assignment Homework

Quantum concentration
bose-einstein gas statistical mechanics Thermal and Statistical Physics 2020 Consider one particle confined to a cube of side \(L\); the concentration in effect is \(n=L^{-3}\). Find the kinetic energy of the particle when in the ground state. There will be a value of the concentration for which this zero-point quantum kinetic energy is equal to the temperature \(kT\). (At this concentration the occupancy of the lowest orbital is of the order of unity; the lowest orbital always has a higher occupancy than any other orbital.) Show that the concentration \(n_0\) thus defined is equal to the quantum concentration \(n_Q\) defined by (63): \begin{equation} n_Q \equiv \left(\frac{MkT}{2\pi\hbar^2}\right)^{\frac32} \end{equation} within a factor of the order of unity.

assignment Homework

Light bulb in a refrigerator
Carnot refridgerator Work Entropy Thermal and Statistical Physics 2020 A 100W light bulb is left burning inside a Carnot refridgerator that draws 100W. Can the refridgerator cool below room temperature?

face Lecture

30 min.

Introducing entropy
Contemporary Challenges 2022 (4 years)

entropy multiplicity heat thermodynamics

This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between changes in entropy and heat.

assignment Homework

Boltzmann probabilities
Energy Entropy Boltzmann probabilities Thermal and Statistical Physics 2020 (3 years) Consider a three-state system with energies \((-\epsilon,0,\epsilon)\).
  1. At infinite temperature, what are the probabilities of the three states being occupied? What is the internal energy \(U\)? What is the entropy \(S\)?
  2. At very low temperature, what are the three probabilities?
  3. What are the three probabilities at zero temperature? What is the internal energy \(U\)? What is the entropy \(S\)?
  4. What happens to the probabilities if you allow the temperature to be negative?

assignment Homework

Free energy of a two state system
Helmholtz free energy entropy statistical mechanics Thermal and Statistical Physics 2020
  1. Find an expression for the free energy as a function of \(T\) of a system with two states, one at energy 0 and one at energy \(\varepsilon\).

  2. From the free energy, find expressions for the internal energy \(U\) and entropy \(S\) of the system.

  3. Plot the entropy versus \(T\). Explain its asymptotic behavior as the temperature becomes high.

  4. Plot the \(S(T)\) versus \(U(T)\). Explain the maximum value of the energy \(U\).

group Small Group Activity

60 min.

Ice Calorimetry Lab

heat entropy water ice

The students will set up a Styrofoam cup with heating element and a thermometer in it. They will measure the temperature as a function of time, and thus the energy transferred from the power supply, from which they compute changes in entropy.

assignment Homework

Heat of vaporization of ice
Vaporization Heat Thermal and Statistical Physics 2020 The pressure of water vapor over ice is 518 Pa at \(-2^\circ\text{C}\). The vapor pressure of water at its triple point is 611 Pa, at 0.01\(^\circ\text{C}\) (see Estimate in \(\text{J mol}^{-1}\) the heat of vaporization of ice just under freezing. How does this compare with the heat of vaporization of water?

assignment Homework

Bottle in a Bottle 2
heat entropy ideal gas Energy and Entropy 2021 (2 years)

Consider the bottle in a bottle problem in a previous problem set, summarized here.

A small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle, which is insulated.

The volume of the small bottle is 0.001 m23 and the volume of the big bottle is 0.01 m3. The initial state of the gas in the small bottle was \(p=106\) Pa and its temperature \(T=300\) K. Approximate the helium gas as an ideal gas of equations of state \(pV=Nk_BT\) and \(U=\frac32 Nk_BT\).

  1. How many molecules of gas does the large bottle contain? What is the final temperature of the gas?

  2. Compute the integral \(\int \frac{{\mathit{\unicode{273}}} Q}{T}\) and the change of entropy \(\Delta S\) between the initial state (gas in the small bottle) and the final state (gas leaked in the big bottle).

  3. Discuss your results.

group Small Group Activity

30 min.

Heat and Temperature of Water Vapor

Thermo Heat Capacity Partial Derivatives

In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.

face Lecture

5 min.

Wavelength of peak intensity
Contemporary Challenges 2022 (3 years)

Wein's displacement law blackbody radiation

This very short lecture introduces Wein's displacement law.

face Lecture

30 min.

Energy and heat and entropy
Energy and Entropy 2021 (2 years)

latent heat heat capacity internal energy entropy

This short lecture introduces the ideas required for Ice Calorimetry Lab or Microwave oven Ice Calorimetry Lab.

assignment Homework

Adiabatic Compression
ideal gas internal energy engine Energy and Entropy 2020

A diesel engine requires no spark plug. Rather, the air in the cylinder is compressed so highly that the fuel ignites spontaneously when sprayed into the cylinder.

In this problem, you may treat air as an ideal gas, which satisfies the equation \(pV = Nk_BT\). You may also use the property of an ideal gas that the internal energy depends only on the temperature \(T\), i.e. the internal energy does not change for an isothermal process. For air at the relevant range of temperatures the heat capacity at fixed volume is given by \(C_V=\frac52Nk_B\), which means the internal energy is given by \(U=\frac52Nk_BT\).

Note: in this problem you are expected to use only the equations given and fundamental physics laws. Looking up the formula in a textbook is not considered a solution at this level.

  1. If the air is initially at room temperature (taken as \(20^{o}C\)) and is then compressed adiabatically to \(\frac1{15}\) of the original volume, what final temperature is attained (before fuel injection)?

  2. By what factor does the pressure increase?

assignment Homework

Active transport
Active transport Concentration Chemical potential Thermal and Statistical Physics 2020

The concentration of potassium \(\text{K}^+\) ions in the internal sap of a plant cell (for example, a fresh water alga) may exceed by a factor of \(10^4\) the concentration of \(\text{K}^+\) ions in the pond water in which the cell is growing. The chemical potential of the \(\text{K}^+\) ions is higher in the sap because their concentration \(n\) is higher there. Estimate the difference in chemical potential at \(300\text{K}\) and show that it is equivalent to a voltage of \(0.24\text{V}\) across the cell wall. Take \(\mu\) as for an ideal gas. Because the values of the chemical potential are different, the ions in the cell and in the pond are not in diffusive equilibrium. The plant cell membrane is highly impermeable to the passive leakage of ions through it. Important questions in cell physics include these: How is the high concentration of ions built up within the cell? How is metabolic energy applied to energize the active ion transport?

David adds
You might wonder why it is even remotely plausible to consider the ions in solution as an ideal gas. The key idea here is that the ideal gas entropy incorporates the entropy due to position dependence, and thus due to concentration. Since concentration is what differs between the cell and the pond, the ideal gas entropy describes this pretty effectively. In contrast to the concentration dependence, the temperature-dependence of the ideal gas chemical potential will not be so great.

face Lecture

10 min.

Basics of heat engines
Contemporary Challenges 2022 (4 years) This brief lecture covers the basics of heat engines.

assignment Homework

Ideal gas calculations
Ideal gas Entropy Sackur-Tetrode Thermal and Statistical Physics 2020

Consider one mole of an ideal monatomic gas at 300K and 1 atm. First, let the gas expand isothermally and reversibly to twice the initial volume; second, let this be followed by an isentropic expansion from twice to four times the original volume.

  1. How much heat (in joules) is added to the gas in each of these two processes?

  2. What is the temperature at the end of the second process?

  3. Suppose the first process is replaced by an irreversible expansion into a vacuum, to a total volume twice the initial volume. What is the increase of entropy in the irreversible expansion, in J/K?

assignment Homework

Distribution function for double occupancy statistics
Orbitals Distribution function Thermal and Statistical Physics 2020

Let us imagine a new mechanics in which the allowed occupancies of an orbital are 0, 1, and 2. The values of the energy associated with these occupancies are assumed to be \(0\), \(\varepsilon\), and \(2\varepsilon\), respectively.

  1. Derive an expression for the ensemble average occupancy \(\langle N\rangle\), when the system composed of this orbital is in thermal and diffusive contact with a resevoir at temperature \(T\) and chemical potential \(\mu\).

  2. Return now to the usual quantum mechanics, and derive an expression for the ensemble average occupancy of an energy level which is doubly degenerate; that is, two orbitals have the identical energy \(\varepsilon\). If both orbitals are occupied the toal energy is \(2\varepsilon\). How does this differ from part (a)?

assignment Homework

Free energy of a harmonic oscillator
Helmholtz free energy harmonic oscillator Thermal and Statistical Physics 2020

A one-dimensional harmonic oscillator has an infinite series of equally spaced energy states, with \(\varepsilon_n = n\hbar\omega\), where \(n\) is an integer \(\ge 0\), and \(\omega\) is the classical frequency of the oscillator. We have chosen the zero of energy at the state \(n=0\) which we can get away with here, but is not actually the zero of energy! To find the true energy we would have to add a \(\frac12\hbar\omega\) for each oscillator.

  1. Show that for a harmonic oscillator the free energy is \begin{equation} F = k_BT\log\left(1 - e^{-\frac{\hbar\omega}{k_BT}}\right) \end{equation} Note that at high temperatures such that \(k_BT\gg\hbar\omega\) we may expand the argument of the logarithm to obtain \(F\approx k_BT\log\left(\frac{\hbar\omega}{kT}\right)\).

  2. From the free energy above, show that the entropy is \begin{equation} \frac{S}{k_B} = \frac{\frac{\hbar\omega}{kT}}{e^{\frac{\hbar\omega}{kT}}-1} - \log\left(1-e^{-\frac{\hbar\omega}{kT}}\right) \end{equation}

    Entropy of a simple harmonic oscillator
    Heat capacity of a simple harmonic oscillator
    This entropy is shown in the nearby figure, as well as the heat capacity.

assignment Homework

Heat pump
Carnot efficiency Work Entropy Heat pump Thermal and Statistical Physics 2020
  1. Show that for a reversible heat pump the energy required per unit of heat delivered inside the building is given by the Carnot efficiency: \begin{align} \frac{W}{Q_H} &= \eta_C = \frac{T_H-T_C}{T_H} \end{align} What happens if the heat pump is not reversible?

  2. Assume that the electricity consumed by a reversible heat pump must itself be generated by a Carnot engine operating between the even hotter temperature \(T_{HH}\) and the cold (outdoors) temperature \(T_C\). What is the ratio \(\frac{Q_{HH}}{Q_H}\) of the heat consumed at \(T_{HH}\) (i.e. fuel burned) to the heat delivered at \(T_H\) (in the house we want to heat)? Give numerical values for \(T_{HH}=600\text{K}\); \(T_{H}=300\text{K}\); \(T_{C}=270\text{K}\).

  3. Draw an energy-entropy flow diagram for the combination heat engine-heat pump, similar to Figures 8.1, 8.2 and 8.4 in the text (or the equivalent but sloppier) figures in the course notes. However, in this case we will involve no external work at all, only energy and entropy flows at three temperatures, since the work done is all generated from heat.

face Lecture

120 min.

Work, Heat, and cycles
Thermal and Statistical Physics 2020

work heat engines Carnot thermodynamics entropy

These lecture notes covering week 8 of Thermal and Statistical Physics include a small group activity in which students derive the Carnot efficiency.

group Small Group Activity

30 min.

“Squishability” of Water Vapor (Contour Map)

Thermo Partial Derivatives

Students determine the “squishibility” (an extensive compressibility) by taking \(-\partial V/\partial P\) holding either temperature or entropy fixed.