PHYSICS 241E - Test No. 1
September 30, 1996
1) On the planet of Tatouine the distribution of molecular speeds in a gas is NOT the Maxwell-Boltzmann distribution
but rather is
as shown in the figure (not included here).
a) (7 points) What is C in terms of 
if f(v) is normalized to 1 as is
the Maxwell-Boltzmann distribution?
b) (7 points) What is the average or mean speed of the molecules?
c) (7 points) What is the root mean square speed of the molecules?
d) (4 points) If the gas is helium, what is the average kinetic energy of the molecules? (Let the molecular mass be m)
2) One mole of ideal gas at T = 300 K expands reversibly and isothermally
between 
= 30 L and 
= 60 L. (R = 8.314 J/mol 
K)
a) (10 points) Find the entropy change in the gas.
b) (5 points) What is the total (gas+surroundings) entropy change?
c) (10 points) If the gas is then compressed at the same temperature until it
reaches 
= 50 L, what has been its change in entropy from point 1 to
point 3?
Entropy is a state variable so changes depend only upon the final and initial state of the system. Both steps occur at same temperature so
3) An engine uses a monoatomic gas to perform a three step cycle:
From 1 
2 it performs an adiabatic expansion (P 
= 4 atm,
V = 10 L; P 
= 1.75 atm, V = 15 L.
From 2 3 it undergoes an isobaric compression and
from 3 1 it undergoes an isochoric transformation.
Indicate the direction of heat flow in the diagram.
1 2: Q = 0 (adiabatic); W outwards
(work done by gas). 3: W inwards (work done on gas); work adds energy to gas
but internal energy ( 
T) decreases Q outwards
( 
).
3 1: W = 0 ( 
); U ( T) increases
Q inwards ( 
).a) (15 points) Calculate the heat flow between points 1 and 2, Q 
,
between 2 and 3, Q 
, and
between 3 and 1, Q 
.
b) (5 points) Calculate the net work performed by the engine.
By First Law of Thermodynamics applied to a cyclic process
c) (5 points) Calculate the efficiency of the engine.
4) A 3 
C charge is uniformly distributred on a ring of radius 8 cm with
its center at z = 0 and central axis along the z axis.
A point charge of -2 C is placed at z = -4 cm, as shown in the figure
(not included).
a) (10 points) What is the force acting on a 1 C point charge at the
origin by the -2 C point charge?
What is the force acting on the 1 C charge by the ring?
Force must be given in vec- tor form.
b) (13 points) The 1 C charge is now moved to z = 4 cm.
What is the force on it due to the ring alone?
What is the total force acting on it due to the ring and the -2 C charge?
where d = distance from point on ring to the 1 C charge and 
is
the angle between a line from a point on the ring to the 1 C charge and
the z axis.
c) (2 points) Is there a point on the z axis at (i) z 
0, (ii) z 
0
where the 1 C charge feels no force due to the ring and -2 C charge?
Give yes or no answer to both (i) and (ii).
Let the position of a (positive) test charge be z.
As z 
the test charge feels a force in the +z
direction due to the net charge of (3-2) C = 1 C near the origin.
For z = 0 the test charge feels a force in the -z direction since at
that point the ring exerts no net force and the -2 C charge exerts a
force in the -z direction.
Hence, somewhere between the force must be zero:
(i) YES
C charge the test
charge feels a force in the +z direction.
As z 
the test charge feels a force in the -z
direction due to the net charge of (3-2) C = 1 C near the origin.
Hence, somewhere between the force must be zero:
(i) YES