Question

Derive the expression for the work done by one mole of van der Waals gas during the isothermal expansion from volume V1, to V2 at temperature T.

Answer 1

Van der Waals equation of state for one mole of a real gas is

$$\left( P+\frac{a}{V_{m}^{2}} \right)\left( {{V}_{m}}-b \right)=RT$$

where V_m is the molar volume of the gas, R is the universal gas constant, T is temperature, P is pressure, a and b are the Van der Waals' constants

Find $P\left( {{V}_{m}} \right)$

$$P\left( {{V}_{m}} \right)=\frac{RT}{{{V}_{m}}-b}-\frac{a}{V_{m}^{2}}$$

Work done by one mole of gas during the isothermal expansion from volume V₁, to V₂ at temperature T is

$$W=\int\limits_{{{V}_{_{1}}}}^{{{V}_{2}}}{P\left( {{V}_{m}} \right)d{{V}_{m}}}=\int\limits_{{{V}_{_{1}}}}^{{{V}_{2}}}{\left( \frac{RT}{{{V}_{m}}-b}-\frac{a}{V_{m}^{2}} \right)d{{V}_{m}}}$$

$$W=\left. \left( RT\ln \left( {{V}_{m}}-b \right)+\frac{a}{{{V}_{m}}} \right) \right|_{{{V}_{1}}}^{{{V}_{2}}}$$

 

$$W=RT\left( \ln \left( {{V}_{2}}-b \right)-\ln \left( {{V}_{1}}-b \right) \right)+\frac{a}{{{V}_{2}}}-\frac{a}{{{V}_{1}}}$$

$$W=RT\ln \frac{{{V}_{2}}-b}{{{V}_{1}}-b}+a\frac{{{V}_{1}}-{{V}_{2}}}{{{V}_{1}}{{V}_{2}}}$$