Student handout: 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.
What students learn
  • Partial derivatives
  • Physical representation
  • Thermodynamic variables
  • Practicing changing certain variables while holding others constant

Your group will be given one of the following partial derivatives: \begin{align} a)\quad\left(\frac{\partial {V}}{\partial {p}}\right)_{T}\quad b)\quad\left(\frac{\partial {U}}{\partial {p}}\right)_{S}\quad c)\quad\left(\frac{\partial {T}}{\partial {V}}\right)_{S}\quad d)\quad\left(\frac{\partial {V}}{\partial {T}}\right)_{p}\quad e)\quad\left(\frac{\partial {U}}{\partial {T}}\right)_{V}\\ f)\quad\left(\frac{\partial {p}}{\partial {V}}\right)_{T}\quad g)\quad\left(\frac{\partial {V}}{\partial {T}}\right)_{S}\quad h)\quad\left(\frac{\partial {T}}{\partial {V}}\right)_{p}\quad i)\quad\left(\frac{\partial {T}}{\partial {U}}\right)_{V}\quad j)\quad\left(\frac{\partial {V}}{\partial {p}}\right)_{S} \end{align} In your group, design an experiment to measure this derivative. Draw a sketch of the apparatus and describe how to convert directly measured data into a numerical value for the derivative.

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

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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.

    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.)

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    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.

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Learning Outcomes