Problem Set 9 Solutions for Thermal Physics | PHYS 224, Assignments of Thermal Physics

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224 problem set 9 solutions

224 problem set 9 solutions Problem 5.30. The slope of a graph of G vs. T is —S, so the slope of each graph should be negative, and become more negative with increasing temperature. Furthermore, at any given temperature, the stable phase should have the lowest. value of G. For H,O at atmospheric pressure, the stable phase is ice below (°C, water between or and 100°C, and steam above 100°C. The plot at left below shows these features qualitatively. Gc Water 0.001 bar | Steam 4+——_+ ® —- T oc 100°C oc 100°C As the pressure is reduced, the relation (0G/8P). = V tells us that the Gibbs free energy of each phase will decrease. However, that of the gas phase will decrease the fastest by far, since its volume is so much greater. At pressures below the triple point (0,006 bar), there is no temperature at which liquid water is stable. So at 0.001 bar, the graphs of G vs. T would look something like the plot above right. Problem 6,16, Starting from the definition of Z, zs = 4 Sheree = s were — SH Bla, Now just multiply by —1/2, cancel the minus signs, and move the Z inside the sum: eSBs) az . “Fag 2 LECH e)yerre = de) z= Y E(s)P(s) = E. Problem 6.31. If state q has energy c{q|, then the Boltzmann factor for this state is e~* ca. co (Since the integral is symmetric under g + —g, I've written it as twice the integral over only positive g values. This trick gets rid of the absolute value bars.) The average energy is therefore Laz 8 1 BE =-5— = -=(-cg-) = = =kT. Za ol ca) 3 T.