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.
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.
 group Ice Calorimetry Lab
group Small Group Activity
60 min.
heat entropy water ice
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.
 face Energy and heat and entropy
 assignment Ice calorimetry lab questions
assignment Homework
This question is about the lab we did in class:
Ice Calorimetry Lab.
 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.
 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.

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 DulongPetit law?

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?
 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?

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.
 group Name the experiment (changing entropy)
group Small Group Activity
30 min.
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.
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
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
Central Forces Spring 2021
Consider the frictionless motion of a hockey puck of mass \(m\) on a perfectly
circular bowlshaped 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 \(r = a\).

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.

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?

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?

In situation 2, what is the angular momentum of the puck immediately after it is struck?

Draw a sketch of the effective potential for situation 2.

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.
Energy and Entropy Fall 2020
Energy and Entropy Fall 2021
work heat first law energy
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 Power from the Ocean
assignment Homework
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.
What is the maximum possible efficiency of an engine operating
between these two temperatures?
 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.
 assignment Power Plant on a River
assignment Homework
efficiency heat engine carnot
Energy and Entropy Fall 2020
Energy and Entropy Fall 2021
At a power plant that produces 1 GW
(\(10^{9} \text{watts}\)) of electricity, the steam turbines take in steam at a
temperature of \(500^{o}C\), and the waste energy is expelled into the
environment at \(20^{o}C\).
What is the maximum possible efficiency of this plant?
Suppose you arrange the power plant to expel its waste energy
into a chilly mountain river at \(15^oC\). Roughly how much money can
you make in a year by installing your improved hardware, if you sell
the additional electricity for 10 cents per kilowatthour?
At what rate will the plant expel waste energy into this river?
Assume the river's flow rate is 100 m\(^{3}/\)s. By how much will
the temperature of the river increase?
 To avoid this “thermal pollution” of the river the plant could
instead be cooled by evaporation of river water. This is more
expensive, but it is environmentally preferable. At what rate must
the water evaporate? What fraction of the river must be evaporated?
In this lab, we will be measuring how much energy it takes to melt ice
and heat water. I have modified this lab to work with a microwave oven that you're likely to have at home.
Before the lab
Before the day of the lab, you will need to collect your equipment. If you do
not have a microwave oven, or do not have a liquid measuring cup, you will need
to use the data from another student in your group. If you do have a
microwave and a way to measure the volume or mass of water, please be ready to
do this lab during class.
You will need the materials below. Ideally the day before the lab you will put about
500 mL (or even more) of water into a container (not a glass jar, which could break)
and put it into the freezer so you will have a large chunk of ice. This will make
it easier to separate the ice from the melt water when you melt it. If you do not
do this you can still use ice from an ice cube tray, but then it will really help to
also have a collander or seive to help separate ice from water.
Materials:
 Microwavesafe bowl or mug
 Microwave oven
 Kitchen scale or liquid measuring cup
 Ice and water
 Thermometer (if available)
This is the home version of Ice Calorimetry Lab for when there is a pandemic, or if you want students to do the activity at home. The accuracy students can get with a microwave oven is surprising, so I may adapt this to be a "home lab" even when we are teaching in person.
While the measurements are being taken, I asked the students a couple of smallwhiteboardquestions, “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: \(\Delta S = \int \frac{dQ}{T}\).
We will be using a microwave oven to measure how much energy it
takes to melt ice and how much energy it takes to boil water. If you have
a thermometer, you can also measure how much energy it takes to raise its temperature.
Our experimental unit of energy will be “seconds in the microwave.”
It's not a great unit of measure. A microwave may not always output equal power
independent of what is in it, so we'll try not to change too much the
amount of ice or water in the oven.
In the analysis, you can use the nominal power of the microwave (as written somewhere
on the inside or outside of your oven) to approximately convert this to Joules, but
keep in mind that this conversion is dubious on many fronts. There will be energy
wasted in the electronics of the microwave, which is not transmitted to the ice.
Also the documented power is probably rounded up, because it is used by electricians
to determine whether the device is safe to put on a circuit, and for that purpose
it is acceptable to use less than the documented power, but not more. We could do
better by measuring the current drawn (and the line voltage), but I doubt you have
that equipment in your kitchen.
Melting ice
We will start by measuring how much energy it takes to melt a gram of ice.
You will want maybe 500 mL of ice, or perhaps two ice cube trays.
Prepare your ice for this experiment by putting it in a bowl of water for a bit.
This will raise its temperature to zero Celsius.
Put a bunch of ice (500 mL or two ice cube trays) into a microwavesafe bowl.
 Microwave the ice for a minute.
 Pour out the liquid water (possibly through a seive or a collander) into a liquid measuring cup to see how much of the ice was melted. Alternatively you could measure the mass of the melt water with a scale.
 Repeat, but increase the time significantly if you didn't melt an eaasily
measurable amount of water. Stop when you've melted at least half of your ice.
Boiling water
We will now measure how much energy it takes to boil water. If you have a glass measuring cup,
fill it to the highest mark with water. Add a pebble to the measuring cup (or bowl).
Then microwave it until it reaches a full rolling boil.
At this point the water should be 100° Celsius.
 Put the water in the microwave, and heat it for three minutes.
 Measure the amount of water remaining either by volume or mass.
 Repeat, but increase the time significantly if you didn't boil an eaasily
measurable amount of water. Stop when you've boiled away at least half of your water.
Heating water (optional)
Now if you have a thermometer, you can find out how much energy it takes to raise
the temperature of water.
Start by filling a cup with water, making sure to measure its mass or volume.
 Measure the temperature of the water and write it down.
 Put the cup in the microwave for a little while (start with 30 seconds or a minute probably), and write down how
long you heated it.
 Repeat until it reaches boiling. Increase the time interval if your water does not change
temperature by an easily measurable amount.