Thermal Equilibrium: Definition, Formula & Example

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  • 0:01 What Is Thermal Equilibrium?
  • 0:51 Mechanism
  • 1:26 Formula
  • 1:58 Example
  • 2:28 Lesson Summary
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Lesson Transcript
David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

Expert Contributor
Elaine Chan

Dr. Chan has a Ph.D. from the U. of California, Berkeley. She has done research and teaching in mathematics and physical sciences.

In this lesson, you will learn what thermal equilibrium is, discover an equation that relates to thermal equilibrium, and go over an example of its use. A short quiz will follow.

What Is Thermal Equilibrium?

There are some concepts in physics that people understand intuitively before they've even heard of them. Like thermal equilibrium, which is so integral to our lives that we tend to understand it by intuition.

For instance, let's say you got a hot mug of tea and put it in the freezer. What will happen to the tea? The tea will, of course, cool down. Everyone knows that. And, you also probably know the tea will keep cooling down until it is the same temperature as the freezer - until it is frozen solid and can't get any colder. Even if you are familiar with this concept, what you may not know is that it is a solid example of thermal equilibrium.

Thermal equilibrium is the state in which two objects connected by a permeable barrier don't have any heat transfer between them. This happens when the two objects have the same temperature.

Illustration of Thermal Equilibrium.


Temperature is the average kinetic, or movement, energy of the molecules in a substance. When you put two objects of different temperatures in contact with one another, the faster-moving molecules in one material will collide with the slower-moving molecules in the other. The heat energy will gradually spread out until the two objects have the same temperature - until they have reached thermal equilibrium. This is basically the same as the second law of thermodynamics,which states that heat only spontaneously moves from hotter places to colder places, never the other way around.


If an object (or system) is in thermodynamic equilibrium, then it can be said that the system has minimized its thermodynamic potential. There are many types of thermodynamic potential quoted in physics, but perhaps the most common one is the Helmholtz free energy, which measures the total amount of useful work that could be extracted from the system. The equation for Helmholtz free energy is as follows:

Equation for Helmholtz Free Energy

As mentioned previously, if the system has reached thermodynamic equilibrium, this number will be at its minimum possible value.


So, let's say you mix tea and milk until they reach thermodynamic equilibrium, and you want to know what is the maximum amount of useful energy you can take out of that cup of tea, if your body was perfectly efficient.

The temperature of the tea and milk (in Kelvin) is 373 K, the total internal energy of the system is 3000 Joules, and the entropy of the system is 6 Joules per Kelvin. Plug those numbers into the equation:

Helmholtz Substitution

And, you get 762 Joules of useful work.

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Thermal Equilibrium, Heat, and Temperature

Thermal equilibrium is the condition under which two substances in physical contact with each other exchange no heat energy. Two substances in thermal equilibrium are said to be at the same temperature.

Zeroth Law of Thermodynamics

The law is called zeroth because the first and second laws were already named, and it was realized that this obvious postulate needed to be stated first. The zeroth law states that if two systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with each other.

Problem 1:

If two systems are in thermal equilibrium and contact, do their temperatures increase?

Answer 1:


Problem 2:

Do two systems in thermal equilibrium with a third system and a fourth system have the same temperature?

Answer 2:


Problem 3:

Is it true that a higher temperature object which is in contact with a lower temperature object will transfer heat to the lower temperature object?

Answer 3:

Yes. The objects will approach the same temperature, and in the absence of loss to other objects, they will then keep a constant temperature.

Problem 4:

Suppose you want to determine if two systems, A and B, are in thermal equilibrium. How do you do it?

Answer 4:

Use a third system, C, a thermometer, for instance. If A and C are in thermal equilibrium as are B and C, then according to the zeroth law, A and B are in thermal equilibrium.

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