Van der Waals Equation and Critical Points
(a) In p V- diagrams the slope ∂p/∂V along an isotherm is never positive. Explain why.
(b) Regions where ∂p/∂V = 0 represent equilibrium between two phases; volume can change with no change in pressure, as when water boils at atmospheric pressure.
We can use this to determine the temperature, pressure, and volume per mole at the critical point using the equation of state P = P (v, T, n). If T > T., then p (V) has no maximum along an isotherm, but if T < Tc, then p (V) has a maximum. Explain bow this leads to the following condition for determining the critical point:
∂p/∂V =0 and ∂2P/∂V2=0 at the critical point
(c) Solve the van der Waals equation (Eq. 18.7) for p; that is, find p (v, T, n). Find ∂p/∂V and ∂2p/∂V2. Set these equal to zero to obtain two equations for V, T, and n.
(d) Simultaneous solution of the two equations obtained in part (c) gives the temperature and volume per mole at the critical point, To and (V/n)c. Find these constants in terms of a and b. (Hint: Divide one equation by the other to eliminate T.)
(e) Substitute these values into the equation of state to find Pc, the pressure at the critical point.
(f) Use the results from parts (d) and (e) to find the ratio RTc/Pc (V/n) c. This should not contain either a or b and so should have the same value for all gases.
(g) Compute the ratio RTc/Pc (V/n) c for the gases H2, N2, and H2O using the critical point data given in the table.
(h) Discuss how well the results of part (g) compare to the prediction of part (I) based on the van der Waals equation. What do you conclude about the accuracy of the van der Waals equation as a description of the behavior of gases near the critical point?