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.
Heat transfer as area under a curve in a \(TS\) plot
Reminder that internal energy is a state function
Reminder of First Law
Media
Before starting this activity, students do need to be taught about heat and
work as \({\mathit{\unicode{273}}} Q = TdS\) and \({\mathit{\unicode{273}}} W = -pdV\). If this did not happen recently,
it might be worth reminding students of these relationships. Otherwise, the
handout basically tells them what to do.
\(pV\) rectangle
We'll begin today looking at processes that happen at constant
pressure, volume, temperature or entropy.
We have spent a lot of time discussing partial derivatives and differentials,
and how they relate to measurements. Another sort of measurement involves
integrals rather than derivatives, and measures finite
changes. To discuss this, it's common to use what are called \(pV\)
diagrams. For instance, consider the following square.
Note You cannot assume that the material in question is an ideal
gas. It could equally well be a mixture of ice and water, or a strange
gooey substance that you found on a distant planet. You can
never assume that we are talking about an ideal gas unless we tell
you so!
In your groups, work out the following questions:
What does this figure describe? Is \(p\) a function of \(V\)?
What is the net work done after one cycle of this
process? How much work was done at each step?
What is the net energy transfered by heating over one cycle of this process?
Try to find the energy transfered by heating at each step.
Student Conversations
Some students will claim that because we return to the original state, the net work (or possibly net heat) must be zero.
Many students will not recognize that on a \(pV\) graph, the work is just (the negative of) the area under the curve. While I wouldn't want them to think of this as a definition of work, it is a necessary and useful tool to have in their toolbox.
It is helpful to ask students to describe what is happening in each stage (e.g. “it is being compressed at fixed pressure... so probably its temperature is dropping, unless it is something weird like ice”).
Some students assume that temperature is fixed when working on the \(pV\) plot, because it isn't listed.
Many students will try to invoke or use the ideal gas law.
\(TS\) rectangle
Now let's look at another cycle. Let's consider the following
figure, which looks similar, but is now a plot of \(T\) vs. \(S\), and answer
the following questions:
What is this cycle? How would you go about running a cycle like this?
What is the net heat transfer over one cycle of this process? How much was transfered on each step?
What is the net work done after one cycle of this
process? How much work was done at each step?
Student Conversations
Although the math using \(TS\) diagram is effectively identical to that with the \(pV\) diagram (with heat and work swapped, and a few minus signs), students will still find the \(TS\) diagram challenging.
Wrap-up
Bring the class back together and have a group argue if the work is positive or negative on each leg of either the \(pV\) or \(TS\) curve (instructor's preference of which to analyze first). Help groups resolve any inconsistencies in answers for the work and heat of each leg.
Many students become confused as to if work is being done on or by the system in these plots because of the minus sign associated with the thermodynamic identity term containing \(pdV\). Be sure to point out that if the volume of the system increases, then work is being done by the system; show this by analyzing a leg of the \(pV\) curve where the pressure remains constant and the volume increases and display that the total work of the system would be negative.
The analysis of the \(TS\) curve should be nearly identical to the \(pV\) curve. Be sure to note that the minus sign that appears in the work term is no longer present in the heat expression.
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).
thermodynamicsMaxwell relationEnergy and Entropy 2020
The Gibbs free energy,
\(G\), is given by
\begin{align*}
G = U + pV - TS.
\end{align*}
Find the total differential of \(G\). As always, show your work.
Interpret the coefficients of the total differential \(dG\) in
order to find a derivative expression for the entropy \(S\).
From the total differential \(dG\), obtain a different
thermodynamic derivative that is equal to
\[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]
Stefan-Boltzmannblackbody radiationThermal and Statistical Physics 2020
A black (nonreflective) sheet of metal at high
temperature \(T_h\) is parallel to a cold black sheet of metal at temperature
\(T_c\). Each sheet has an area \(A\) which is much greater than the distance between them. The sheets are in vacuum, so energy can only be transferred by radiation.
Solve for the net power transferred between the two sheets.
A third black metal sheet is
inserted between the other two and is allowed to come to a steady
state temperature \(T_m\). Find the temperature of the middle sheet,
and solve for the new net power transferred between the hot and cold sheets.
This is the principle of the heat shield, and is part of how the James Web telescope shield works.
Optional: Find the power through an \(N\)-layer sandwich.
The internal energy is of any ideal gas can be written as
\begin{align}
U &= U(T,N)
\end{align}
meaning that the internal energy depends only on the number of
particles and the temperature, but not the volume.* The ideal gas law
\begin{align}
pV &= Nk_BT
\end{align}
defines the relationship between \(p\), \(V\) and \(T\). You may take the
number of molecules \(N\) to be constant. Consider the free adiabatic
expansion of an ideal gas to twice its volume. “Free expansion”
means that no work is done, but also that the process is also
neither quasistatic nor reversible.
What is the change in entropy of the gas? How do you know
this?
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).
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.
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 \(P_B\)?
Write down the total differential of your savings, in terms of
\(B\), \(C\), \(P_B\) and \(P_C\).
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.
