?The critical constants of a van der Waals gas can be found by setting the following

Chapter 1, Problem P1C.12

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The critical constants of a van der Waals gas can be found by setting the following derivatives equal to zero at the critical point:

\(\frac{\mathrm{d} p}{\mathrm{d} V_{\mathrm{m}}}=-\frac{R T}{\left(V_{\mathrm{m}}-b\right)^{2}}+\frac{2 a}{V_{\mathrm{m}}^{3}}=0\)

\(\frac{\mathrm{d}^{2} p}{\mathrm{d} V_{\mathrm{m}}^{2}}=\frac{2 R T}{\left(V_{\mathrm{m}}-b\right)^{3}}-\frac{6 a}{V_{\mathrm{m}}^{4}}=0\)

Solve this system of equations and then use eqn 1C.5b to show that \(p_{\mathrm{c}}\), \(V_{\mathrm{c}}\), and \(T_{\mathrm{c}}\) are given by eqn 1C.6.

Text Transcription:

P_c

V_c

T_c

dp/dV_m=-RT/(V_m-b)^3 = 2a/V_m^3 = 0

d^2p/dV_m^2=2RT/(V_m-b)^3 = 6a/V_m^4 = 0