Thermal equilibrium, internal energy, heat and work

Thermal equilibrium

Put a cold metal block into warm water. Heat flows until both reach one temperature, and then nothing more changes: they are in thermal equilibrium.

The zeroth law: if A and B are each in equilibrium with C, they are in equilibrium with each other. That is why a thermometer (C) can compare A and B: equal readings mean equal temperatures.

Internal energy

Every body holds energy in its molecules: their motion, their spinning, and the pull between them. This is its internal energy, $U$.

For an ideal gas the molecules do not pull on each other, so $U$ is all motion energy, and it depends only on the temperature. Each way a molecule can move holds $\tfrac12 k_BT$ on average, so

with $f = 3$ for a monatomic gas (moving along $x$, $y$ and $z$) and $f = 5$ for a diatomic gas (two ways of turning as well).

You should find that doubling the temperature doubles the internal energy, and that a diatomic gas holds $\tfrac53$ times as much as a monatomic gas at the same temperature.

Heat and work

Energy can get into or out of a gas in two ways.

• Heat, $Q$: energy that flows because of a difference of temperature.
• Work, $W$: energy passed by a force moving something, such as a gas pushing a piston.

Neither is something a body has. A body has internal energy; heat and work are energy on the move.

Watch out. Temperature is not internal energy. A bath at $40\ ^\circ\text{C}$ holds far more internal energy than a cup at $40\ ^\circ\text{C}$.

A worked example

Find the internal energy of $2.0$ mol of helium and of $2.0$ mol of nitrogen, both at $300\ \text{K}$.

1. Helium ($f = 3$): $U = \tfrac32 \times 2.0\ \text{mol} \times 8.31\ \text{J mol}^{-1}\text{K}^{-1} \times 300\ \text{K} = 7480\ \text{J}$.
2. Nitrogen ($f = 5$): $U = \tfrac52 \times 2.0\ \text{mol} \times 8.31\ \text{J mol}^{-1}\text{K}^{-1} \times 300\ \text{K} = 12\,470\ \text{J}$.
3. Per molecule, the energy of motion along $x$, $y$, $z$ is $\tfrac32 k_BT = 6.2 \times 10^{-21}\ \text{J}$ in both.

Practise with new numbers

Try it in Python

The program works out the two gases of the example.

Remember these

• Zeroth law: equilibrium with the same body means equal temperatures.
• Ideal gas: $U = \tfrac{f}{2}nRT$; $f = 3$ monatomic, $5$ diatomic.
• Heat and work are energy in transit; internal energy belongs to the body.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What is the internal energy of $1.0$ mol of argon at $500\ \text{K}$?

2. Can you say how much heat a body 'contains'?

3. Does the internal energy of an ideal gas change when it expands at a steady temperature?

4. You rub your hands together and they warm. Was energy passed as heat or as work?