## Activity: Quantifying Change (Remote)

In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.
• group Small Group Activity schedule 30 min. build Contour graph with P,V,S,T,U contours in color , handout for each student, A personal or shared writing space for each student to write/draw/sketch. description Student handout (PDF)
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Thermo Derivatives
What students learn
• To calculate derivatives on a contour map.
• Many different derivatives can be determined at a point.
• Think about derivatives as ratios of small changes.
• Reinforces the idea that a derivative is a ratio of small changes along a path
• In thermo, derivatives are often invertible?
• The sign of a derivative does not depend on the direction you're traveling along the path
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Goals

• To calculate derivatives on a contour map.
• Many different derivatives can be determined at a point.
• Think about derivatives as ratios of small changes.
• Reinforces the idea that a derivative is a ratio of small changes along a path
• In thermo, derivatives are often invertible?
• The sign of a derivative does not depend on the direction you're traveling along the path

Time Estimate: 30 minutes Tools/Equipment

• Contour graph with P,V,S,T,U contours
• Student handout for each student
• A personal or shared writing space for each student to write/draw/sketch.

Intro

• Orient the students to the contour plot - it's busy.
• Exposure to differential calculus

Whole Class Discussion / Wrap Up

• How students report out their answers to Explore.
• Many students will not be willing to call the ratio of changes a derivative until you've taken the limit that the changes become infinitesimal. Talk about linear approximation and/or tables of data.
• Most students will have heard in a math class that derivatives are not invertible. In thermo, they often are as long as you don't change the path.
• Many students will think that if you reverse the path, the derivative will change sign. Discuss how the signs of both the numerator and denominator of the ratio of small changes swaps, leaving the derivative unchanged.

Orient: The plot shows various thermodynamic quantities for water vapor in an insulated piston (cylindrical thermos with a movable top) at different states. From state (point) $A$ to state (point) $B$, estimate the following quantities:

 Verbal Description Symbol Estimate (with Units) Change in volume: $\Delta V_{A\rightarrow B}$ Change in entropy: $\Delta S_{A\rightarrow B}$ Change in temperature: $\Delta T_{A\rightarrow B}$ Change in pressure: $\Delta P_{A\rightarrow B}$ Change in internal energy: $\Delta U_{A\rightarrow B}$

At what stage would you be willing to label these quantities with d's?

Discussion: Small Some students say “small” but not what it is small compared to. The instructor can talk about linearity in the WCD.
Discussion: Infinitesimals “At what stage would you be willing to label these quantities with d's?”

Explore: Pick two of the variables in the table and determine the rate of change of one with respect to the other from $A$ to $B$. What experiment could you do to measure this rate?

Discussion: Derivatives: “Under what circumstances would you be willing to call this rate a derivative?” Have students plot graphs of their functions to see how linear they are. Many students will not be willing to call this a derivative because we aren't taking a limit - talk about linear approximation.
Discussion: Where? If a student is willing to call this rate a derivative, at which point is it the derivative? It is the best approximation to the derivative somewhere between the two points.
Discussion: Slope Is their derivative a slope? Of what?
Follow-up: How would you represent this derivative symbolically? Some students write $\Delta U / \Delta T$ vs. $dU/dT$; some students write subscript $S$ vs. $A \rightarrow B$. A good opportunity to talk about how the derivative ought to depend on the value of $S$.

Interpret: Determine the reciprocal of the rate you calculated. How would you physically interpret this number?

Discussion: Experiment What experiment would you do to measure this quantity?
Discussion: Inversion The derivative can be flipped as long as the path is unchanged.
Follow-up: How would you represent this rate it symbolically?

Reinterpret: How does the rate you previously calculated change if instead you went from state $B$ to state $A$?

Answer: The rate (and therefore the derivative is the same.
Discussion: Sign Many students will initially say that the derivative changes sign.

• group Squishability'' of Water Vapor (Contour Map)

group Small Group Activity

30 min.

##### “Squishability” of Water Vapor (Contour Map)

Students determine the “squishibility” (an extensive compressibility) by taking $-\partial V/\partial P$ holding either temperature or entropy fixed.
• group Heat and Temperature of Water Vapor (Remote)

group Small Group Activity

5 min.

##### Heat and Temperature of Water Vapor (Remote)

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.
• group Ideal Gas Model

group Small Group Activity

30 min.

##### Ideal Gas Model

Students should be familiar with the basic definitions of pressure, volume, temperature, and internal energy.
• 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.

• group Covariation in Thermal Systems

group Small Group Activity

30 min.

##### Covariation in Thermal Systems

Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.
• assignment_ind Partial Derivatives from a Contour Map

assignment_ind Small White Board Question

10 min.

##### Partial Derivatives from a Contour Map
Static Fields 2022 (3 years) Students are shown the contour graph of a function of two variables and asked to find the derivative. They discover that, without a function to differentiate, they must instead think of the derivative as a ratio of small changes. This requires them to pick two nearby points. Which two?
• assignment Contours

assignment Homework

##### Contours
Static Fields 2022 (4 years)

Shown below is a contour plot of a scalar field, $\mu(x,y)$. Assume that $x$ and $y$ are measured in meters and that $\mu$ is measured in kilograms. Four points are indicated on the plot.

1. Determine $\frac{\partial\mu}{\partial x}$ and $\frac{\partial\mu}{\partial y}$ at each of the four points.
2. On a printout of the figure, draw a qualitatively accurate vector at each point corresponding to the gradient of $\mu(x,y)$ using your answers to part a above. How did you choose a scale for your vectors? Describe how the direction of the gradient vector is related to the contours on the plot and what property of the contour map is related to the magnitude of the gradient vector.
3. Evaluate the gradient of $h(x,y)=(x+1)^2\left(\frac{x}{2}-\frac{y}{3}\right)^3$ at the point $(x,y)=(3,-2)$.

• group Charged Sphere

group Small Group Activity

30 min.

##### Charged Sphere

Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.
• assignment Derivatives from Data (NIST)

assignment Homework

##### Derivatives from Data (NIST)
Energy and Entropy 2021 (2 years) Use the NIST web site “Thermophysical Properties of Fluid Systems” to answer the following questions. This site is an excellent resource for finding experimentally measured properties of fluids.
1. Find the partial derivatives $\left(\frac{\partial {S}}{\partial {T}}\right)_{p}$ $\left(\frac{\partial {S}}{\partial {T}}\right)_{V}$ where $p$ is the pressure, $V$ is the volume, $S$ is the entropy, and $T$ is the temperature. Please find these derivatives for one gram of methanol at one atmosphere of pressure and at room temperature.
2. Why does it take only two variables to define the state?
3. Why are the derivatives above different?
4. What do the words isobaric, isothermal, and isochoric mean?
• group Gravitational Force

group Small Group Activity

30 min.

##### Gravitational Force

Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.

Author Information
Raising Physics to the Surface
Keywords
Thermo Derivatives
Learning Outcomes