Specific Heats of Gases

Learn about Cv and Cp and how specific heats differ for monatomic, diatomic and polyatomic gases.

1. Meaning of Specific Heat for a Gas

Specific heat of a gas tells how much heat is needed to raise the temperature of the gas by 1°C (or 1 K). For gases, this depends on whether volume or pressure is kept constant during heating.

2. Specific Heat at Constant Volume (Cv)

Cv is the amount of heat required to raise the temperature of 1 mole of a gas by 1 K while keeping the volume fixed.

At constant volume, the gas cannot expand, so all heat supplied increases only the internal energy.

2.1. Formula

C_V = \left( \dfrac{\partial Q}{\partial T} \right)_V

3. Specific Heat at Constant Pressure (Cp)

Cp is the amount of heat needed to raise the temperature of 1 mole of a gas by 1 K when the pressure is kept constant.

At constant pressure, the gas expands while being heated, so some heat goes into doing work and some into raising internal energy.

3.1. Formula

C_P = \left( \dfrac{\partial Q}{\partial T} \right)_P

3.2. Important Relation

Since Cp includes extra heat to do expansion work:

C_P > C_V

4. Molar Heat Capacities and Degrees of Freedom

Using the law of equipartition of energy, the internal energy of a gas molecule is:

E = \dfrac{f}{2} k_B T

For one mole:

U = \dfrac{f}{2} R T

So the molar specific heat at constant volume becomes:

C_V = \dfrac{f}{2} R

4.1. Then Cp

Using the relation:

C_P = C_V + R

5. Specific Heats for Different Types of Gases

The degrees of freedom determine the specific heats.

5.1. Monatomic Gases (f = 3)

  • Examples: He, Ne, Ar

  • C_V = \dfrac{3}{2} R

  • C_P = \dfrac{5}{2} R

5.2. Diatomic Gases (f = 5 at room temperature)

  • Examples: O2, N2, H2

  • C_V = \dfrac{5}{2} R

  • C_P = \dfrac{7}{2} R

5.3. Polyatomic Gases (f = 6)

  • Examples: CO2, H2O, SO2

  • C_V = 3R

  • C_P = 4R

6. Why Different Gases Have Different Specific Heats

Gases with more degrees of freedom have more ways to store energy. So more heat is required to raise their temperature by the same amount.

6.1. Simple Understanding

More degrees of freedom → more energy storage → higher specific heat.

7. Everyday Examples

  • Helium heats up quickly because it has low specific heat.
  • Steam (H2O gas) does not heat up as quickly because it has more degrees of freedom and a higher specific heat.
  • Car engines heat nitrogen and oxygen, both diatomic gases with moderate specific heats.