Question

Using the first law of thermodynamics, show that for an ideal gas, the difference between the molar specific heat capacities at constant pressure and at constant volume is equal to the molar gas constant R.

Answer 1

Imagine a cylinder with a volume of V that holds n moles of an ideal gas at a pressure of P and is equipped with an area A piston. Assume that the gas is heated to a temperature increase of dT at constant pressure. As seen in the image, the piston moves outward a little distance dx under the overall force F = PA exerted by the gas.

Expansion of a gas at constant pressure

The work done by the force in moving the piston is

dW = Fdx = PAdx = PdV  ...(1)

Where the gas volume increases during expansion is represented by Adx = dV. The work that the expanding gas does on its surroundings is measured in dW. According to the first rule of thermodynamics, if the heat given to the gas is dQ_P and the increase in its internal energy is dE, then

dQ_P = dE + dW = dE +pdv  

If C_P is the molar-specific heat capacity of the gas at constant pressure, dQ_P = nC_P dT.

∴ nC_P dT = dE + PdV  ...(2)

In contrast, dW = 0 occurs if the gas is heated from its original state to a temperature that increases by the same amount of dT at constant volume (as opposed to constant pressure). Since a perfect gas's internal energy is only dependent on its temperature, the internal energy would rise by dE once more. Based on the concept of molar-specific heat capacity at constant volume (C_V) and the first rule of thermodynamics, if dQ_V represents the heat given to the gas in this scenario,

dQ_V = dE = nC_V dT  ...(3)

From Eqs. (2) and (3),

nC_P dT = nC_V dT + PdV

∴ C_P − C_V = `P/n (dV)/(dT)`  ...(4)

The equation of state of an ideal gas is PV = nRT, where R is the molar gas constant. Therefore, at a constant pressure,

PdV = nRdT

∴ `(dV)/(dT) = (nR)/P`   ...(5)

From Eqs. (4) and (5),
C_P − C_V = `P/n. (nR)/P = R`  ...(6)

This is Mayer’s relation between C_P and C_V.