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 Small Group Activity
schedule
30 min.
build
Heat Capacity graph, Student handout for each student, A personal or shared writing space for each student to write/draw/sketch.
description Student handout(PDF)
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
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: the ratio of the amount of heat needed to change the temperature a small amount and the change in temperature (i.e., the amount of heat transferred per change in temperature).
The graph provided by your instructor shows internal energy and volume contours plotted on temperature and pressure axes.
Estimate the Temperature-Responsiveness: 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 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.
Answer: We want the ratio of change in internal energy and the change in temperature along the path of constant volume. It is the ratio of small changes, not a slope on this graph.
Discussion: Which derivative? We haven't yet 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. It's nice to have done the Quantifying Change activity before this one. Some correct answers might look like: \({\mathit{\unicode{273}}} Q/dT\), \(\left(dU/dT\right)_V\), \(Q/\Delta T\), \(\left(\Delta U/\Delta T\right)_V\).
Student Ideas: Students try to use equipartition theorem or other equations, rather than directly measuring the rate on the graph. Students might not always realize that work done is zero, or how this would relate U and Q.
Follow-Up: For what graph would the slope be the rate you're looking for? How is that graph related to the surface?
Reflect on Your Process: Why do you think it matters that you held volume constant in the above estimate?
Answer: 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 Dependencies on State Variables: Does the value of your estimate depend on the value of the volume? the temperature?
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:
How would you estimate the heat capacity at constant PRESSURE?
Answer: The most straightforward way is to account for the work done along the constant pressure path and subtract that from the change in internal energy.)
Which value is bigger: \(C_V\) or \(C_p\)? Why do you think that is?
What procedure would you use to you estimate \(C_v\) from the purple surface?
SUMMARY PAGE
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 contour map
Student worksheet for each student
A personal or shared writing space for each student to write/draw/sketch.
Optional: Blue \(U(T,p)\) plastic surface for each group
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.
Potential energyHeat capacityThermal 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.
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).
Compute the entropy.
Work out the heat capacity at constant pressure \(C_p\).
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).
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.
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 Dulong-Petit 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.
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.
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}
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, from which they compute changes in entropy.
Consider the bottle in a bottle problem in a previous problem set, summarized here.
A small bottle of
helium is placed inside a large bottle, which otherwise contains
vacuum. The inner bottle contains a slow leak, so that the helium
leaks into the outer bottle. The inner bottle contains one tenth
the volume of the outer bottle, which is insulated.
The volume of the small bottle is 0.001 m23 and the volume of the big bottle is 0.01 m3. The initial state of the gas in the small bottle was \(p=106\) Pa and its temperature \(T=300\) K. Approximate the helium gas as an ideal gas of equations of state \(pV=Nk_BT\) and \(U=\frac32 Nk_BT\).
How many molecules of gas does the large bottle contain? What is the final temperature of the gas?
Compute the integral \(\int \frac{{\mathit{\unicode{273}}} Q}{T}\) and the change of entropy \(\Delta S\) between the initial state (gas in the small bottle) and the final state (gas leaked in the big bottle).
