PS1

Physics 230, Fall 2014
Problem Set 1: ideal gases, kinetic theory, Maxwell-Boltzmann distribution
Reading: lecture notes; chapter 17 (sections 4 through 10); chapter 18 (sections 1, 2, 6)
Questions: 2, 4, 5, 13, and 21 from chapter 18.
Problems:
1. The relationship  =  o (1+ α ΔT) is an approximation that works when the
average coefficient of expansion is small. If α is large, one must integrate
the relationship dL/dT = αL to determine the final length. (a) Assuming that
the coefficient of linear expansion is constant as L varies, determine a
general expression for the length,  . (b) Given a rod of length 1.00 m and a
temperature change of 100.0 °C, determine the error caused by the
approximation when α = 2.00 × 10–5 (°C)–1 (a typical value for a metal).
2. The definition of the coefficient of volume expansion is given by β =
(1/V) dV/dT. Use this to determine the volume expansion coefficient for an
ideal gas at constant pressure. (b) What value does this expression predict
for β at 0°C? Compare this result with the experimental values for air in
Table 17-1.
3. A cylinder is closed by a piston connected to a spring of constant
2.00 × 103 N/m. With the spring relaxed, the cylinder is filled with 5.00 L of
gas at a pressure of 1.00 atm and a temperature of 20.0°C. (a) If the piston
has a cross-sectional area of 0.010 0 m2 and negligible mass, how high will it
rise when the temperature is raised to 250°C? (b) What is the pressure of the
gas at 250°C?
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4. Two vessels of volumes 1.22 L and 3.18 L contain krypton gas and are
connected by a thin tube. Initially the vessels are at the same temperature,
16.0 °C, and the same pressure, 1.44 atm. The larger vessel is then heated to
108 °C while the smaller one remains at 16.0 °C. Calculate the final
pressure.
5. Container A contains an ideal gas at a pressure of 5.0 x 105 Pa and
temperature of 300 K. It is connected by a thin tube to container B that has
four times the volume of container A. Container B holds the same ideal gas,
but is at a pressure of 1.0 x 105 Pa and a temperature of 400 K. The valve in
the thin tube is opened and the gas from each container mixes with the
temperature remaining constant. Calculate the final pressure.
6. Consider a vertical cylinder of cross sectional area "A" that is divided into
two sections with equal volumes by a moveable piston of mass mp. The top
section is filled with gas A at pressure PA and temperature TA. The bottom
section is filled with gas B at temperature 2TA and pressure PB. The cylinder
is now turned upside down. To keep the piston in the middle, gas B must be
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cooled to a temperature equal to TA while gas A remains at temperature TA.
3
Find PB, the initial pressure of gas B, in terms of mp, and the cross sectional
area "A". Note: both gases are ideal and there is no friction between the
piston and the cylinder wall.
7. In a period of 1.00 s, 5.00 × 1023 nitrogen molecules strike a wall with an
area of 8.00 cm2. If the molecules move with a speed of 300 m/s and strike
the wall head-on in elastic collisions, what is the pressure exerted on the
wall? (The mass of one N2 molecule is 4.68 × 10–26 kg.)
8. Work problem 21 from chapter 18.
9. Twenty particles, each of mass m and confined to a volume V, have
various speeds: two have speed v; three have speed 2v; five have speed
3v; four have speed 4v; three have speed 5v; two have speed 6v; one has
speed 7v. Find (a) the average speed, (b) the rms speed, (c) the most
probable speed, (d) the pressure the particles exert on the walls of the
vessel, and (e) the average kinetic energy per particle.
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10. Assume that the Earth’s atmosphere has a uniform temperature of 20°C
and uniform composition, with an effective molar mass of 28.9 g/mol. (a)
Show that the number density of molecules depends on height according to
n V (y) = n o e−mgy kT
where n0 is the number density at sea level, where y = 0. This result is
called the law of atmospheres. (b) Commercial jetliners typically cruise at
an altitude of 11.0 km. Find the ratio of the atmospheric density there to
the density at sea level.
11. Using the law of atmospheres from problem 10, derive an expression for
the average height of a molecule in Earth’s atmosphere using the following:
∞
y =
∫ yn
V
(y)dy
0
∞
∫n
V
(y)dy
0
12. Using the Maxwell-Boltzmann distribution function of molecular speeds,
derive (by direct integration or differentiation) expressions for v mp (most
probable speed), v rms (root mean square speed), and v (average speed).
Order the three from slowest to fastest.
13. How much more likely will a molecule of oxygen gas (O2) at 25 °C and
1atm have a speed of 500 m/s than a speed of 1500 m/s.
14. Consider the hypothetical molecular speed distribution given below.
N v = Av2 (100 − 4v 2 ) for 0 ≤ v ≤ 5
N v = 0 for v > 5
(a) Determine N, the total number of molecules. Your answer should be in
terms of the constant A.
(b) Determine the average kinetic energy per molecule. Your answer should
be in terms of m, the mass of a single molecule.

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