Specific heat capacities of gases and solids

From energy to heat capacity

In the last lesson we found the internal energy of an ideal gas:

Warm the gas at a constant volume. It does no work, so all the heat goes into $U$. The heat for each mole and each kelvin is the molar heat capacity at constant volume:

You met Mayer's relation, $C_p - C_v = R$, in the chapter on thermodynamics. So

Now the counting of the last lesson gives numbers we can test.

Gases, one kind at a time

• Monatomic, $f = 3$: $C_v = \tfrac32 R = 12.5\ \text{J/(mol K)}$, $C_p = \tfrac52 R$, $\gamma = \tfrac53$.
• Rigid diatomic, $f = 5$: $C_v = \tfrac52 R = 20.8\ \text{J/(mol K)}$, $C_p = \tfrac72 R$, $\gamma = \tfrac75$.
• Diatomic that vibrates, $f = 7$: $C_v = \tfrac72 R$, $C_p = \tfrac92 R$, $\gamma = \tfrac97$.
• Polyatomic: moving and turning give $\tfrac32 k_BT + \tfrac32 k_BT$. Each active vibration adds $k_BT$. With $v$ vibrations, $C_v = (3 + v)R$ and $C_p = (4 + v)R$.

How good are these numbers? For neon and argon, the measured $C_v$ is $12.5\ \text{J/(mol K)}$, just as predicted. For carbon dioxide at room temperature it is about $28.8\ \text{J/(mol K)}$. That is more than the rigid value, $20.8\ \text{J/(mol K)}$. So some of its vibrations are already active.

A mixture of gases

A gas used in some radiation detectors is $90\ \%$ argon and $10\ \%$ methane, by moles. What is its $C_v$?

The internal energies of the two gases simply add. So the heat capacity of the mixture is an average, weighted by the moles:

Then $C_p = C_v + R$ and $\gamma = C_p/C_v$, as for any ideal gas.

Watch out. Do not average the two values of $\gamma$. Average $C_v$ first.

Start with pure argon. Then add methane bit by bit. Does gamma fall in a straight line? Find the share that gives $\gamma = 1.5$.

Solids: the rule of Dulong and Petit

In a solid, each atom vibrates about a fixed place. It can vibrate along $x$, $y$ and $z$. Each direction has kinetic and potential energy, so each atom has $6 \times \tfrac12 k_BT = 3k_BT$. One mole has

A solid hardly expands, so $C_p$ and $C_v$ are almost the same. This is the rule of Dulong and Petit. It works well for most metals at room temperature. To get the specific heat, divide by the molar mass in kg/mol: $c = 3R/M$.

Water

Treat a water molecule like three atoms in a solid, each with $3k_BT$. Then one molecule has $9k_BT$, and

The measured value is about $75\ \text{J/(mol K)}$. Dividing by $0.018\ \text{kg/mol}$ gives about $4160\ \text{J/(kg K)}$, close to the well-known $4186\ \text{J/(kg K)}$.

Where the classical theory fails

Cool a solid towards $0\ \text{K}$. Its measured heat capacity falls towards zero, far below $3R$. The heat capacity of hydrogen gas also changes with temperature, in steps.

At low temperatures hydrogen behaves like a monatomic gas. Its molecules do not take energy into turning. Near room temperature they turn, and $C_v = \tfrac52 R$. Only when it is very hot do they vibrate.

Equipartition cannot explain this. It says every degree of freedom always takes $\tfrac12 k_BT$. The answer came from quantum theory. Energy is taken in small packets. A molecule at a low temperature does not have enough energy for one packet of turning or vibrating, so those degrees of freedom stay "frozen".

Practise with new numbers

Try it in Python

The program finds $C_v$, $C_p$ and $\gamma$ for three kinds of gas. Then it tests the rule of Dulong and Petit on three metals.

Common mistakes

• Averaging the two values of $\gamma$ for a mixture. Average $C_v$, weighted by the moles. Then find $\gamma$.
• Adding $\tfrac12 R$ for each vibration. A vibration has two squared terms, so it adds $R$ to $C_v$.
• Using the molar mass in g/mol in $c = 3R/M$. Use kg/mol.
• Expecting $3R$ for every solid at every temperature. Near $0\ \text{K}$ the heat capacity falls far below it.

Remember these

• $C_v = \tfrac{f}{2}R$, $C_p = C_v + R$, $\gamma = 1 + \tfrac{2}{f}$.
• Polyatomic with $v$ active vibrations: $C_v = (3 + v)R$.
• Mixture: $C_v = \dfrac{n_1C_{v1} + n_2C_{v2}}{n_1 + n_2}$.
• Solids: $C = 3R$ (Dulong and Petit). Water: about $9R$.
• At low temperatures some degrees of freedom are frozen. Quantum theory explains it.

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 $\gamma$ for steam, if its molecules turn but do not vibrate?

2. One mole of argon is mixed with one mole of a rigid diatomic gas. What is $C_v$ of the mixture?

3. Why is the molar heat capacity of most metals close to $25\ \text{J/(mol K)}$?

4. Hydrogen gas is cooled to $40\ \text{K}$. Its $C_v$ falls to about $\tfrac32 R$. Why?