Specific Heat Capacity

The Concept of Specific Heat Capacity

When you heat a substance, its temperature rises. The amount of heat required to raise the temperature of a unit mass of the substance by one degree (1 °C or 1 K) is called its specific heat capacity. It is a material property, denoted by $s$ or $c$, and its SI unit is $\text{J kg}^{-1} \text{K}^{-1}$.

For a given substance, the specific heat capacity depends on the conditions under which the heat is supplied. In thermodynamics, two conditions are especially important: constant volume and constant pressure. These give rise to two distinct specific heats for gases: $C_V$ (molar specific heat at constant volume) and $C_P$ (molar specific heat at constant pressure).

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Molar Specific Heat at Constant Volume ($C_V$)

Consider one mole of an ideal gas enclosed in a rigid container (volume fixed). If we supply a small amount of heat $\Delta Q$ to it, the temperature rises by $\Delta T$. Since the volume does not change, no work is done by the gas ($\Delta W = 0$). From the first law of thermodynamics:

Thus, all the heat supplied goes into increasing the internal energy of the gas. The molar specific heat at constant volume is defined as:

For an ideal gas, the internal energy $U$ depends only on temperature, not on volume. So we can drop the subscript and write:

This is a fundamental relation: $C_V$ is the rate of change of internal energy with temperature for one mole of an ideal gas.

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Molar Specific Heat at Constant Pressure ($C_P$)

Now consider one mole of the same ideal gas in a cylinder fitted with a movable, frictionless piston. The external pressure is kept constant. When heat $\Delta Q$ is supplied, the gas expands, doing work against the external pressure. The temperature rises by $\Delta T$. The molar specific heat at constant pressure is:

From the first law:

For one mole of an ideal gas, the equation of state is $PV = RT$. At constant pressure, $P \Delta V = R \Delta T$. Also, $\Delta U = C_V \Delta T$. Therefore:

Cancelling $\Delta T$ (non-zero), we obtain the Mayer's relation:

This is a key result for ideal gases. It tells us that $C_P$ is always greater than $C_V$ by the universal gas constant $R \approx 8.314 \, \text{J mol}^{-1} \text{K}^{-1}$.

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Deriving $C_V$ from Kinetic Theory

From kinetic theory, the internal energy of one mole of a monatomic ideal gas (like helium or argon) is purely translational kinetic energy:

Differentiating with respect to temperature:

Then using Mayer's relation:

The ratio of specific heats, $\gamma = C_P / C_V$, is:

These values match experimental data for monatomic gases very well.

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Diatomic and Polyatomic Gases

For diatomic molecules (like $N_2$, $O_2$, $H_2$), the molecule can also rotate and vibrate. At ordinary temperatures (around room temperature), the vibrational modes are usually "frozen out" — they do not contribute to the specific heat. Only translational and rotational degrees of freedom are active.

A diatomic molecule has 3 translational and 2 rotational degrees of freedom (rotation about the axis joining the atoms has negligible moment of inertia). According to the law of equipartition of energy, each degree of freedom contributes $\frac{1}{2}RT$ to the molar internal energy. So:

Thus:

These values agree well with experimental data for diatomic gases at moderate temperatures.

At very high temperatures, vibrational modes become active, adding two more degrees of freedom (kinetic and potential energy of vibration), giving $C_V = \frac{7}{2}R$ and $\gamma \approx 1.29$.

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Polyatomic Gases

For polyatomic molecules (like $CO_2$, $NH_3$, $CH_4$), the number of degrees of freedom depends on the molecular structure. A non-linear polyatomic molecule has 3 translational and 3 rotational degrees of freedom. If vibrational modes are also active, the total can be larger.

For a general polyatomic gas with $f$ active degrees of freedom:

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Comparing Predicted and Measured Specific Heats

The values derived above from kinetic theory and the equipartition of energy are predictions — they ignore vibrational modes, which are mostly frozen out at ordinary temperatures. It is worth comparing these predictions directly against what is actually measured in the laboratory.

Table 12.1 — Predicted values of specific heat capacities of gases (ignoring vibrational modes)