Chem 534: Problem Set #2 Due in class: Monday, Feb. 3rd (1) Starting from TdS = ∑ E j dPj as discussed in class, show that S = −k ∑ Pj ln Pj , where j Pj = that e −E j /kT Q j and Q = ∑ e −E j /kT . Note that at some point you will have to explicitly show j ⎛ ⎞ ln P dP = d P ln P ⎜ ∑ j j ∑ j j⎟ . ⎝ j ⎠ j (2) Starting from A = −kT lnQ and the Gibbs Fundamental equation for dA, derive the expressions ⎛ ∂ lnQ ⎞ S = kT ⎜ + k lnQ ⎝ ∂T ⎟⎠ V,N ⎛ ∂ lnQ ⎞ p = kT ⎜ ⎝ ∂V ⎟⎠ T,N ⎛ ∂ lnQ ⎞ U = kT 2 ⎜ ⎝ ∂T ⎟⎠ V,N (3) Show that for a particle in a cubical box with infinitely thick walls with sides of length L, the 2 Ej pressure in quantum state j, pj, is . 3V (4) For a monatomic ideal gas Q = 1 ⎛ 2π mkT ⎞ ⎜ ⎟ N! ⎝ h 2 ⎠ 3N /2 V N . Derive expressions for the pressure and energy from this partition function. Also show that in general if Q is of the form f (T )V N where f(T) is any function of temperature alone, the ideal gas equation of state is recovered. ⎛ e −hν /2kT ⎞ (5) Given the partition function of a crystal, Q = ⎜ −hν /kT ⎟ ⎝1− e ⎠ 3N eU0 /kT where hν = Θ E is a k constant and U0 is the sublimation energy, calculate the heat capacity Cv and show that at high temperatures Cv → 3Nk as T → ∞ (Dulong & Petit law). (6) Derive an expression for the fluctuation in the pressure in a canonical ensemble.
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