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
    • activity_media/pvsquare_JuBDPxH.svg
    • activity_media/TSsquare_YwdkzFg.svg
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.


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.

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    Bottle in a Bottle 2
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    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.

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    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).

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  • assignment Coffees and Bagels and Net Worth

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    Coffees and Bagels and Net Worth
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    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.

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    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.

Learning Outcomes