van der Waals gas has interactions between its molecules, so naturally its internal energy is not...

 

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van der Waals gas has interactions between its molecules, so naturally its internal energy is not purely a function of its temperature, N U(TV) = NCVT-av where Cy is the heat capacity at constant volume of an ideal gas. You can derive this yourself by using some MATH203 and something called Maxwell relations, but just take it granted for this problem. (a) Find the analog of the adiabatic formula pV = const of ideal gases for the van der Waals gas i.e. f(p.V) = const for some function f of the pressure and volume. (b) Find Cp of the van der Waals gas. It is not a constant unlike the ideal gas. (c) We mentioned in class that the efficiency of the Carnot engine is uni- versal, any system that has a cycle of two adiabatic and two isothermal processes gives exactly the same result. Explicitly show this for the van der Waals gas running between two temperatures Te and TH above the critical temperature. van der Waals gas has interactions between its molecules, so naturally its internal energy is not purely a function of its temperature, N U(TV) = NCVT-av where Cy is the heat capacity at constant volume of an ideal gas. You can derive this yourself by using some MATH203 and something called Maxwell relations, but just take it granted for this problem. (a) Find the analog of the adiabatic formula pV = const of ideal gases for the van der Waals gas i.e. f(p.V) = const for some function f of the pressure and volume. (b) Find Cp of the van der Waals gas. It is not a constant unlike the ideal gas. (c) We mentioned in class that the efficiency of the Carnot engine is uni- versal, any system that has a cycle of two adiabatic and two isothermal processes gives exactly the same result. Explicitly show this for the van der Waals gas running between two temperatures Te and TH above the critical temperature.