課程名稱︰熱物理
課程性質︰物理系大三必修
課程教師︰郭光宇
開課學院：理學院
開課系所︰物理系
考試日期（年月日）︰105/01/12
考試時限（分鐘）：120分鐘
試題 :
(Please note : 9 problems on 2 pages; Answers in both Chinese and English are
OK.)
1.(12%) An ideal monatomic gas undergoes a reversible expansion from specific
volume v1 to specific volume v2.
(a)(4%) Calculate the change in specific entropy Δs if the expansion is
isobaric.
(b)(4%) Calculate Δs if the process is isothermal.
(c)(4%) Which is larger? By how much?
2.(10%) Consider a van der Waals gas.
(a)(4%) Show that c_v is a function of T only.
(b)(3%) Show that the specific internal energy is u = ∫c_vdT-a/v+u_0.
(c)(3%) Show that the specific entropy is s=∫(c_v/T)dT+Rln(v-b)+s_0.
3.(a)(4%) Show that the Joule coefficient may be written η≡(∂T/∂v)_u
= (1/c_v)(P-Tβ/κ).
(b)(4%) Show that Joule-Thomson coefficient may be written μ≡(∂T/∂P)_h
= (v/c_p)(Tβ-1).
(c)(4%) Using these results, find η and μ for a van der Waals gas and
show that both are zero for an ideal gas.
4.(10%) The Gibbs function of a certain gas is G = nRTlnP + A + BP + CP^2/2
+ DP^3/3, where A, B, C, and D are constants. Find the equation of
state of the gas.
5.(12%) The equations of the sublimation and the vaporization curves of a
particular material are given by ln P = 0.04 - 6/T (sublimation) and
ln P = 0.03 - 4/T (vaporization), where P is in atmospheres,
respectively.
(a)(4%) Find the temperature of the triple point.
(b)(4%) Show that the specific latent heats of vaporization and sublimation
are 4R and 6R, respectively. (You may assume that the specific
volume in the vapor phase is much larger than the specfic volume in
the liquid and solid phases.)
(c)(4%) Find the latent heat of fusion.
6.(a)(6%) Express the chemical potential of an ideal gas in terms of the
temperature T and the volume V : μ = c_p T -c_v TlnT - RTlnV - s_0
+ constant.
(b)(6%) Similarly, find μ in terms of T and P. Show that the chemical
potential at the fixed temperature T varies with pressure as μ=μ_0
+ RTln(P/P_0), where μ_0 = is the value of μ at the reference
point (P_0, T). This expression is of great use in chemistry.
7.(a)(5%) A mixture of gold and thallium can exist in equilibrium with four
phases present : solution, vapor, solid gold, and solid thallium.
What is the variance?
(b)(5%) The Gibbs phase rule can be generalized to systems in which there
occur r chemical reactions : f = k - π- r + 2. Determine the number
of degrees of freedom at equilibrium of a chemically reactive system
containing solid sulfur S and the three gases O2, SO2 and SO3. The
elements S and O2 appear in the reactions S + O2 → SO2 and
S + (3/2)O2 → SO3.
8.Show that, during a first-order phase transition :
(a)(5%) The change of entropy of the system undergoing the transition is a
linear function of the volume change.
(b)(5%) The change of internal energy is given by ΔU = L(1-(dlnT/dlnP)),
where L is the latent heat of transformation.
9.(a)(4%) Show that the volume expansion coefficient β≡(1/V)(∂V/∂T)_P
tends to zero as the temperature tends to absolute zero.
(b)(4%) Use this result to show that van der Waals' equation cannot be valid
at low temperatures. What does the third law say about the existence
of the ideal gas?
(c)(4%) Consider a solid whose equation of state is PV + f(V) = AU, where
f(V) is a function of the volume only and A is a constant. Show that
C_v → 0 as T → 0.