## Activity: 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.
What students learn
• Heat capacity at constant volume relates to changes in internal energy, i.e. ${\mathit{\unicode{273}}} Q = dU$.
• A conceptual definition of heat capacity at constant volume is the derivative of internal energy with respect to temperature without changing the volume.
• Not all derivatives are slopes but they are all ratios of small changes.
• Heat capacity at constant volume depends on the value of the volume. optional: Heat capacity at constant pressure is NOT $\left(dU/dT\right)_p$ - you have to account for work done.
• Media

Goals

• Heat capacity at constant volume relates to changes in internal energy, i.e. ${\mathit{\unicode{273}}} Q = dU$.
• A conceptual definition of heat capacity at constant volume is the derivative of internal energy with respect to temperature without changing the volume.
• Not all derivatives are slopes but they are all ratios of small changes.
• Heat capacity at constant volume depends on the value of the volume. optional: Heat capacity at constant pressure is NOT $\left(dU/dT\right)_p$ - you have to account for work done.

Equipment Needed:

• Heat Capacity graph
• Optional: Blue $U(T,p)$ plastic surface for each group
• Student worksheet for each student
• A personal or shared writing space for each student to write/draw/sketch.

Introduction

• Students should have seen 1st Law of Thermodynamics and the Thermodynamic Identity
• Students should know how to compute work as $p\;dV$

Whole Class Discussion:

• Students need to figure out what derivative they're trying to estimate. Having an early WCD about this would be useful.
• The derivative they are trying to estimate is tantalizingly close to a slope, especially for the blue dot and the green triangle. Not all partial derivatives are slopes but they are all ratios of small changes.
• For an ideal gas, the heat capacities are constant. Water vapor is not an ideal gas but it is pretty close. In general, the heat capacity is a state variable.

Heat & Temperature of Water Vapor

A pressure cooker is an enclosed pot that expels air and traps water vapor, which increases the internal pressure. This in turn raises the boiling point of water and allows food to cook at high temperatures.

Imagine you have a large industrial pressure cooker that holds 1 kg of water vapor. You would like to know how responsive the system is to changes in temperature. To do this, you need to determine a characteristic rate: how much heat is needed to change the temperature by a small amount.

The graph shows internal energy and volume contours plotted on temperature and pressure axes.

 Internal Energy 2cm $\rightarrow$ 170. kJ Temperature 2cm $\rightarrow$ 70 K Pressure 2cm $\rightarrow$ 128000 Pa Entropy Contours Curves $\rightarrow$ 0.33 kJ/K apart Volume Contours Line Segments $\rightarrow$ 0.7 m3 apart

Estimate: Use the graph to determine this temperature-responsiveness when the volume is held fixed. The initial state of the system corresponds to the black square. Describe your process.

Student Ideas: Students tend toward trying to use equipartition theorem or other equations, rather than directly measuring the rate on the surface. Students might not always realize that work done is zero, or how this would relate U and Q.

Solution Want the ratio of change in internal energy and the change in temperature. This is NOT the slope of the surface. It is the ratio of small changes.

Discussion: We haven't told students what derivative to estimate. Have a whole class discussion about this after most groups have struggled for a few minutes and have had some insights.

Student Ideas: Students might need help in recognizing that for a pressure cooker, you can control the pressure and temperature but the volume of water vapor is constant.

Follow-Up: For what graph would the slope be the rate you're looking for? How is that graph related to the surface?

Explain: Why does it matter that you are holding volume constant in the above estimate?

Two reasons: (1) you need to specify a path in order to take the derivative, and (2) holding the volume constant means the work done is zero, so the heat added is equal to the change in internal energy.

Explore: Does the value of your estimate depend on the value of the volume?

Any answer is productive here. For an ideal gas, the answer is no. Water vapor is nearly an ideal gas (but not quite). The heat capacity at constant volume is not the directional derivative.

Optional Extensions:
• Can you estimate the heat capacity at constant PRESSURE? (Yes, there are several ways, including estimating the work done along the constant pressure path and subtracting that from the change in internal energy. Also, chain rule (try to avoid).)

• Can you estimate $C_v$ using a different graph?

• group Number of Paths

group Small Group Activity

30 min.

##### Number of Paths

Student discuss how many paths can be found on a map of the vector fields $\vec{F}$ for which the integral $\int \vec{F}\cdot d\vec{r}$ is positive, negative, or zero. $\vec{F}$ is conservative. They do a similar activity for the vector field $\vec{G}$ which is not conservative.
• group Calculating Coefficients for a Power Series

group Small Group Activity

30 min.

##### Calculating Coefficients for a Power Series
Static Fields 2022 (5 years)

Power Series Sequence (E&M)

This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the coefficients of a power series for a particular function:

$c_n={1\over n!}\, f^{(n)}(z_0)$

After a brief lecture deriving the formula for the coefficients of a power series, students compute the power series coefficients for a $\sin\theta$ (around both the origin and $\frac{\pi}{6}$). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up.

• assignment Heat capacity of vacuum

assignment Homework

##### Heat capacity of vacuum
Heat capacity entropy Thermal and Statistical Physics 2020
1. Solve for the heat capacity of a vacuum, given the above, and assuming that photons represent all the energy present in vacuum.
2. Compare the heat capacity of vacuum at room temperature with the heat capacity of an equal volume of water.
• assignment Potential energy of gas in gravitational field

assignment Homework

##### Potential energy of gas in gravitational field
Potential energy Heat capacity Thermal and Statistical Physics 2020 Consider a column of atoms each of mass $M$ at temperature $T$ in a uniform gravitational field $g$. Find the thermal average potential energy per atom. The thermal average kinetic energy is independent of height. Find the total heat capacity per atom. The total heat capacity is the sum of contributions from the kinetic energy and from the potential energy. Take the zero of the gravitational energy at the bottom $h=0$ of the column. Integrate from $h=0$ to $h=\infty$. You may assume the gas is ideal.
• assignment Using Gibbs Free Energy

assignment Homework

##### Using Gibbs Free Energy
thermodynamics entropy heat capacity internal energy equation of state Energy and Entropy 2021 (2 years)

You are given the following Gibbs free energy: \begin{equation*} G=-k T N \ln \left(\frac{a T^{5 / 2}}{p}\right) \end{equation*} where $a$ is a constant (whose dimensions make the argument of the logarithm dimensionless).

1. Compute the entropy.

2. Work out the heat capacity at constant pressure $C_p$.

3. Find the connection among $V$, $p$, $N$, and $T$, which is called the equation of state (Hint: find the volume as a partial derivative of the Gibbs free energy).

4. Compute the internal energy $U$.

• group Static Fields Equation Sheet

group Small Group Activity

5 min.

##### Static Fields Equation Sheet
Static Fields 2022 (3 years)

assignment Homework

Energy and Entropy 2021 (2 years)

The isothermal compressibility is defined as \begin{equation} K_{T}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T} \end{equation} $K_T$ is be found by measuring the fractional change in volume when the the pressure is slightly changed with the temperature held constant. In contrast, the adiabatic compressibility is defined as \begin{equation} K_{S}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{S} \end{equation} and is measured by making a slight change in pressure without allowing for any heat transfer. This is the compressibility, for instance, that would directly affect the speed of sound. Show that \begin{equation} \frac{K_{T}}{K_{S}} = \frac{C_{p}}{C_{V}} \end{equation} Where the heat capacities at constant pressure and volume are given by \begin{align} C_{p} &= T \left(\frac{\partial S}{\partial T}\right)_{p} \\ C_{V} &= T \left(\frac{\partial S}{\partial T}\right)_{V} \end{align}

• biotech Microwave oven Ice Calorimetry Lab

biotech Experiment

60 min.

##### Microwave oven Ice Calorimetry Lab
Energy and Entropy 2021 (2 years)

In this remote-friendly activity, students use a microwave oven (and optionally a thermometer) to measure the latent heat of melting for water (and optionally the heat capacity). From these they compute changes in entropy. See also Ice Calorimetry Lab.
• assignment Power from the Ocean

assignment Homework

##### Power from the Ocean
heat engine efficiency Energy and Entropy 2021 (2 years)

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.

• 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.

Learning Outcomes