## Activity: Using $pV$ and $TS$ Plots

Energy and Entropy 2021 (2 years)
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.
What students learn
• Work as area under curve in a $pV$ plot
• Heat transfer as area under a curve in a $TS$ plot
• Reminder that internal energy is a state function
• Reminder of First Law
• Media
Before starting this activity, students do need to be taught about heat and work as ${\mathit{\unicode{273}}} Q = TdS$ and ${\mathit{\unicode{273}}} W = -pdV$. If this did not happen recently, it might be worth reminding students of these relationships. Otherwise, the handout basically tells them what to do.

## $pV$ rectangle

We'll begin today looking at processes that happen at constant pressure, volume, temperature or entropy.

We have spent a lot of time discussing partial derivatives and differentials, and how they relate to measurements. Another sort of measurement involves integrals rather than derivatives, and measures finite changes. To discuss this, it's common to use what are called $pV$ diagrams. For instance, consider the following square.

Note You cannot assume that the material in question is an ideal gas. It could equally well be a mixture of ice and water, or a strange gooey substance that you found on a distant planet. You can never assume that we are talking about an ideal gas unless we tell you so!

In your groups, work out the following questions:

1. What does this figure describe? Is $p$ a function of $V$?

2. What is the net work done after one cycle of this process? How much work was done at each step?

3. What is the net energy transfered by heating over one cycle of this process? Try to find the energy transfered by heating at each step.

Student Conversations
• Some students will claim that because we return to the original state, the net work (or possibly net heat) must be zero.
• Many students will not recognize that on a $pV$ graph, the work is just (the negative of) the area under the curve. While I wouldn't want them to think of this as a definition of work, it is a necessary and useful tool to have in their toolbox.
• It is helpful to ask students to describe what is happening in each stage (e.g. “it is being compressed at fixed pressure... so probably its temperature is dropping, unless it is something weird like ice”).
• Some students assume that temperature is fixed when working on the $pV$ plot, because it isn't listed.
• Many students will try to invoke or use the ideal gas law.

## $TS$ rectangle

Now let's look at another cycle. Let's consider the following figure, which looks similar, but is now a plot of $T$ vs. $S$, and answer the following questions:

1. What is this cycle? How would you go about running a cycle like this?

2. What is the net heat transfer over one cycle of this process? How much was transfered on each step?

3. What is the net work done after one cycle of this process? How much work was done at each step?

Student Conversations
1. Although the math using $TS$ diagram is effectively identical to that with the $pV$ diagram (with heat and work swapped, and a few minus signs), students will still find the $TS$ diagram challenging.

## Wrap-up

Bring the class back together and have a group argue if the work is positive or negative on each leg of either the $pV$ or $TS$ curve (instructor's preference of which to analyze first). Help groups resolve any inconsistencies in answers for the work and heat of each leg.

Many students become confused as to if work is being done on or by the system in these plots because of the minus sign associated with the thermodynamic identity term containing $pdV$. Be sure to point out that if the volume of the system increases, then work is being done by the system; show this by analyzing a leg of the $pV$ curve where the pressure remains constant and the volume increases and display that the total work of the system would be negative.

The analysis of the $TS$ curve should be nearly identical to the $pV$ curve. Be sure to note that the minus sign that appears in the work term is no longer present in the heat expression.

• face Review of Thermal Physics

face Lecture

30 min.

##### Review of Thermal Physics
Thermal and Statistical Physics 2020

These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.
• face Work, Heat, and cycles

face Lecture

120 min.

##### Work, Heat, and cycles
Thermal and Statistical Physics 2020

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

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 Gibbs free energy

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 Entropy, energy, and enthalpy of van der Waals gas

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}

• 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 Heat shields

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.

• 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 Free Expansion

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?

• assignment Bottle in a Bottle 2

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).