课程名称︰统计物理（一）
课程性质︰物理系、物理所必修 / 应物所、天文物理所选修
课程教师︰蔡政达
开课学院：理学院
开课系所︰物理学系、物理学研究所、应用物理所、天文物理研究所
考试日期（年月日）︰2017/4/10
考试时限（分钟）：120 min
试题 :
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]