Calculate the entropy change per mole of gas.

### 1 Answer

If we define the entropy as a function of T and V, then the total differential of the entropy is given by:

dS = (∂S/∂V)_T dV + (∂S/∂T)_V dT

For an isothermal process, dT = 0, so:

dS = (∂S/∂V)_T dV

Using a Maxwell’s relation (see source) we can write (∂S/∂V)_T = (∂p/∂T)_V, so:

dS = (∂p/∂T)_V dV

For an ideal gas, p = n*R*T/V, so (∂p/∂T)_V = n*R/V, and:

dS = (n*R) * dV/V

Integrating this, we get:

ΔS = n*R*ln(V_final/V_initial)

This is the equation that defines the entropy change for an isothermal expansion of an ideal gas.

In this case, we have that n = 1 mol, and V_final/V_initial = 1/6, so:

ΔS = (1mol)*(8.314 J/(mol*K))*ln(1/6)

ΔS/mol = -14.9 J/(mol*K)