PHY 3513 Fall 1999 – Final Exam Information
1. The final exam will be held in NPB 1011 from 8:00 pm to 10:00 pm on Monday,
December 13.
2. The scope of the exam will be all material covered in the course relating to Callen
Chapters 1–7. The following sections of Callen were completely skipped in the course
and can be ignored for purposes of the exam: 2.9, 3.8, 4.1, 4.3, 4.8, 4.9, 5.4, 6.4–7,
7.2–5.
The exam will consist of two long problems worth 15 points each, and a number of
smaller problems (or independent parts of problems) worth approximately 5 points
each. It is likely that one of the long problems will address the material covered
recently in Chapters 5 and 6 of Callen.
3. You will be permitted to use the course text (Callen) and your lecture notes in the exam.
However, but you must not consult any other written materials, such as homework
solutions (either your own or others’).
4. The sample exam below is designed to give an idea of the level of the questions on
the Final. (The questions on this sample are over-weighted towards the material covered most recently in order to provide you with extra practice in these areas.) The
sample exam will not be graded. Solutions will be distributed in class on Wednesday,
December 8.
PHY 3513 Fall 1999 – Sample Final Exam
This exam lasts 2 hours. Answer all questions. The maximum points for each question or
part question are shown in brackets. To obtain full credit, please explain your reasoning
and show all working. Please write neatly and include your name on the front page of your
answers.
You are permitted to use the course text (Callen) and your lecture notes, but you may not
consult any other written materials (e.g., homework solutions).
You must not seek or obtain help on this exam from anyone other than the
proctor, nor must you assist anyone else.
1. Each part of this problem is completely independent of the other parts.
(a) A simple, one-component system in contact with a suitable reservoir is held at
constant temperature, volume, and chemical potential. What thermodynamic
potential A describing just the system (not the reservoir) will be minimized at
thermodynamic equilibrium? Specify A in terms of U , S, T , V , P , N , and µ. Use
the Euler equation to simplify your answer. [5 points]
(b) Convert (∂S/∂P )T,N to an expression involving one or more of cP , α, and κT
(plus, possibly, other standard thermodynamic variables such as P , V , and T ).
[5 points]
(c) Two subsystems, having heat capacities C1 = γ1 T and C2 = γ2 T , start at temperatures T1i and T2i , respectively. Find the final common temperature Tf when the
subsystems are placed in contact to form a closed composite system. [5 points]
(d) Calculate the overall entropy change for the process described in (c), assuming
that it is carried out quasistatically. (The variable Tf may appear in your answer.)
[5 points]
(e) It is found that a particular system obeys the equations
U = P V,
P = AT 2 ,
where A is a constant. Find the fundamental equation S = S(U, V ) for this
system. [5 points]
(f) A system is claimed to satisfy the fundamental equation
2
u = Aeb(v−v0 ) s4/3 es/3R ,
where A, B, and v0 are positive constants. Check whether this equation satisfies
each of Callen’s entropy postulates. [5 points]
2. A soap film (the gray area in the figure at
right) is supported by a wire frame. Due
to its surface tension, γ = γ0 − αT (where
γ0 and α are positive constants), the film
exerts a force F = 2γL on the cross-wire,
directed so as to reduce the area of the film.
The entropy of the film depends on its internal energy and total area, but the shape
and total volume have negligible effect, i.e.,
S = S(U, A, N ).
F
L
cross−wire
(a) Write down the quasistatic work d¯W that must be performed on the film to
increase its area by an infinitesimal amount dA. (Consider the work needed to
move the cross-wire a distance dx to the right. Then express dx in terms of dA.)
[3 points]
(b) Calculate the total work done on the film in increasing its area from A1 to A2 at
a constant temperature T . [3 points]
(c) Calculate the change in Helmholtz free energy for the process described in (b).
Hint: Start by writing down the total differential dF for this system. Then
integrate dF to get ∆F . [3 points]
(d) Calculate the change in entropy for the process described in (b). Hint: How is
the entropy related to the Helmholtz free energy? [3 points]
(e) Calculate the change in internal energy for the process described in (b). [3 points]
3. A system is described by the molar Helmholtz free energy
f (T, v) = RT ln 1 − e−A/T ,
A = θ0 − θ1 v/v0 .
(a) Calculate the molar entropy s(T, v) and the pressure P (T, v). [6 points]
(b) Calculate the molar energy u(T, v). [4 points]
(d) Calculate the fundamental relation s = s(u, v). [5 points]
(1)