## Thermodynamics and Statistics

- Dulong Petit Law
- Mach number

## 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,

3*kTN _{A}* is the energy per mole where

*k*is the Boltzmann constant*T*is the temperature in Kelvin*N*is Avogadro's Number_{A}

The specific heat at constant volume

*C _{V}* is the derivative with respect to T of 3

*kTN*. When T = 1 Kelvin, 3

_{A}*kTN*= 3

_{A}*kN*or 3

_{A}*R*

*C _{V}* = 3

*R*is the Dulong-Petit Law, where

*C*is measured in J/kg/K

_{V}*R*is the universal gas constant*M*is the molecular mass in kg/mol

### Problem 1

Calculate the specific heat of a copper coin using the law of Dulong and Petit, which states that *C _{V}* = 3

*R*

### Solution 1

Answer### Problem 2

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

to estimate the thermionic emission current density for a tungsten filament at 2000 K. Take φ = 4.55 eV as the work function and A_{0} = 120 A/cm^{2}K^{2} as the Richardson constant.

### Solution 2

The Richardson-Dushman Law states that

### 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

### Problem 4

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

- TV
- T
^{γ}V^{-1} - TV
^{γ} - T
^{γ}V - TV
^{γ - 1}

### Solution 4

E### Problem 5

A thermodynaic system, initially at absolute temperature T_{1}, contains a mass m of water with specific heat capacity c. Heat is added until temperature rises to T_{2}. What is the change in entropy of the water?

- T
_{2}- T_{1} - mcT
_{2} - mc(T
_{2}- T_{1}) - 0
- mc ln(T
_{2}/ T_{1})

### 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:

- 500J
- 1000J
- 1100J
- 2000J
- 2200J

### Solution 7

### Problem 8

A particle can occupy two possible states with energies E_{1} and E_{2} , where E_{2} > E_{1}. 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 T_{0} from volume V_{1} to volume V_{2} , 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 ?

- It is proportional to 1/m
- It is proportional to
^{4}√m - It is proportional to √m
- It is proportional to m
- It is proportional to m
^{2}

### 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 T_{1} = 0°C and the other is initially at a temperature T_{2} = 100°C. The 2 blocks are initially separated. When the blocks are placed in contact, they come to equilibrium at a final temperature *T _{f}*. 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

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

### Problem 12

In an ideal monoatomic adiabatic expansion, if the volume of the gas doubles, from V_{0} to 2V_{0} then what happens to the temperature?

### Solution 12

The adiabatic gas law is

The ideal gas equation of state for the final situation

*γ*= 5/3 for a monatomic ideal gas with 3 degrees of freedom. Thus,

_{0}

### Problem 13

The rms-speed and the internal energy of an ideal gas are V_{rms} 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

Answer### 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### 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

- 600 kJ
- 300 kJ
- 0
- -300 kJ
- -600 kJ

### Solution 15

Answer### 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?

- 10K
- 50K
- 250K
- 1500K
- 6250K

### Solution 16

Answer### 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?

- Molecule I has a specific heat c
_{v}= 3/2 N k - Model II has a smaller specific heat than Model I
- Model I is always correct
- Model II is always correct
- The choice between Models I and II depends on the temperature

### Solution 17

Answer### 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 Q_{2} from the hot reservoir T_{2} and releasing heat Q_{1} to the cold reservoir T_{1}. All of the following statements are true EXCEPT:

- The work W done is equal to the net heat absorbed, Q
_{2}- Q_{1} - The efficiency of the cycle is independent of the working substance
- Q
_{1}/T_{1}= Q_{2}/T_{2} - The entropy of the hot reservoir decreases
- The entropy of the system increases

### Solution 18

Answer### 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

Answer### 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

Answer### 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 C_{p} at constant pressure and C_{v} at constant volume. The net heat added to the gas during the cycle is equal to

- -C
_{p}(T_{h}- T_{c}) - C
_{v}(T_{h}- T_{c}) - RT
_{h}lnV_{2}/V_{1}- C_{p}(T_{h}- T_{c}) - RT
_{h}lnV_{2}/V_{1}- R(T_{h}- T_{c}) - RT
_{h}V_{2}/V_{1}

### Solution 21

Answer### 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?

- 25/64
- √(5/8)
- 13/16
- 5/8
- 17/64

### Solution 22

Answer### 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

Answer### 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?

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

### Solution 24

Answer### 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

- 2π√(m/k) + 4√(2E/mg
^{2}) - √(k/m)
- 2π√(m/k)
- 2√(2E/mg
^{2}) - π√(m/k) + 2√(2E/mg
^{2})

### Solution 25

Answer### Problem 26

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