The virial equation of state is [p v=Re Tleft(mathrm{~b}_{1}+frac{mathrm{b}_{2}}{v}+frac{mathrm{b}_{3}}{v^{2}}+ldots . . ight)] Compare this equation with van

The virial equation of state is

\[p v=\Re T\left(\mathrm{~b}_{1}+\frac{\mathrm{b}_{2}}{v}+\frac{\mathrm{b}_{3}}{v^{2}}+\ldots . .\right)\]

Compare this equation with van der Waals equation of state and determine the first two virial coefficients, \(b_{1}\) and \(b_{2}\), as a function of temperature and the van der Waals constants.

Determine the critical temperature and volume \(\left(T_{\mathrm{c}}, v_{\mathrm{c}}\right)\) for the van der Waals gas, and show that

\[\mathrm{b}_{2}=\frac{v_{\mathrm{c}}}{3}\left(1-\frac{27 T_{\mathrm{c}}}{8 v_{\mathrm{c}}}\right)\]

\(\left[\mathrm{b}_{1}=1 ; \mathrm{b}_{2}=(\mathrm{b}-\mathrm{a} / \Re T)\right]\)