# Derivation of work done by real gas

According to the Wikipedia on real gases, it is stated that the expansion work done by a real gas differs from the ideal gas by $$\int\left(V-\frac{RT}p\right)\,\mathrm dp$$.

Firstly, does this mean $$W_\text{real} = W_\text{ideal} + \Delta W$$ or $$W_\text{real} = W_\text{ideal} - \Delta W$$?

Secondly, how is this formula derived?

• There's something awfully fishy about that equation. Regarding $\Delta W$ it depends on what convention (definition) you use. Mar 2 '19 at 11:28
• That equation was added on 19:33, 14 April 2017. It looks like it may be vandalism. Mar 2 '19 at 12:36
• That statement in Wikipedia is incorrect. That expression is the residual gibbs free energy at constant temperature for a real gas. Mar 2 '19 at 12:39
• Wikipedia's "Real gas" article gives various equations used to model gases besides the ideal gas law. For example the Van der Waals model etc.
– MaxW
Mar 2 '19 at 16:03

For a reversible process in a closed system (no mass entering or leaving), the general equation for the expansion work done by a real gas on the surroundings is the same for an ideal gas, namely $$W = \int p\,\mathrm{d}V$$ However, for a real gas, we use the equation of state for that gas $$p=p(n,V,T)$$ rather than $$p=\frac{nRT}{V}$$, the equation of state for the ideal gas.

That expression is incorrect.

For one mole of an ideal gas,

$$V_\mathrm{m} = \frac{RT}{p}$$

where $$p$$ is the pressure of the gas.

$$pV$$ work is defined as

$$W_{pV} = -\int_{V_i}^{V_f} p_\mathrm{ext}\mathrm{d}V_\mathrm{m}$$

where $$p_\mathrm{ext}$$ is the applied pressure against which work must be done. By this sign convention work done by the system is negative.

Therefore $$W_{pV,\mathrm{real}} - W_{pV,\mathrm{ideal}} = -\int_{V_i}^{V_f} (p_\mathrm{ext,real} - p_\mathrm{ext,ideal})\mathrm{d}V_\mathrm{m}$$

Assuming mechanical equilibrium between the applied pressure and the gas,

$$W_{pV,\mathrm{real}} - W_{pV,\mathrm{ideal}} = \int_{V_i}^{V_f} (\frac{RT}{V_\mathrm{m}}-p_\mathrm{real})\mathrm{d}V_\mathrm{m}$$

where $$p_\mathrm{real}$$ is described by the equation of state for the real gas.