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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?

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    $\begingroup$ There's something awfully fishy about that equation. Regarding $\Delta W$ it depends on what convention (definition) you use. $\endgroup$ – Buck Thorn Mar 2 at 11:28
  • $\begingroup$ That equation was added on 19:33, 14 April 2017. It looks like it may be vandalism. $\endgroup$ – Buck Thorn Mar 2 at 12:36
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    $\begingroup$ That statement in Wikipedia is incorrect. That expression is the residual gibbs free energy at constant temperature for a real gas. $\endgroup$ – Chet Miller Mar 2 at 12:39
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    $\begingroup$ 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. $\endgroup$ – MaxW Mar 2 at 16:03
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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.

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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.

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