Van Der Waals Equation - Other Thermodynamic Parameters

Other Thermodynamic Parameters

We reiterate that the extensive volume V is related to the volume per particle v=V/N where N = nN_A is the number of particles in the system.

The equation of state does not give us all the thermodynamic parameters of the system. We can take the equation for the Helmholtz energy A

$A = -kT \ln Q.\,$

From the equation derived above for lnQ, we find

$A(T,V,N)=-NkT\left -\frac{a' N^2}{V}.$

Where Φ is an undetermined constant, which may be taken from the Sackur-Tetrode equation for an ideal gas to be:

This equation expresses A in terms of its natural variables V and T, and therefore gives us all thermodynamic information about the system. The mechanical equation of state was already derived above

$p = -\left(\frac{\partial A}{\partial V}\right)_T = \frac{NkT}{V-Nb'}-\frac{a' N^2}{V^2}.$

The entropy equation of state yields the entropy (S )

$S = -\left(\frac{\partial A}{\partial T}\right)_V =Nk\left$

from which we can calculate the internal energy

Similar equations can be written for the other thermodynamic potential and the chemical potential, but expressing any potential as a function of pressure p will require the solution of a third-order polynomial, which yields a complicated expression. Therefore, expressing the enthalpy and the Gibbs energy as functions of their natural variables will be complicated.

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