas generally increases its internal energy.
• Compressing a gas can transfer energy into its internal energy store.
• An expanding gas can do work on its surroundings and lose internal energy.
• The particle model connects microscopic behaviour with macroscopic observations such as temperature and pressure.

Check Your Understanding

1. Define internal energy.
2. What two microscopic forms of energy contribute to the internal energy of a substance?
3. Explain the difference between microscopic and macroscopic kinetic energy.
4. A sealed container of gas is sitting motionless on a table. Explain why the gas still has internal energy.
5. Explain why a 10 L container of water can have more internal energy than a 100 mL cup of water at the same temperature.
6. What generally happens to the average kinetic energy of particles when temperature increases?
7. Explain how internal energy can increase during melting even though the temperature remains constant.
8. A gas is heated inside a rigid container. Describe what happens to its particle motion, internal energy, temperature, and pressure.
9. Explain why rapidly compressing air in a bicycle pump can make the pump warmer.
10. A gas expands and does work on its surroundings without receiving energy by heating. Predict what may happen to its internal energy and temperature.
11. Explain why saying "an object contains heat" is less accurate than describing heating as an energy-transfer process.
12. A moving container of hot gas suddenly stops. Explain the difference between the change in the container's macroscopic kinetic energy and the gas's internal energy.