## Activity: Ice Calorimetry Lab

The students will set up a Styrofoam cup with heating element and a thermometer in it. They will measure the temperature as a function of time, and thus the energy transferred from the power supply.
What students learn This lab gives students a chance to take data on the first day of class (or later, but I prefer to do it the first day of class). It provides an immediate context for thermodynamics, and also gives them a chance to experimentally measure a change in entropy. Students are required to measure the energy required to melt ice and raise the temperature of water, and measure the change in entropy by integrating the heat capacity.
• Media
• biotech Microwave oven Ice Calorimetry Lab

biotech Experiment

60 min.

##### Microwave oven Ice Calorimetry Lab
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

The students will set up a Styrofoam cup with heating element and a thermometer in it. They will measure the temperature as a function of time, and thus the energy transferred from the power supply.
• assignment Ice calorimetry lab questions

assignment Homework

##### Ice calorimetry lab questions
This question is about the lab we did in class: Ice Calorimetry Lab.
1. Plot your data I Plot the temperature versus total energy added to the system (which you can call $Q$). To do this, you will need to integrate the power. Discuss this curve and any interesting features you notice on it.
2. Plot your data II Plot the heat capacity versus temperature. This will be a bit trickier. You can find the heat capacity from the previous plot by looking at the slope. \begin{align} C_p &= \left(\frac{\partial Q}{\partial T}\right)_p \end{align} This is what is called the heat capacity, which is the amount of energy needed to change the temperature by a given amount. The $p$ subscript means that your measurement was made at constant pressure. This heat capacity is actually the total heat capacity of everything you put in the calorimeter, which includes the resistor and thermometer.
3. Specific heat From your plot of $C_p(T)$, work out the heat capacity per unit mass of water. You may assume the effect of the resistor and thermometer are negligible. How does your answer compare with the prediction of the Dulong-Petit law?
4. Latent heat of fusion What did the temperature do while the ice was melting? How much energy was required to melt the ice in your calorimeter? How much energy was required per unit mass? per molecule?
5. Entropy of fusion The change in entropy is easy to measure for a reversible isothermal process (such as the slow melting of ice), it is just \begin{align} \Delta S &= \frac{Q}{T} \end{align} where $Q$ is the energy thermally added to the system and $T$ is the temperature in Kelvin. What is was change in the entropy of the ice you melted? What was the change in entropy per molecule? What was the change in entropy per molecule divided by Boltzmann's constant?
6. Entropy for a temperature change Choose two temperatures that your water reached (after the ice melted), and find the change in the entropy of your water. This change is given by \begin{align} \Delta S &= \int \frac{{\mathit{\unicode{273}}} Q}{T} \\ &= \int_{t_i}^{t_f} \frac{P(t)}{T(t)}dt \end{align} where $P(t)$ is the heater power as a function of time and $T(t)$ is the temperature, also as a function of time.
• face Energy and heat and entropy

face Lecture

30 min.

##### Energy and heat and entropy
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

This short lecture introduces the ideas required for Ice Calorimetry Lab or Microwave oven Ice Calorimetry Lab.
• group Name the experiment (changing entropy)

group Small Group Activity

30 min.

##### Name the experiment (changing entropy)
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021 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.
• group Name the experiment

group Small Group Activity

30 min.

##### Name the experiment
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021 Energy and Entropy Fall 2021 Students will design an experiment that measures a specific partial derivative.
• assignment Icecream Mass

assignment Homework

##### Icecream Mass
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Use integration to find the total mass of the icecream in a packed cone (both the cone and the hemisphere of icecream on top).

• assignment Hockey

assignment Homework

##### Hockey
Central Forces Spring 2021

Consider the frictionless motion of a hockey puck of mass $m$ on a perfectly circular bowl-shaped ice rink with radius $a$. The central region of the bowl ($r < 0.8a$) is perfectly flat and the sides of the ice bowl smoothly rise to a height $h$ at $r = a$.

1. Draw a sketch of the potential energy for this system. Set the zero of potential energy at the top of the sides of the bowl.
2. Situation 1: the puck is initially moving radially outward from the exact center of the rink. What minimum velocity does the puck need to escape the rink?
3. Situation 2: a stationary puck, at a distance $\frac{a}{2}$ from the center of the rink, is hit in such a way that it's initial velocity $\vec v_0$ is perpendicular to its position vector as measured from the center of the rink. What is the total energy of the puck immediately after it is struck?
4. In situation 2, what is the angular momentum of the puck immediately after it is struck?
5. Draw a sketch of the effective potential for situation 2.
6. In situation 2, for what minimum value of $\vec v_0$ does the puck just escape the rink?

• group Using $pV$ and $TS$ Plots

group Small Group Activity

30 min.

##### Using $pV$ and $TS$ Plots
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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.
• assignment Coffees and Bagels and Net Worth

assignment Homework

##### Coffees and Bagels and Net Worth
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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 Power from the Ocean

assignment Homework

##### Power from the Ocean
heat engine efficiency Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

It has been proposed to use the thermal gradient of the ocean to drive a heat engine. Suppose that at a certain location the water temperature is $22^\circ$C at the ocean surface and $4^{o}$C at the ocean floor.

1. What is the maximum possible efficiency of an engine operating between these two temperatures?

2. If the engine is to produce 1 GW of electrical power, what minimum volume of water must be processed every second? Note that the specific heat capacity of water $c_p = 4.2$ Jg$^{-1}$K$^{-1}$ and the density of water is 1 g cm$^{-3}$, and both are roughly constant over this temperature range.

In this lab, we will be measuring how much energy it takes to melt ice and heat water.
Students will set up a Styrofoam cup with heating element and a thermometer in it. They will measure the temperature as a function of time, and thus the energy transferred from the power supply.
##### Materials:
• Styrofoam cup
• Heating element
• Scale
• 2 digital multimeters
• Temperature guage
• Ice and water

##### Introduction

This lab is very simple to run, but to get it done in one period you'll need to get students working quickly to measure the ice and water and get their heating started. Once their labs are going, there is time to give a middle-of-lab lecture introducing thermodynamics and thermal measurements.

There are two things to keep in mind for this lab. One is that the ice-water cups need to be vigorously stirred, otherwise the hot water (around 4 degrees Celsius) will settle at the bottom while the cold ice floats on the top. The other is that the ice should be cubes (not crushed) and the water should be ice-cold before it is massed out, otherwise too much ice will melt immediately on being added to the water.

Once the measurements are taken, I asked the students a couple of small-whiteboard-questions, “What is heat?” and “What is entropy?”. I then lecture on what the heat capacity $C_p$ is, and how they could extract it from their data, and on how they can calculate entropy from their measurements: $\Delta S = \int \frac{dQ}{T}$.

##### Student Conversations

Many students have difficulties simply measuring the water and ice correctly. Perhaps smaller containers for retrieving water would be good (around the same size as necessary). Or perhaps some basic lab procedures need to be gone over at some point in the Paradigms. Once the basic lab setup was accomplished, students seemed able to do the rest of the lab with no difficulty. -Amanda Abbott

##### Wrap-up

At the end of the lab, students should know how to calculate the entropy from the change in temperature. Students should be given a few days to do the analysis, and the data that they collect should be distributed to each member of the group.

## The setup

You will put some mass of ice (about 50g) and ice-cold water (about 150g) into your styrofoam cup. Use the scale to record the mass of the ice and water as you add them to the cup. Finally, add your ice-cold heating element and thermometer through the lid of the cup.

## Collect data

We will be measuring the temperature of the water and the power dissipated in the heating element (which is just a resistor). Thus we can find out how much energy was added to the water, and how this changes the temperature. In order to keep the temperature measurement reasonable, we will need to periodically stir the cup and heat it moderately slowly.

You will be collecting temperature data using the computer, so before you turn on the heater, you should make sure the computer is taking data. Turn on the heater, and write down the time you do so as well as the current and voltage, from which you can find the power dissipated in the resistor. If the current or voltage changes during the course of the experiment, take note of the new values---and the time.

##### Question 1: Plot your data I
Plot the temperature versus total energy added to the system (which you can call $Q$). To do this, you will need to integrate the power. Discuss this curve and any interesting features you notice on it.

##### Question 2: Plot your data II
Plot the heat capacity versus temperature. This will be a bit trickier. You can find the heat capacity from the previous plot by looking at the slope. \begin{align} C_p &= \left(\frac{\partial Q}{\partial T}\right)_p \end{align} This is what is called the heat capacity, which is the amount of energy needed to change the temperature by a given amount. The $p$ subscript means that your measurement was made at constant pressure. This heat capacity is actually the total heat capacity of everything you put in the calorimeter, which includes the resistor and thermometer.

##### Question 3: Specific heat
From your plot of $C_p(T)$, work out the heat capacity per unit mass of water. You may assume the effect of the resistor and thermometer are negligible. How does your answer compare with the prediction of the Dulong-Petit law?

##### Question 4: Latent heat of fusion
1. What did the temperature do while the ice was melting? How much energy was required to melt the ice in your calorimeter? How much energy was required per unit mass? per molecule?
2. The change in entropy is easy to measure for a reversible isothermal process (such as the slow melting of ice), it is just \begin{align} \Delta S &= \frac{Q}{T} \end{align} where $Q$ is the energy thermally added to the system and $T$ is the temperature in Kelvin. What is was change in the entropy of the ice you melted? What was the change in entropy per molecule? What was the change in entropy per molecule divided by Boltzmann's constant?

##### Question 5: Entropy for a temperature change
Choose two temperatures that your water reached (after the ice melted), and find the change in the entropy of your water. This change is given by \begin{align} \Delta S &= \int \frac{{\mathit{\unicode{273}}} Q}{T} \\ &= \int_{t_i}^{t_f} \frac{P(t)}{T(t)}dt \end{align} where $P(t)$ is the heater power as a function of time and $T(t)$ is the temperature, also as a function of time.

Author Information
David Roundy
Learning Outcomes