Specific heats of a gas and Mayer's relation

Two ways to warm a gas

Warm a gas in a sealed rigid can: all the heat raises its internal energy. Warm it under a free piston: the gas expands, pushing the piston, so some heat leaves as work. It needs more heat for the same rise in temperature.

So a gas has two molar heat capacities:

• $C_v$, at a steady volume: $Q = nC_v\,\Delta T$;
• $C_p$, at a steady pressure: $Q = nC_p\,\Delta T$.

Mayer's relation

At a steady pressure, the extra heat is the work $p\,\Delta V = nR\,\Delta T$. So

From the ways molecules move

Since $U = \tfrac{f}{2}nRT$, $C_v = \tfrac{f}{2}R$ and $C_p = \left(\tfrac{f}{2} + 1\right)R$. Their ratio is $\gamma = \dfrac{C_p}{C_v} = 1 + \dfrac{2}{f}$.

You should find that $C_p - C_v$ stays $8.31\ \text{J mol}^{-1}\text{K}^{-1}$, while $\gamma$ falls toward $1$ as molecules gain more ways to move.

Watch out. $\Delta U = nC_v\,\Delta T$ in every process of an ideal gas, even one at a steady pressure: $U$ depends only on $T$.

A worked example

$0.30$ mol of oxygen (diatomic) is warmed by $40\ \text{K}$. Find the heat needed at a steady volume and at a steady pressure.

1. Steady volume. $0.30\ \text{mol} \times 20.8\ \text{J mol}^{-1}\text{K}^{-1} \times 40\ \text{K} = 249\ \text{J}$.
2. Steady pressure. $0.30\ \text{mol} \times 29.1\ \text{J mol}^{-1}\text{K}^{-1} \times 40\ \text{K} = 349\ \text{J}$.
3. The difference, $100\ \text{J}$, is the work $nR\,\Delta T = 0.30\ \text{mol} \times 8.31\ \text{J mol}^{-1}\text{K}^{-1} \times 40\ \text{K}$.

Practise with new numbers

Try it in Python

The program finds the heat capacities for three kinds of molecule.

Remember these

• $C_p - C_v = R$ (Mayer).
• $C_v = \tfrac{f}{2}R$; $\gamma = 1 + \dfrac{2}{f}$.
• $\Delta U = nC_v\,\Delta T$ always, for an ideal gas.

Check yourself

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

1. What are $C_v$ and $C_p$ for helium?

2. Why is $C_p$ larger than $C_v$?

3. A gas has $C_v = 25\ \text{J mol}^{-1}\text{K}^{-1}$. What is its $\gamma$?

4. $2.0$ mol of a diatomic gas is warmed by $14\ \text{K}$ at a steady pressure. How much of the heat raises its internal energy?