The van der Waals equation of state is a modification of the ideal gas law that

Question

The van der Waals equation of state is a modification of the ideal gas law that attempts to account for the non-zero size of molecules and the interactions between them. The associated internal energy of a gas of N molecules is E_int = 3/2 NkT - aN^2 / V, where a is a constant. Initially the gas is at temperature T and occupies a volume V. The gas is allowed to expand adiabatically into a vacuum (a process known as "free expansion") so that it eventually occupies a volume 2V. What is the final temperature of the gas?

Explanation / Answer

From the first law of thermodynamics, Law of conservation of energy states that

dQ = dU +dW;

Change in heat provided to gas will change the internal energy of gas by rising temeperature and changing the work done by gas by expansion or compression.

So dQ = 0 in an adiabatic process.

dW = 0 because, the gas is allowed to expand against vaccum means resisting pressure p = 0

So pdV is the work done by gas which is zero.

So change in the internal Energy will also be zero.

So dE = 0 implies 3/2 Nk dT + (a N2 / V2  ) dV = 0

Constant internal energy will give 3/2 NkT - a N2 / V   =    3/2 Nk T(final) - a N2 / 2V

Since T initial = T and V initial = V and V final = 2V

T final = T - (a N / 3kV)

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