PHY 3513 Fall 1999 – Mid-Term Exam 2 Information
1. The exam will be held in NPB 1011 from 8:00 to 10:00 pm on Thursday, November 18.
2. The scope of the exam will be all material covered in the course relating to Callen
Chapter 3 (excluding Section 3.8) and Sections 4-7 and 10 of Chapter 4 (plus the
discussion of quasistatic and reversible processes given in lectures). Only reversible
heat engines, refrigerators, and heat pumps will be covered (thus the exclusion of
Callen Section 4.9).
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
Mid-Term. The sample exam will not be graded. Solutions will be distributed in class
on Monday, November 15.
PHY 3513 Fall 1999 – Sample Mid-Term Exam 2
This exam lasts 2 hours. Answer all four questions, which carry equal weight. 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. A system obeys the equations of state
u = av 1/3 T,
3P v = (s + bv)T.
(a) Find the fundamental relation s = s(u, v). (Don’t worry if your answer violates
the Nernst postulate.)
(b) Calculate the molar chemical potential µ(u, v).
2. Consider an ideal van der Waals fluid, described by the fundamental equation
s = s0 + R ln [(v − b)(u + a/v)c ] ,
where a and b are positive constants.
Calculate κs (T, v) = −v −1 (∂v/∂P )s for this fluid.
3. A room (at temperature Th ) is warmed by a heat pump which draws heat from the
outside (at temperature Tc ). The heat pump consumes electrical power P and operates
at a fraction f of the ideal heat-pump efficiency.
(a) Calculate the rate at which heat is delivered to the room. (Your answer should
have dimensions of power.)
(b) Suppose that heat escapes from the room back to the outside at a rate A(Th −Tc ),
where A is a positive constant. What is the steady-state temperature of the room
with the heat-pump operating continuously? Your answer should be a root of a
quadratic equation. Specify which root is the physical one.
(c) Consider the limit f P ATc . By what factor does Th − Tc increase when P is
doubled at fixed f , A and Tc ?
4. The Joule cycle consists of four steps, alternately isentropic (∆S = 0) and isobaric
(∆P = 0), as represented by the path
ABCDA on the S-P diagram at right.
S
D
C
A
B
Consider an implementation of this cycle,
using for the auxiliary system a simple
ideal gas which satisfies
s = s0 + R ln
u
u0
c
v
.
v0
Assume that each step in the cycle is carried out quasistatically.
P
(a) Sketch the Joule cycle on the P -V and T -S planes (with V and S forming the
horizontal axes). Label on both diagrams the vertices corresponding to A through
D on the S-P diagram. Draw each line so that the sign of both the slope and the
curvature is clear (e.g., distinguish straight lines from those curving up or down).
(b) For each of the four steps in the cycle, indicate whether each of the following
quantities is positive, negative, or zero: (i) Q, the heat entering the auxiliary
system; (ii) W , the work done on the auxiliary system; (iii) ∆U , the change in
the internal energy of the auxiliary system.
(c) Explain carefully why the area enclosed by the cycle on the P -V diagram equals
the area enclosed on the T -S diagram.