# Molar Specific Heats and Degrees of Freedom Revision Notes

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13.8Molar Specific Heats and Degrees of Freedom

In these revision notes for Molar Specific Heats and Degrees of Freedom, we cover the following key points:

• What is/are the factor(s) affecting the internal energy of an ideal gas?
• What is molar specific heat at constant volume?
• How to calculate the molar specific heat at constant volume for an ideal monoatomic gas?
• What about the molar specific heat at constant volume if the gas is diatomic or polyatomic?
• The same for the molar specific heat at constant pressure.
• How do these two molar specific heats relate to each other?
• What are degrees of freedom and how they are related to the energy absorbed by ideal gas?
• How to find the number of degrees of freedom in various ideal gases?

## Molar Specific Heats and Degrees of Freedom Revision Notes

The internal energy of an ideal gas is a function of temperature only. In other words, it depends only on the temperature of gas.

Mathematically, we have

U = 3/2 × n × R × T

By definition, Molar specific heat of an ideal gas is the heat we provide to the gas to raise the temperature of one mole through 1K or 1°C. It differs from the specific heat capacity c discussed earlier, as the molar specific heat is calculated for 1 mole instead of 1 kg of material. We represent it as C and its unit is J mol/K.

Molar specific heat of a monoatomic ideal gas is

Cv = 3/2 R

Since R = 8.31 J/K, we obtain for the value of molar specific heat of a monoatomic ideal gas:

Cv = 3/2 × 8.31 J/mol × K
≈12.5 J/mol × K

When dealing with two or more atomic gases, we observe that the motion of molecules is not that simple, as atoms in monoatomic gases, because molecules can also spin or vibrate, not only move in a translational way. Therefore, the value of CV will be greater than that obtained above for monoatomic gases. Thus,

The actual internal energy of an ideal gas at constant volume is

U = n × Cv × T

and the change in the internal energy of an ideal gas at constant volume is

∆U = n × Cv × ∆T

The above formula implies that:

Any change in the internal energy of an ideal gas enclosed within a container depends only on the change in temperature; it is independent from the type of process that has brought that change.

Similarly as in the process with constant volume, we have for the heat absorbed by the ideal gas at constant pressure

Q = n × Cp × ∆T

where Cp is known as the molar specific heat at constant pressure.

The value of Cp is numerically greater than the corresponding value of CV for the same change in temperature as in the process with constant volume, a part of heat energy supplied goes for doing work for lifting the piston.

From the First Law of Thermodynamics, we know that

∆U = Q - W

Substituting, we obtain

n × Cv × ∆T = n × Cp × ∆T-n × R × ∆T

After simplifying and rearranging, we obtain for the molar specific heat at constant pressure.

Cp = Cv+R

For a monoatomic gas, we have

Cp = 3/2 R + R
= 5/2 R
≈ 20.8 J/mol × K

Therefore, we can write for the heat absorbed by a monoatomic gas at constant pressure to increase its temperature by ΔT:

∆Q = 5/2 n × R × ∆T

With degrees of freedom, we understand independent ways of a particle's motion.

All atoms in monoatomic gases are assumed to move only in a translational way as even if they spin around themselves, this is not observable and relevant.

The internal energy of monoatomic ideal gases is equal to

U = 3/2 × n × R × T

The number 3 is because there are three degrees of freedom in translational motion. The division by 2 is because every individual direction includes two sub-directions in itself. (left-right, back-forth, up-down).

In diatomic (two-atomic) molecules, there is a spinning effect produced which causes a double direction of rotation.

Therefore, in total we have 5 degrees of freedom for the internal energy, i.e.

U = 5/2 × n × R × T

If molecules are three or more atomic (polyatomic), there is another spinning direction added because two points form a line (one direction) while three point form a plane (two directions). Therefore, the total internal energy becomes

U = 5 + 1/2 × n × R × T
= 6/2 × n × R × T
= 3 × n × R × T

Maxwell formulated the theorem of the equipartition of energy, which says:

Every molecule has a certain number of degrees of freedom, which represent independent ways of storing energy. A portion of energy associated with a single degree of freedom is given by the expression

U = 1/2 kT (for a single molecule) or U = 1/2 RT (for a mole)

Thus, we can obtain a general form of the relationship between the number of degrees of freedom f and the molar specific heat at constant volume CV for an ideal gas in terms of the ideal gas constant R:

Cv = f/2 × R = f × R/2
= f × 8.31/2 J/mol × K
= 4.16 × f J/mol × K

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