assignment Homework

Gibbs free energy
thermodynamics Maxwell relation Energy and Entropy 2020 The Gibbs free energy, \(G\), is given by \begin{align*} G = U + pV - TS. \end{align*}
  1. Find the total differential of \(G\). As always, show your work.
  2. Interpret the coefficients of the total differential \(dG\) in order to find a derivative expression for the entropy \(S\).
  3. From the total differential \(dG\), obtain a different thermodynamic derivative that is equal to \[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]

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?

face Lecture

30 min.

A coarse-grained model for transportation
Contemporary Challenges 2022 (3 years)

energy flow diagram energy car

A short lecture introducing the idea that most of the energy loss when driving is going into the kinetic energy of the air.

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?

assignment Homework

Free Expansion
Energy and Entropy 2021 (2 years)

The internal energy is of any ideal gas can be written as \begin{align} U &= U(T,N) \end{align} meaning that the internal energy depends only on the number of particles and the temperature, but not the volume.*

The ideal gas law \begin{align} pV &= Nk_BT \end{align} defines the relationship between \(p\), \(V\) and \(T\). You may take the number of molecules \(N\) to be constant. Consider the free adiabatic expansion of an ideal gas to twice its volume. “Free expansion” means that no work is done, but also that the process is also neither quasistatic nor reversible.
  1. What is the change in entropy of the gas? How do you know this?

  2. What is the change in temperature of the gas?

group Small Group Activity

30 min.

Work By An Electric Field (Contour Map)

E&M Path integrals

Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.

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.

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.

assignment Homework

Heat shields
Stefan-Boltzmann blackbody radiation Thermal and Statistical Physics 2020 A black (nonreflective) sheet of metal at high temperature \(T_h\) is parallel to a cold black sheet of metal at temperature \(T_c\). Each sheet has an area \(A\) which is much greater than the distance between them. The sheets are in vacuum, so energy can only be transferred by radiation.
  1. Solve for the net power transferred between the two sheets.

  2. A third black metal sheet is inserted between the other two and is allowed to come to a steady state temperature \(T_m\). Find the temperature of the middle sheet, and solve for the new net power transferred between the hot and cold sheets. This is the principle of the heat shield, and is part of how the James Web telescope shield works.
  3. Optional: Find the power through an \(N\)-layer sandwich.

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

Centrifuge
Centrifugal potential Thermal and Statistical Physics 2020 A circular cylinder of radius \(R\) rotates about the long axis with angular velocity \(\omega\). The cylinder contains an ideal gas of atoms of mass \(M\) at temperature \(T\). Find an expression for the dependence of the concentration \(n(r)\) on the radial distance \(r\) from the axis, in terms of \(n(0)\) on the axis. Take \(\mu\) as for an ideal gas.

assignment Homework

Photon carnot engine
Carnot engine Work Energy Entropy Thermal and Statistical Physics 2020

In our week on radiation, we saw that the Helmholtz free energy of a box of radiation at temperature \(T\) is \begin{align} F &= -8\pi \frac{V(kT)^4}{h^3c^3}\frac{\pi^4}{45} \end{align} From this we also found the internal energy and entropy \begin{align} U &= 24\pi \frac{(kT)^4}{h^3c^3}\frac{\pi^4}{45} V \\ S &= 32\pi kV\left(\frac{kT}{hc}\right)^3 \frac{\pi^4}{45} \end{align} Given these results, let us consider a Carnot engine that uses an empty metalic piston (i.e. a photon gas).

  1. Given \(T_H\) and \(T_C\), as well as \(V_1\) and \(V_2\) (the two volumes at \(T_H\)), determine \(V_3\) and \(V_4\) (the two volumes at \(T_C\)).

  2. What is the heat \(Q_H\) taken up and the work done by the gas during the first isothermal expansion? Are they equal to each other, as for the ideal gas?

  3. Does the work done on the two isentropic stages cancel each other, as for the ideal gas?

  4. Calculate the total work done by the gas during one cycle. Compare it with the heat taken up at \(T_H\) and show that the energy conversion efficiency is the Carnot efficiency.

group Small Group Activity

60 min.

Systems of Equations Compare and Contrast

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?

keyboard Computational Activity

120 min.

Electrostatic potential of a square of charge
Computational Physics Lab II 2022 (2 years)

integration electrostatic potential surface charge density

Students write python programs to compute the potential due to a square of surface charge, and then to visualize the result. This activity can be used to introduce students to the process of integrating numerically.

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

Using Gradescope (AIMS)
AIMS Maxwell 2021 (2 years)

Task: Draw a right triangle. Put a circle around the right angle, that is, the angle that is \(\frac\pi2\) radians.

Preparing your submission:

  • Complete the assignment using your choice of technology. You may write your answers on paper, write them electronically (for instance using xournal), or typeset them (for instance using LaTeX).
  • If using software, please export to PDF. If writing by hand, please scan your work using the AIMS scanner if possible. You can also use a scanning app; Gradescope offers advice and suggested apps at this URL. The preferred format is PDF; photos or JPEG scans are less easy to read (and much larger), and should be used only if no alternative is available.)
  • Please make sure that your file name includes your own name and the number of the assignment, such as "Tevian2.pdf."

Using Gradescope: We will arrange for you to have a Gradescope account, after which you should receive access instructions directly from them. To submit an assignment:

  1. Navigate to https://paradigms.oregonstate.eduhttps://www.gradescope.com and login
  2. Select the appropriate course, such as "AIMS F21". (There will likely be only one course listed.)
  3. Select the assignment called "Sample Assignment"
  4. Follow the instructions to upload your assignment. (The preferred format is PDF.)
  5. You will then be prompted to associate submitted pages with problem numbers by selecting pages on the right and questions on the left. (In this assignment, there is only one of each.) You may associate multiple problems with the same page if appropriate.
  6. When you are finished, click "Submit"
  7. After the assignments have been marked, you can log back in to see instructor comments.

assignment Homework

Electric Field of a Finite Line

Consider the finite line with a uniform charge density from class.

  1. Write an integral expression for the electric field at any point in space due to the finite line. In addition to your usual physics sense-making, you must include a clearly labeled figure and discuss what happens to the direction of the unit vectors as you integrate.Consider the finite line with a uniform charge density from class.
  2. Perform the integral to find the \(z\)-component of the electric field. In addition to your usual physics sense-making, you must compare your result to the gradient of the electric potential we found in class. (If you want to challenge yourself, do the \(s\)-component as well!)

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.