1.3 - Thermal Energy Transfer and Equilibrium

Thermal Contact

Two systems are in thermal contact if they can transfer energy through thermal processes. Notice that this means they can exchange energy because of a temperature difference, and they don’t necessarily have to be physically touching.

Heating and Cooling

Heating is energy transferred into a system through thermal processes: - The system gains energy. - Its temperature usually goes up.

Cooling is energy transferred out of a system through thermal processes: - The system loses energy. - Its temperature usually goes down.

Heat is a transfer, not something an object has. In physics, heat (

Q

on the equation sheet) is the amount of energy transferred into or out of a system by thermal processes, the same way work is energy transferred by forces. An object has internal energy and a temperature, but it doesn't "contain heat." That's why AP Physics talks about heating and cooling.

Heating usually raises the temperature, but not always. If a gas does work on its surroundings while it's being heated, its temperature can stay the same (an isothermal expansion, Topic 1.4).

Three Thermal Processes

Energy can move between systems at different temperatures in three ways.

1. Conduction

Happens through direct molecular contact.

Energy passes along through collisions between neighboring atoms and molecules.

Works in solids, liquids, and gases.

Example: a metal spoon in hot coffee gets warm.

2. Convection

Involves the bulk motion of a fluid (a liquid or a gas).

Warmer fluid rises and cooler fluid sinks because of density differences.

This sets up circulation currents.

Example: boiling water, or the way the atmosphere circulates.

3. Radiation

Travels as electromagnetic waves, mostly infrared.

Needs no medium, so it even works through a vacuum.

Every object gives off thermal radiation.

Example: the Sun warming Earth, or heat coming off a fire.

Which Way Energy Moves

Energy always flows on its own from a higher-temperature system to a lower-temperature system. It’s always hot to cold, and never spontaneously cold to hot.

This is the Second Law of Thermodynamics in action: heat naturally flows from hot to cold, which increases the total entropy.

Why It Works at the Microscopic Level

When atoms from two different systems collide, energy is most likely to move from the higher-energy atom to the lower-energy one. Here’s the reasoning:

In the hot system, atoms have higher average kinetic energy.

In the cold system, atoms have lower average kinetic energy.

At the boundary, the fast atoms tend to hand energy over to the slow ones.

It’s statistical, so fast-to-slow is just far more probable than slow-to-fast.

Example: one head-on collision. Two identical helium atoms (

m = 6.64 \times 10^{-27}

kg) collide head-on. Atom A, from the hot gas, moves at

+9.0 \times 10^2

m/s, and atom B, from the cold gas, moves at

-3.0 \times 10^2

m/s. In an elastic head-on collision between equal masses, the atoms swap velocities: A leaves at

-3.0 \times 10^2

m/s and B leaves at

+9.0 \times 10^2

m/s.

Check with conservation of momentum: before,

p = m(900 - 300) = m(600 \text{ m/s})

. After,

p = m(-300 + 900) = m(600 \text{ m/s})

. Kinetic energy is conserved too, since the two speeds just traded places.

The energy transferred from A to B is

\Delta K = \frac{1}{2}m\left(v_{A,i}^2 - v_{A,f}^2\right) = \frac{1}{2}(6.64 \times 10^{-27} \text{ kg})\left[(900 \text{ m/s})^2 - (300 \text{ m/s})^2\right] = 2.4 \times 10^{-21} \text{ J}

The faster atom handed energy to the slower one. A single collision can go the other way (a glancing hit can speed up an atom that's already fast), but averaged over billions of collisions, the net flow is always from the higher-temperature system to the lower-temperature one.

After a whole lot of these collisions between atoms from the two systems, the most probable end state is both systems ending up at the same temperature. Their average kinetic energies even out, and so do their temperatures.

Thermal Equilibrium

Thermal equilibrium is what you get when no net energy moves between two systems that are in thermal contact.

At equilibrium: - Both systems are at the same temperature. - Energy is still being exchanged, but equal amounts flow both ways. - There’s no net energy flow. - The state is stable.

This connects to the Zeroth Law of Thermodynamics: if system A is in thermal equilibrium with system C, and system B is also in thermal equilibrium with system C, then A and B are in thermal equilibrium with each other.

Practice Problems

1. A hot metal block is dropped into cool water. Describe what happens to the energy at the microscopic level.

2. Why does heat flow on its own from hot to cold, but never cold to hot?

3. Two systems in thermal contact are at 400 K and 300 K. Which way does the net energy flow?

4. A metal rod connects a hot reservoir (500 K) and a cold reservoir (300 K). After a long time, what does the rod’s temperature distribution look like if the reservoirs are held at constant temperature?

5. Explain why thermal equilibrium doesn’t mean energy has stopped being exchanged.

Solutions

1. At the boundary between the metal and the water: - The metal atoms have high average kinetic energy because the metal is hot. - The water molecules have lower average kinetic energy because the water is cooler. - When they collide, the fast metal atoms pass energy to the slower water molecules. - The metal atoms slow down, so the metal cools. - The water molecules speed up, so the water heats. - This keeps going until the average kinetic energies, and therefore the temperatures, are equal. - At that point they’ve reached thermal equilibrium.

2. It comes down to statistical probability. In collisions between high-energy and low-energy atoms, energy moving from high to low is far more likely than low to high.

Once you scale that up to huge numbers of atoms, the overwhelmingly most probable result of all those collisions is that the hot system loses energy and the cold system gains it, until they even out.

Heat flowing cold to hot would require entropy to spontaneously drop, meaning the fast atoms would have to keep gaining energy from the slow ones, collision after collision, which is statistically almost impossible.

3. The net energy flows from the 400 K system to the 300 K system, so hot to cold.

The hotter system has higher average molecular kinetic energy, so collisions at the boundary tend to move energy from hot to cold.

4. This is a steady state, which is not the same as equilibrium: - Energy flows continuously from the hot end (500 K) to the cold end (300 K). - There’s a temperature gradient along the rod. - Each point stays at a constant temperature, but it’s different from its neighbors. - The hot end sits near 500 K and the cold end near 300 K. - If the rod is uniform, the gradient is linear.

It’s not thermal equilibrium, because energy keeps flowing through the rod.

5. At thermal equilibrium, energy is still being exchanged between the systems, but the rates are equal in both directions.

The hot-to-cold transfer rate equals the cold-to-hot transfer rate.

The net energy transfer is zero.

The molecular collisions never stop, since the atoms keep moving.

It’s a dynamic equilibrium, with ongoing processes that balance out.

Think of two rooms with people walking through a doorway at equal rates. There’s no net flow, but people are constantly moving.

Common Mistakes

Thinking equilibrium means no energy is exchanged. Energy is always being exchanged. What’s actually zero is the net transfer.

Confusing temperature and energy. Temperature is average kinetic energy per particle, while total energy depends on both temperature and how much substance there is.

Assuming everything is conduction. There are three separate mechanisms: conduction, convection, and radiation.

Thinking cold gets transferred. Energy flows from hot to cold. “Cold” isn’t a substance that moves.

Forgetting radiation needs no medium. Conduction and convection need matter, but radiation works in a vacuum.

Assuming thermal contact means physical contact. Systems can be in thermal contact through radiation without ever touching.

Thinking thermal equilibrium means equal total energy. Equilibrium means equal temperature, not equal total energy.

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