Problem 1: In a monolayer of gas adsorbed onto a solid surface, a ...

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Problem 1: In a monolayer of gas adsorbed onto a solid surface, a two-dimensional assumption is approximately valid. Derive the partition function for a system of N monatomic, non-interacting molecules confined to a two-dimensional region of area A at fixed temperature T. (For non- interacting particles, the interaction potential is everywhere zero.) What is the expression for two-dimensional pressure in this system? How would your expressions for the partition function and the 2D pressure change if the molecules were nonspherical? Problem 2: Assume argon, xenon, water, and n-octane all obey the same 2-parameter corresponding states relation, based on the usual reduced pressure and reduced temperature. Estimate the saturation pressure of xenon at the temperature corresponding to argon at 86.302 K. Repeat for water and n-octane. Comment on the accuracy of each estimate and explain why simple CST succeeds or fails in each case. Reference data are given in Table 1. Table 1. Experimental vapor pressure data for various substances. T (K) Psal (atm) Tc (K) Pc (bar) argon 86.302 0.9 150.8 48.7 xenon 165.9 1.05 289.7 58.4 water 370.7 0.898 647.3 221.2 n-octane 325.82 0.075 568.8 24.9 Problem 3: Consider the following PT projection of a phase diagram typical of H2O/CO2 systems. Plot T-xj- y1 and P-x1-y1 diagrams for the indicated pressures and temperatures. Clearly indicate any three-phase lines, azeotropes, critical end points, and lines for coexisting vapor and liquid phases on your diagrams. P. P3 P, , vP) P. vp2 T, , T3 T4 4 T5 Figure 1. A P-T projection showing class III behavior Problem 4: Starting from the canonical (T, V,NI, N2) ensemble for a binary mixture, for which the thermodynamic potential is A (Helmholtz free energy), use the Legendre transform method to derive expressions in the osmotic ensemble, with independent variables (T, P, N1, 122) for: (a) thermodynamic potential; (b) partition function; (c) probability distribution law.

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