Problem 1

plumleyj 11 Jan 2007 17:08

Two forms of the virial expansion are as follows;

(1)\begin{align} \frac {pV} {RT} = 1 + \frac {B(T)} {V} + \frac {C(T)}{V^2} + ... \end{align}

(2)
\begin{align} \frac {pV} {RT} = 1 + B'(T)p + C'(T)p^2 + ... \end{align}

Relations between the two sets of virial coefficients can be determined as follows;

First solve equation 1 for p and substitute into equation 2 as follows;

(3)\begin{align} \begin {flalign*} \frac {pV} {RT} = & 1 + \frac {B'(T)RT} {V} \left(1 + \frac {B(T)} {V} + \frac {C(T)} {V^2} + ...\right) + \frac {C'(T)RT} {V} \left(1 + \frac {B(T)} {V} + \frac {C(T)} {V^2} + ...\right)^2 + ... \\ = & 1 + \frac {B'(T)RT} {V} + \frac {B(T)B'(T)RT} {V^2} + \frac {C(T)B'(T)RT} {V^3}+... +\frac {C'(T)RT} {V} + \frac {2B(T)C'(T)RT} {V^2} + \frac {2C(T)C'(T)RT} {V^3} + \\ & \frac {B(T)^2C'(T)RT} {V^3} + \frac {2B(T)C(T)C'(T)RT} {V^3} + ... \end {flalign*} \end{align}

Collecting terms within equation 3 with common powers of V,

(4)\begin{align} \frac {pV} {RT} = 1 + \frac {RT} {V} \left[ B'(T) + C'(T)\right] + \frac {B(T)RT} {V^2} \left[B'(T) + 2C'(T)\right] + ... \end{align}

In order to write equation 4 as equation 1, the following must be true,

(5)\begin{align} \begin {flalign*} B(T) & = RT[B'(T) + C'(T)] \\ C(T) & = B(T)RT[(B'(T) + 2C'(T)] \\ & = R^2T^2[B'(T)^2 +3B'(T)C'(T)+2C'(T)^2] \end {flalign*} \end{align}