## Activity: Name the experiment (changing entropy)

Energy and Entropy 2021 (2 years)
Students are placed into small groups and asked to create an experimental setup they can use to measure the partial derivative they are given, in which entropy changes.
What students learn
• Partial derivatives
• Physical representation
• Thermodynamic variables
• Practicing changing certain variables while holding others constant
• Measurement and control of entropy

## Instructor's Guide

### Prerequisite Knowledge

• The ability to interpret partial derivatives.
• The ability to physically interpret partial derivatives.
• Students should be familiar with the thermodynamic definition of entropy.

### Introduction

Little introduction is necessary for this activity; however, the first Name the experiment activity should be performed before entering this Name the Experiment activity. Be sure to state to the groups that they must measure the partial derivatives they are given for a rubber band, providing both a description and a picture of their experiment.

If you finish with your derivative, you can try designing an experiment for the next derivative in the list.

For this activity we have just four partial derivatives:

Heat capacity measurement
$\left(\frac{\partial S}{\partial T}\right)_{V}\;\left(\frac{\partial S}{\partial T}\right)_{p}$
Isothermal (challenging)
$\left(\frac{\partial S}{\partial V}\right)_{T}\;\left(\frac{\partial S}{\partial p}\right)_{T}$
The first two derivatives are “simple” heat capacity measurements, and the second two are extremely challenging.

If a group finishes their experiment early, have them create an experiment for a more challenging partial derivative.

### Student Conversations

• The heat capacity derivatives (the derivatives with respect to $T$) are very similar to the experiment students did in the [[courses:activities:eeact:eeice|ice calorimetry lab]], and students often recognize this, which provides a nice review and synergy, and gives students who weren't solid on that lab previously a chance to feel more comfortable with it.
• The two isothermal derivatives are very challenging, and can serve as a motivation for why we care about the derivative tricks that students hate in thermodynamics: it's great to be able to write a hard-to-measure quantity in terms of an easy-to-measure one. This is precisely what students do in the [[courses:activities:eeact:eeice|next name-the-experiment activity]], so this difficult problem can provide a good foreshadowing.
• Some groups are able to find experiments for the isothermal derivatives in this activity. Student solutions most often involve the use of a thermostat combined with a measurement of how much heating that thermostat required. Another student solution involves using ice water to maintain the temperature and then measuring how much water was converted into ice or vice versa.

### Wrap-up

Have each group present their experiment. If any groups had a difficult time creating an experimental setup, have other students state any ideas that they would have for measuring the partial derivative. Mention that the partials that have terms of entropy in them or the constant are difficult to measure directly, and that there are ways of measuring a different partial derivative as an alternative to ones with entropy in them. This comment is a good precursor to the Maxwell relations. Here is a [[whitepapers:narratives:entropy|narrative]] for this activity (the second name the experiment).
• group Name the experiment

group Small Group Activity

30 min.

##### Name the experiment
Energy and Entropy 2021 (3 years)

Student groups design an experiment that measures an assigned partial derivative. In a compare-and-contrast wrap-up, groups report on how they would measure their derivatives.
• face Energy and Entropy review

face Lecture

5 min.

##### Energy and Entropy review
Thermal and Statistical Physics 2020 (3 years)

This very quick lecture reviews the content taught in Energy and Entropy, and is the first content in Thermal and Statistical Physics.
• assignment Coffees and Bagels and Net Worth

assignment Homework

##### Coffees and Bagels and Net Worth
Energy and Entropy 2021 (2 years)

In economics, the term utility is roughly related to overall happiness. Many things affect your happiness, including the amount of money you have and the amount of coffee you drink. We cannot directly measure your happiness, but we can measure how much money you are willing to give up in order to obtain coffee or bagels. If we assume you choose wisely, we can thus determine that your happiness increases when you decrease your amount of money by that amount in exchange for increasing your coffee consumption. Thus money is a (poor) measure of happiness or utility.

Money is also a nice quantity because it is conserved---just like energy! You may gain or lose money, but you always do so by a transaction. (There are some exceptions to the conservation of money, but they involve either the Fed, counterfeiters, or destruction of cash money, and we will ignore those issues.)

In this problem, we will assume that you have bought all the coffee and bagels you want (and no more), so that your happiness has been maximized. Thus you are in equilibrium with the coffee shop. We will assume further that you remain in equilibrium with the coffee shop at all times, and that you can sell coffee and bagels back to the coffee shop at cost.*

Thus your savings $S$ can be considered to be a function of your bagels $B$ and coffee $C$. In this problem we will also discuss the prices $P_B$ and $P_C$, which you may not assume are independent of $B$ and $C$. It may help to imagine that you could possibly buy out the local supply of coffee, and have to import it at higher costs.

1. The prices of bagels and coffee $P_B$ and $P_C$ have derivative relationships between your savings and the quantity of coffee and bagels that you have. What are the units of these prices? What is the mathematical definition of $P_C$ and $P_B$?

2. Write down the total differential of your savings, in terms of $B$, $C$, $P_B$ and $P_C$.

3. Solve for the total differential of your net worth. Your net worth $W$ is the sum of your total savings plus the value of the coffee and bagels that you own. From the total differential, relate your amount of coffee and bagels to partial derivatives of your net worth.

• assignment Rubber Sheet

assignment Homework

##### Rubber Sheet
Energy and Entropy 2021 (2 years)

Consider a hanging rectangular rubber sheet. We will consider there to be two ways to get energy into or out of this sheet: you can either stretch it vertically or horizontally. The distance of vertical stretch we will call $y$, and the distance of horizontal stretch we will call $x$.

If I pull the bottom down by a small distance $\Delta y$, with no horizontal force, what is the resulting change in width $\Delta x$? Express your answer in terms of partial derivatives of the potential energy $U(x,y)$.

• group Expectation Value and Uncertainty for the Difference of Dice

group Small Group Activity

60 min.

##### Expectation Value and Uncertainty for the Difference of Dice
Quantum Fundamentals 2022 (2 years)
• assignment Pressure and entropy of a degenerate Fermi gas

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 Unknowns Spin-1/2 Brief

assignment Homework

##### Unknowns Spin-1/2 Brief
Quantum Fundamentals 2022 (2 years) With the Spins simulation set for a spin 1/2 system, measure the probabilities of all the possible spin components for each of the unknown initial states $\left|{\psi_3}\right\rangle$ and $\left|{\psi_4}\right\rangle$.
1. Use your measured probabilities to find each of the unknown states as a linear superposition of the $S_z$-basis states $\left|{+}\right\rangle$ and $\left|{-}\right\rangle$.
2. Articulate a Process: Write a set of general instructions that would allow another student in next year's class to find an unknown state from measured probabilities.
3. Compare Theory with Experiment: Design an experiment that will allow you to test whether your prediction for each of the unknown states is correct. Describe your experiment here, clearly but succinctly, as if you were writing it up for a paper. Do the experiment and discuss your results.
4. Make a Conceptual Connection: In general, can you determine a quantum state with spin-component probability measurements in only two spin-component-directions? Why or why not?
• group Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute

group Small Group Activity

30 min.

##### Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute
Quantum Fundamentals 2022 (2 years)
• group Projectile with Linear Drag

group Small Group Activity

120 min.

##### Projectile with Linear Drag
Theoretical Mechanics 2021 (2 years)

Students consider projectile motion of an object that experiences drag force that in linear with the velocity. Students consider the horizontal motion and the vertical motion separately. Students solve Newton's 2nd law as a differential equation.
• assignment Quantum harmonic oscillator

assignment Homework

##### Quantum harmonic oscillator
Entropy Quantum harmonic oscillator Frequency Energy Thermal and Statistical Physics 2020
1. Find the entropy of a set of $N$ oscillators of frequency $\omega$ as a function of the total quantum number $n$. Use the multiplicity function: $$g(N,n) = \frac{(N+n-1)!}{n!(N-1)!}$$ and assume that $N\gg 1$. This means you can make the Sitrling approximation that $\log N! \approx N\log N - N$. It also means that $N-1 \approx N$.

2. Let $U$ denote the total energy $n\hbar\omega$ of the oscillators. Express the entropy as $S(U,N)$. Show that the total energy at temperature $T$ is $$U = \frac{N\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}$$ This is the Planck result found the hard way. We will get to the easy way soon, and you will never again need to work with a multiplicity function like this.

Learning Outcomes