課程名稱︰統計物理（一） 課程性質︰物理系、物理所必修 / 應物所、天文物理所選修 課程教師︰蔡政達 開課學院：理學院 開課系所︰物理學系、物理學研究所、應用物理所、天文物理研究所 考試日期（年月日）︰2017/4/10 考試時限（分鐘）：120 min 試題 : Statistical Physics Midterm Examination (I) Date: April 10, 2017 1. Assuming that the entropy S and the statistical number Ω of a physical system are related through an arbitrary functional form S = f(Ω), show that the additive character of S and the multiplicative character of Ω necessarily require that the function f(Ω) be of the form S = klnΩ. [20 points] 2. Four moles of nitrogen and one mole of oxygen at P = 1atm and T = 300K are mixed together to form air at the same pressure and temperature. Calculate the entropy of mixing per mole of the air formed. [20 points] 3. The generalized coordinates of a simple pendulum are the angular displacement θ and the angular momentum m(l^2)(dθ/dt). Study, both mathematically and graphically, the nature of the corresponding trajectories in the phase space of the system, and show that the area A enclosed by a trajectory is equal to the product of the total energy E and the period τ of the pendulum. [20 points] 4. Making use of the fact that the Helmhiltz free energy A(N,V,T) of a thermodynamic system is an extensive property of the system, show that dA dA N(──)_V,T + V(──)_N,T = A. dN dV [Note that this result implies the well-known relationship:Nμ = A + PV = G] [20 points] 5. Show that, for a classical ideal gas, S Q_1 dlnQ_1 ── = ln(──) + T(───)_P. Nk N dT [20 points] --

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