Question

Prove the Mayer's relation `C_p - C _v = R/J`

Answer 1

Consider one mole of an ideal gas that is enclosed in a cylinder by a light, frictionless, airtight piston.

Let P, V, and T be the pressure volume and temperature of the gas, respectively. If the gas is heated so that its temperature rises by dT, but volume remains constant.

Let dQ₁ → be the amount of heat supplied to the gas,

dE → be the increase in internal energy. Since, volume of gas is constant, no work is done in moving the piston.

dQ₁ = dE + dW

∴ dQ₁ = dE = C_VdT    ...(i)

(As dW = 0)

If the gas is heated to the same temperature at constant pressure.

Let dV → be the increase in volume.

Therefore work is done and it is given by

dW = PdV

Let dQ₂ → be the heat supplied to the gas

∴ dQ₂ = dE + dW

∴ C_pdT = dE + dW    ...(ii)

Putting dE = C_vdT and dW = PdV

∴ Equation (ii) becomes

C_PdT = C_vdT + PdV

(C_p − C_v)dT = PdV    ...(iii)

For one mole of gas

PV = RT

∴ pdV = RdT

∴ (C_p − C_v)dT = RdT

C_p − C_v = R    ...(iv)

This is known as Mayer's relation.

Generally, heat supplied is measured in calories, and work done is measured in joules. Therefore, equation (iv) is modified as follows:.

`C_p - C _v = R/J`