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Thermodynamics and Statistics

Dulong Petit Law

For an atom in a crystal, 3 translational degress of freedom and additionally 3 degrees of freedom for atomic vibrations give the total energy per atom of 3kT. Thus,

3kTNA is the energy per mole where

The specific heat at constant volume

CV is the derivative with respect to T of 3kTNA. When T = 1 Kelvin, 3kTNA = 3kNA or 3R

CV = 3R is the Dulong-Petit Law, where CV is measured in J/kg/K

Problem 1

Calculate the specific heat of a copper coin using the law of Dulong and Petit, which states that CV = 3R

Solution 1


Problem 2

A common laboratory experiment involves the themionic emission of electrons from metal surfaces. Use the Richarson-Dushman law,

J = A_0 T^2 e^{-\phi/kT}

to estimate the thermionic emission current density for a tungsten filament at 2000 K. Take φ = 4.55 eV as the work function and A0 = 120 A/cm2K2 as the Richardson constant.

Solution 2

\phi = 4.55 eV
kT = 1.381 \times 10^{-16} (2000) \frac{1eV}{1.602 \times 10^{-12}\mathrm{erg}} = 0.1724 \mathrm{eV}

The Richardson-Dushman Law states that

J = A_0 T^2 e^{-\phi/kT}
J = (120)(2000)^2 e^{-4.55/0.172k}
Q = (E-)0.00166 \mathrm{A/cm}^2

Problem 3

What is the root-mean-square speed of molecules of mass m in an ideal gas at temperature T?

Solution 3

Root mean square speed is the measure of speed of particles in a gas that is most convenient for solving problems in kinetic theory of gases

v_{rms} = \sqrt{ \frac{3RT}{M_m} } = \sqrt{ \frac{3kT}{m} }

Problem 4

For an adiabatic process involving in ideal gas, having volume V and temperature T, which of the following is constant? γ = CP/CV

  1. TγV-1
  2. TγV
  3. TVγ
  4. TVγ - 1
  5. TV

Solution 4


Problem 5

A thermodynaic system, initially at absolute temperature T1, contains a mass m of water with specific heat capacity c. Heat is added until temperature rises to T2. What is the change in entropy of the water?

  1. mcT2
  2. mc ln(T2 / T1)
  3. 0
  4. T2 - T1
  5. mc(T2 - T1)

Solution 5

Problem 6

Heat Q is added to a monatomic ideal gass under condition of constant volume, resulting in a temperature change Δ T. How much heat, in relation to Q, will be required to produce the same temperature change if it is added under conditions of constant pressure?

Solution 6

Problem 7

A heat pump is to extract heat from an outdoor environment at 7°C and heat the environment indoors to 27°C. For each 15000 J of heat delivered indooors, the smallest amount of work that must be supplied to the heat pump is approximately:

  1. 500J
  2. 1000J
  3. 1100J
  4. 2000J
  5. 2200J

Solution 7

Problem 8

A particle can occupy two possible states with energies E1 and E2 , where E2 > E1. At temperature T, the probability of finding the particle in state 2 is given by which of the following?

Solution 8

Problem 9

Consider 1 mole of a real gas that obeys the van der Waals equation of state shown above. If the gas undergoes an isothermal expansion at temperature T0 from volume V1 to volume V2 , which of the following gives the work done by the gas?

Solution 9

Problem 10

A gas at temperature T is composed of molecules of mass m . Which of the following describes how the average time between intermolecular collisions varies with m ?

  1. It is proportional to 1/m
  2. It is proportional to 4√m
  3. It is proportional to √m
  4. It is proportional to m
  5. It is proportional to m2

Solution 10

Problem 11

There are 2 identical 1.0-kg blocks of copper metal enclosed in a perfectly insultating container. One is one initially at a temperature T1 = 0°C and the other is initially at a temperature T2 = 100°C. The 2 blocks are initially separated. When the blocks are placed in contact, they come to equilibrium at a final temperature Tf. What is the amount of heat exchanged between the 2 blocks in this process? Specific heat of copper metal = 0.1 kilocalorie/kilogram °K

Solution 11

The final temperature is 50°C. The heat exchanged from the hot block to the cool block is

Q = m c \Delta T

Both bodies are involved in the heat transfer, but the mass is only 1kg not 2kg because you're only looking at the energy needed to heat the 1 block (the one at a lower temperature) since heat "flows" from high temp to low temp, so you calculate the energy using only the colder block and it's specific heat and the change in it's temperature

Q = (1) (0.1) (50)
Q = 5 \mathrm{kcal}

Problem 12

In an ideal monoatomic adiabatic expansion, if the volume of the gas doubles, from V0 to 2V0 then what happens to the temperature?

Solution 12

The adiabatic gas law is

pV^{\gamma} = p_0V_0^{\gamma}
pV = nRT

The ideal gas equation of state for the final situation

p_0 V_0 = nRT_0
\frac{nRTV^{\gamma}}{V} = \frac{nRT_0V_0^{\gamma}}{V_0}
\frac{T}{T_0} = (\frac{V_0}{V})^{\gamma - 1}
\frac{T}{T_0} = (\frac{V_0}{V})^{2/3}
Take γ = 5/3 for a monatomic ideal gas with 3 degrees of freedom. Thus,
\frac{T}{T_0} = (\frac{1}{2})^{2/3} = 0.63
T = 0.63 T_0
The temperature falls to 0.63 T0

Problem 13

The rms-speed and the internal energy of an ideal gas are Vrms and U, respectively. If the absolute temperature of the gas were decreased to 1/4 the original value, what would be the new values for rms-speed and internal energy?

Solution 13


Problem 14


The classical model of a diatomic molecule is a springy dumbbell, as shown above, where the dumbbell is free to rotate about axes perpendicular to the spring. In the limit of high temperature, what is the specific heat per mole at constant volume?

Solution 14

Note that this problem wants the regime of high temperatures, so the answer is not
from classical thermodynamics, but rather
\frac{7}{2} R
The problem suggests that a quantized linear oscillator is used. From the energy relation
\epsilon = \left(j+\frac{1}{2}\right)\hbar \nu
you can write a partition function and do the usual Stat Mech jig. Since you're probably too lazy to calculate entropy, you can find the specific heat (at constant volume) from
c_v=\left.\frac{\partial U}{\partial T}\right|_v
U=NkT^2\left(\frac{\partial Z}{\partial T}\right)_V
where N is the number of particles, k is the Boltzmann constant. There are actually three contributions to the specific heat at constant volume.
Chunk out the math and take the limit of high temperature to find that

Problem 15


A constant amount of an ideal gas undergoes the cyclic process ABCA in the PV diagram shown above. The path BC is isothermal. The work done by the gas during one complete cycle beginning and ending at A, is most nearly

  1. 600 kJ
  2. 300 kJ
  3. 0
  4. -300 kJ
  5. -600 kJ

Solution 15


Problem 16


The distribution of relative intensity I(λ) of blackbody radiation from a solid object versus the wavelength λ is shown in the figure above. If the Wien displacement law constant is 2.9E-3 mK, what is the approximate temperature of the object?

  1. 10K
  2. 50K
  3. 250K
  4. 1500K
  5. 6250K

Solution 16


Problem 17

In a gas of N diatomic molecules, two possible models for a classical description of a diatomic molecule are:


Which of the following statements about this gas is true?

  1. Model II is always correct
  2. The choice between Models I and II depends on the temperature
  3. Molecule I has a specific heat cv = 3/2 N k
  4. Model I is always correct
  5. Model II has a smaller specific heat than Model I

Solution 17


Problem 18


In the cycle above, KL and NM represent isotherms, while KN and LM represent reversible adiabats. A system is carried through the Carnot cycle KLMN, taking in heat Q2 from the hot reservoir T2 and releasing heat Q1 to the cold reservoir T1. All of the following statements are true EXCEPT:

  1. Q1/T1 = Q2/T2
  2. The work W done is equal to the net heat absorbed, Q2 - Q1
  3. The efficiency of the cycle is independent of the working substance
  4. The entropy of the system increases
  5. The entropy of the hot reservoir decreases

Solution 18


Problem 19


Isotherms and coexistence curves are shown in the pV diagram above for a liquid-gas system. The dashed lines are the boundaries of the labeled regions.

Which numbered curve is the critical isotherm?

Solution 19


Problem 20


Isotherms and coexistence curves are shown in the pV diagram above for a liquid-gas system. The dashed lines are the boundaries of the labeled regions.

In which region are the liquid and the vapor in equilibrium with each other?

Solution 20


Problem 21


Suppose one mole of an ideal gas undergoes the reversible cycle ABCA shown in the P-V diagram above, where AB is an isotherm. The molar heat capacities are Cp at constant pressure and Cv at constant volume. The net heat added to the gas during the cycle is equal to

  1. RThlnV2/V1 - Cp(Th - Tc)
  2. RThlnV2/V1 - R(Th - Tc)
  3. RThV2/V1
  4. Cv(Th - Tc)
  5. -Cp(Th - Tc)

Solution 21


Problem 22


The wave function for a particle constrained to move in one dimension is shown in the graph above (Ψ = 0 for x ≤ 0 and x ≥ 5). What is the probability that the particle would be found between x = 2 and x = 4?

  1. 5/8
  2. 25/64
  3. 13/16
  4. 17/64
  5. √(5/8)

Solution 22


Problem 23


Window A is a pane of glass 4 millimeters thick, as shown above. Window B is a sandwich consisting of two extremely thin layers of glass separated by an air gap 2 millimeters thick, as shown above. If the thermal conductivities of glass and air are 0.8 watt/meter degree Celcius and 0.025 watt/meter degree Celcius, respectively, then what is the ratio of the heat flow through window A to the heat flow through window B?

Solution 23


Problem 24


An experimenter needs to heat a small sample to 900K but the only available oven has a maximum temperature of 600K. Could the experimenter heat the sample to 900K by using a large lens to concentrate the radiation from the oven onto the sample, as shown above?

  1. Yes, if the area of the front of the oven is at least 3/2 the area of the front of the sample
  2. Yes, if the volume of the oven is at least 3/2 the volume of the sample.
  3. No, because it would violate conservation of energy
  4. Yes, if the sample is placed at the focal point of the lens
  5. No, because it would violate the second law of thermodynamics

Solution 24


Problem 25


A particle of mass m moves in the potential shown above. The period of the motion when the particle has energy E is

  1. 2π√(m/k) + 4√(2E/mg2)
  2. π√(m/k) + 2√(2E/mg2)
  3. √(k/m)
  4. 2√(2E/mg2)
  5. 2π√(m/k)

Solution 25


Problem 26

Which of the following curves is characteristic of the specific heat Cv of a metal such as lead, tin, or aluminum in the temperature region where it becomes superconducting?

  1. 995
  2. 995
  3. 995
  4. 995
  5. 995

Solution 26