# How do you calculate the internal energy of a mixture of real gases?

Is there an equation to calculate the internal energy of a gas mixture, or is it too complicated to be computed so simply? I know that there's $$E_{internal}=N(degrees of freedom)\frac12kT$$ and $$(P+a\frac{n^2}{V^2})(V-nb)=nRT$$, but are there any formulas for mixtures of real gases. I'm aware of the Wan der Waals equation, $$a=\sum_{i=1}^{i=n}\sum_{j=1}^{j=n}(x_ix_j\sqrt{a_ia_i}), b=\sum_{i=1}^{i=n}\sum_{j=1}^{j=n}(x_ix_j\sqrt{b_ib_i})$$, though I have no idea what it means or how to use it, so don't know whether or not it's what I'm looking for.

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• You need to know their fugacity. Then multiply the molar ratio (or value depending on the units you need and information you have) of each real gas by its fugacity and internal energy and add those internal energies up to get the internal energy of the system. Jun 24 at 2:30
• Is the composition changing, or is it constant between the initial and final states? Jun 24 at 11:18
• @PulkitSharma How do you calculate the fugacity? The internet seems to suggest that you would need to know the fugacity coefficient (φ), but it also seems that the value of φ is dependent upon both the temperature and pressure. I feel like I'm way out of my depth with this.
– Qaos
Jun 24 at 16:14
• eng.uc.edu/~beaucag/Classes/ChEThermoBeaucage/… Jun 24 at 19:28
• Read page 373 onwards. Fugacity and other reduced properties in a mixture is a real headache in itself I just don't want to get in again. I had to design a gas system once. Never again. I deal in liquids now. Even then electrolytes fuck me daily. I hate Dimethyl Acetamide. Jun 24 at 19:30

For an gas mixture of constant composition, you have $$dU=C_vdT-\left(P-T\left(\frac{\partial P}{\partial T}\right)_V\right)dV$$where U is the molar internal energy and V is the molar volume. For a van Der Waals gas, you can show that $$C_v$$ is independent of volume, and is thus equal to the ideal gas heat capacity.