Mayer's Relation: Cp - Cv = R

Mayer's Relation for an Ideal Gas

In thermodynamics, the relationship between the specific heat capacity of an ideal gas at constant pressure (Cp) and at constant volume (Cv) is given by Mayer's Relation:

C_p - C_v = R

Where:

• C_p = Molar heat capacity at constant pressure
• C_v = Molar heat capacity at constant volume
• R = Universal gas constant

Brief Derivation

According to the First Law of Thermodynamics for 1 mole of an ideal gas:

dQ = dU + dW

Since dW = P·dV, we can write:

dQ = dU + P·dV

1. At Constant Volume (dV = 0):

No work is done, so the heat added changes only the internal energy:

dQv = dU = Cv·dT
Therefore, dU = Cv·dT (Since internal energy of an ideal gas depends only on temperature).

2. At Constant Pressure:

The heat added changes both internal energy and performs work:

dQp = Cp·dT

Substituting dQp and dU back into the first law equation:

Cp·dT = Cv·dT + P·dV

3. Using the Ideal Gas Equation:

For 1 mole of an ideal gas: PV = RT

Differentiating both sides at constant pressure gives:

P·dV = R·dT

4. Final Substitution:

Substitute P·dV back into the thermodynamic equation:

Cp·dT = Cv·dT + R·dT

Dividing the entire equation by dT:

Cp = Cv + R
or
C_p - C_v = R