Question

I need help with all parts of question one so any help on any question is very much appreciated please help!!!!

In p - V phase diagrams, the slope. (partialdifferential p/partialdifferential V)_T, n, along an isotherm is never positive. Why? Regions where (partialdifferential p/partialdifferential V)_T, n = 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_e, then p (V) has no extrema along an isotherm, but if T = T_e, then p (V) has an extrema. Explain how this leads to the following condition for determining the critical point: (partialdifferential p/partialdifferential V)_T, n = 0 and (partialdifferential^2 p/partialdifferential V^2)_T, n = 0 Solve the van der Waals equation for p. That is, find p (V, T, n). Find (partialdifferential p/partialdifferential V)T, n and (partialdifferential^2 p/partialdifferential V^2)_T, n. Set these equal to zero to obtain two equations for V, T, and n. [2an^2 (V - nb)^2 = nRTV^3; 3an^2 (V - nb)^3 = nRTV^4] Simultaneous solution of the two equations obtained in part (c) gives the volume per mole and temperature at the critical point: (V/n)_c, and T_c. Find these constants in terms of a and b. Substitute these values into the equation of state to find p_c, the pressure at the critical point. [P_c = a/27b^2] Use the results from parts (d) and (e) to find the ratio RT_e/p_c (V/n)_c. This should not contain either a or b and so should have the same value for all gases. [2.7] Compute the ratio RT_c/p_c (V/n)_c for the gases Ne, H_2, N_2 and H_2 0 using the following critical-point data: (3.2, 3.3, 3.4, 4.4] Discuss how well the results of part (g) compare to the prediction of part (f) 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? Is it a better description for gases near the critical point than the IG EOS?

Answer 1

answer 1A

slope of isothermal and adiabotic curves:-

for isothermal process we know PV=constant(at constant tempeture)

PdV+VdP=0

so from this expression we can find the value of

dP/dV=-P/V

dP/dV called the slope of isothermal indicator diagram.which is always negetive as shown by the expression given above.