assignment Homework

Pressure of thermal radiation
Thermal radiation Pressure Thermal and Statistical Physics 2020

(modified from K&K 4.6) We discussed in class that \begin{align} p &= -\left(\frac{\partial F}{\partial V}\right)_T \end{align} Use this relationship to show that

  1. \begin{align} p &= -\sum_j \langle n_j\rangle\hbar \left(\frac{d\omega_j}{dV}\right), \end{align} where \(\langle n_j\rangle\) is the number of photons in the mode \(j\);

  2. Solve for the relationship between pressure and internal energy.

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

Isothermal/Adiabatic Compressibility
Energy and Entropy 2021 (2 years)

The isothermal compressibility is defined as \begin{equation} K_{T}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T} \end{equation} \(K_T\) is be found by measuring the fractional change in volume when the the pressure is slightly changed with the temperature held constant. In contrast, the adiabatic compressibility is defined as \begin{equation} K_{S}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{S} \end{equation} and is measured by making a slight change in pressure without allowing for any heat transfer. This is the compressibility, for instance, that would directly affect the speed of sound. Show that \begin{equation} \frac{K_{T}}{K_{S}} = \frac{C_{p}}{C_{V}} \end{equation} Where the heat capacities at constant pressure and volume are given by \begin{align} C_{p} &= T \left(\frac{\partial S}{\partial T}\right)_{p} \\ C_{V} &= T \left(\frac{\partial S}{\partial T}\right)_{V} \end{align}

assignment Homework

Vapor pressure equation
phase transformation Clausius-Clapeyron Thermal and Statistical Physics 2020 Consider a phase transformation between either solid or liquid and gas. Assume that the volume of the gas is way bigger than that of the liquid or solid, such that \(\Delta V \approx V_g\). Furthermore, assume that the ideal gas law applies to the gas phase. Note: this problem is solved in the textbook, in the section on the Clausius-Clapeyron equation.
  1. Solve for \(\frac{dp}{dT}\) in terms of the pressure of the vapor and the latent heat \(L\) and the temperature.

  2. Assume further that the latent heat is roughly independent of temperature. Integrate to find the vapor pressure itself as a function of temperature (and of course, the latent heat).

assignment Homework

Derivatives from Data (NIST)
Energy and Entropy 2021 (2 years) Use the NIST web site “Thermophysical Properties of Fluid Systems” to answer the following questions. This site is an excellent resource for finding experimentally measured properties of fluids.
  1. Find the partial derivatives \[\left(\frac{\partial {S}}{\partial {T}}\right)_{p}\] \[\left(\frac{\partial {S}}{\partial {T}}\right)_{V}\] where \(p\) is the pressure, \(V\) is the volume, \(S\) is the entropy, and \(T\) is the temperature. Please find these derivatives for one gram of methanol at one atmosphere of pressure and at room temperature.
  2. Why does it take only two variables to define the state?
  3. Why are the derivatives above different?
  4. What do the words isobaric, isothermal, and isochoric mean?

group Small Group Activity

5 min.

Heat and Temperature of Water Vapor (Remote)

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.

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?

group Small Group Activity

30 min.

Ideal Gas Model

Ideal Gas surfaces thermo

Students should be familiar with the basic definitions of pressure, volume, temperature, and internal energy.

assignment Homework

Pressure and entropy of a degenerate Fermi gas
Fermi gas Pressure Entropy Thermal and Statistical Physics 2020
  1. Show that a Fermi electron gas in the ground state exerts a pressure \begin{align} p = \frac{\left(3\pi^2\right)^{\frac23}}{5} \frac{\hbar^2}{m}\left(\frac{N}{V}\right)^{\frac53} \end{align} In a uniform decrease of the volume of a cube every orbital has its energy raised: The energy of each orbital is proportional to \(\frac1{L^2}\) or to \(\frac1{V^{\frac23}}\).

  2. Find an expression for the entropy of a Fermi electron gas in the region \(kT\ll \varepsilon_F\). Notice that \(S\rightarrow 0\) as \(T\rightarrow 0\).

assignment Homework

Bottle in a Bottle
irreversible helium internal energy work first law Energy and Entropy 2021 (2 years)

The internal energy of helium gas at temperature \(T\) is to a very good approximation given by \begin{align} U &= \frac32 Nk_BT \end{align}

Consider a very irreversible process in which 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. What is the change in temperature when this process is complete? How much of the helium will remain in the small bottle?

group Small Group Activity

30 min.

Using \(pV\) and \(TS\) Plots
Energy and Entropy 2021 (2 years)

work heat first law energy

Students work out heat and work for rectangular paths on \(pV\) and \(TS\) plots. This gives with computing heat and work, applying the First Law, and recognizing that internal energy is a state function, which cannot change after a cyclic process.

assignment Homework

Using Gibbs Free Energy
thermodynamics entropy heat capacity internal energy equation of state Energy and Entropy 2021 (2 years)

You are given the following Gibbs free energy: \begin{equation*} G=-k T N \ln \left(\frac{a T^{5 / 2}}{p}\right) \end{equation*} where \(a\) is a constant (whose dimensions make the argument of the logarithm dimensionless).

  1. Compute the entropy.

  2. Work out the heat capacity at constant pressure \(C_p\).

  3. Find the connection among \(V\), \(p\), \(N\), and \(T\), which is called the equation of state (Hint: find the volume as a partial derivative of the Gibbs free energy).

  4. Compute the internal energy \(U\).

assignment Homework

Entropy, energy, and enthalpy of van der Waals gas
Van der Waals gas Enthalpy Entropy Thermal and Statistical Physics 2020

In this entire problem, keep results to first order in the van der Waals correction terms \(a\) and $b.

  1. Show that the entropy of the van der Waals gas is \begin{align} S &= Nk\left\{\ln\left(\frac{n_Q(V-Nb)}{N}\right)+\frac52\right\} \end{align}

  2. Show that the energy is \begin{align} U &= \frac32 NkT - \frac{N^2a}{V} \end{align}

  3. Show that the enthalpy \(H\equiv U+pV\) is \begin{align} H(T,V) &= \frac52NkT + \frac{N^2bkT}{V} - 2\frac{N^2a}{V} \\ H(T,p) &= \frac52NkT + Nbp - \frac{2Nap}{kT} \end{align}

Effects of High Altitude by Randall Munroe, at xkcd.

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

Entropy of mixing
Entropy Equilibrium Sackur-Tetrode Thermal and Statistical Physics 2020

Suppose that a system of \(N\) atoms of type \(A\) is placed in diffusive contact with a system of \(N\) atoms of type \(B\) at the same temperature and volume.

  1. Show that after diffusive equilibrium is reached the total entropy is increased by \(2Nk\ln 2\). The entropy increase \(2Nk\ln 2\) is known as the entropy of mixing.

  2. If the atoms are identical (\(A=B\)), show that there is no increase in entropy when diffusive contact is established. The difference has been called the Gibbs paradox.

  3. Since the Helmholtz free energy is lower for the mixed \(AB\) than for the separated \(A\) and \(B\), it should be possible to extract work from the mixing process. Construct a process that could extract work as the two gasses are mixed at fixed temperature. You will probably need to use walls that are permeable to one gas but not the other.

Note

This course has not yet covered work, but it was covered in Energy and Entropy, so you may need to stretch your memory to finish part (c).

assignment Homework

Ice calorimetry lab questions
This question is about the lab we did in class: Ice Calorimetry Lab.
  1. Plot your data I Plot the temperature versus total energy added to the system (which you can call \(Q\)). To do this, you will need to integrate the power. Discuss this curve and any interesting features you notice on it.
  2. Plot your data II Plot the heat capacity versus temperature. This will be a bit trickier. You can find the heat capacity from the previous plot by looking at the slope. \begin{align} C_p &= \left(\frac{\partial Q}{\partial T}\right)_p \end{align} This is what is called the heat capacity, which is the amount of energy needed to change the temperature by a given amount. The \(p\) subscript means that your measurement was made at constant pressure. This heat capacity is actually the total heat capacity of everything you put in the calorimeter, which includes the resistor and thermometer.
  3. Specific heat From your plot of \(C_p(T)\), work out the heat capacity per unit mass of water. You may assume the effect of the resistor and thermometer are negligible. How does your answer compare with the prediction of the Dulong-Petit law?
  4. Latent heat of fusion What did the temperature do while the ice was melting? How much energy was required to melt the ice in your calorimeter? How much energy was required per unit mass? per molecule?
  5. Entropy of fusion The change in entropy is easy to measure for a reversible isothermal process (such as the slow melting of ice), it is just \begin{align} \Delta S &= \frac{Q}{T} \end{align} where \(Q\) is the energy thermally added to the system and \(T\) is the temperature in Kelvin. What is was change in the entropy of the ice you melted? What was the change in entropy per molecule? What was the change in entropy per molecule divided by Boltzmann's constant?
  6. Entropy for a temperature change Choose two temperatures that your water reached (after the ice melted), and find the change in the entropy of your water. This change is given by \begin{align} \Delta S &= \int \frac{{\mathit{\unicode{273}}} Q}{T} \\ &= \int_{t_i}^{t_f} \frac{P(t)}{T(t)}dt \end{align} where \(P(t)\) is the heater power as a function of time and \(T(t)\) is the temperature, also as a function of time.

group Small Group Activity

30 min.

Changes in Internal Energy (Remote)

Thermo Internal Energy 1st Law of Thermodynamics

Students consider the change in internal energy during three different processes involving a container of water vapor on a stove. Using the 1st Law of Thermodynamics, students reason about how the internal energy would change and then compare this prediction with data from NIST presented as a contour plot.

group Small Group Activity

30 min.

A glass of water
Energy and Entropy 2021 (2 years)

thermodynamics intensive extensive temperature volume energy entropy

Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.

group Small Group Activity

30 min.

Optical depth of atmosphere
Contemporary Challenges 2022 (3 years) In this activity students estimate the optical depth of the atmosphere at the infrared wavelength where carbon dioxide has peak absorption.

face Lecture

120 min.

Phase transformations
Thermal and Statistical Physics 2020

phase transformation Clausius-Clapeyron mean field theory thermodynamics

These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.