Question

Prove Mayer's relation: \( C_p - C_v = \frac{R}{J} \)

Hint:

Mayer's relation connects the specific heat capacities at constant pressure and volume for an ideal gas. It is derived from the first law of thermodynamics and the ideal gas law.

Answer 1

Mayer's relation is a thermodynamic identity that links the specific heat capacities at constant pressure \( C_p \) and constant volume \( C_v \) for an ideal gas. We will derive this relation using the first law of thermodynamics. 
Step 1: First law of thermodynamics
The first law of thermodynamics states: \[ dQ = dU + pdV \] where: - \( dQ \) is the heat added to the system,
- \( dU \) is the change in internal energy,
- \( p \) is the pressure,
- \( dV \) is the change in volume. 
Step 2: Define heat capacities
Heat capacities are defined as the amount of heat added per unit change in temperature: - \( C_p = \frac{dQ}{dT} \) at constant pressure,
- \( C_v = \frac{dQ}{dT} \) at constant volume. Thus, for an ideal gas, at constant pressure and constant volume, we can express \( dQ \) in terms of \( C_p \) and \( C_v \). 
Step 3: Relationship between heat capacities
For an ideal gas, the first law can be rewritten in terms of the heat capacities: \[ dQ = dU + pdV \] Using the thermodynamic identity and the ideal gas law \( pV = nRT \), we get: \[ dQ = C_v dT + nR dT \] This gives the relationship between the heat capacities at constant pressure and volume: \[ C_p - C_v = \frac{R}{J} \] where \( R \) is the universal gas constant, and \( J \) is the number of moles. Thus, we have derived Mayer's relation.