Grand Potential - Solid State Physics - Solved Paper, Exams of Solid State Physics

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Solution NAME: 1 March 2005 Physics 112 Spring 2005 Midterm 1 (80 minutes =80 points) Time yourself and move to the next problem after a number of minutes equal to the number of points. 1. Thermodynamic identity (20 points) We consider a system with only one species of particles and define with our usual notation the “Grand Potential” Q=F-pN =U-to-uN a) (5 points) Write down the fundamental (differential) thermodynamic identity for @ Aol: 4U- dite) ~ddan} AU - tae edt ~ pd ~ MAyA QW (che ~pAVepAN) -tde ~ode AN SN hi zl-pdV ~edt ~ Nadu ( What independent variables are the most natural To use Tor s27-———> Vi, pd b) (3 points) Show that when expressed as a function of temperature, volume and chemical potential, the pressure is independent of the volume (Vt) = a(t) (Hint: divide the system into two unequal volumes in equilibrium) (rotpprrey bee) pisPa, Gate Mem, =? ’ ae Gry . | Ag, | PCY, T A}= Pi > Pe= ep (v., ty) So f es cadepederr sf vOllime wher ¢) (5 points) Deduce from b) and the fact that Q is zero at zero volume that Re P regs od ay - : Q=—PV : . aw acto, of (Since we have three variables, be careful in defining the path of integration) A, Te, SUV, p, T= SLOY, we)-SL OO, we) Vv © (AWA ala she path “fzerrimery Qo Trea rah, (pcm) AVY tye fon On Q raximan value, a u iM, vy > (CpaV-ode-Mads 9 = ~ pV as desired —