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### Unit: Math Bits

**Hour 1: Math Bits - Differentials**- Differentials
- Zapping with d
*Surfaces Activity: Covariation in Thermal Systems*

- QUIZ
*Small Group Activity: Exploring the Partial Derivative Machine*- Contour Graphs
*Surfaces Activity: Thermodynamic States*

*Small Group Activity: Exploring the Partial Derivative Machine*- Partial Derivative Machine dictionary

**Hour 4: Math Bits - Energy and Integrals***dU*- Numerical Integration
*Surfaces Activity: Quantifying Change in Thermal Systems*

**Hours 5-6: Math Bits - Derivatives and Chain Rules**- Chain Rules
*Small Group Activity: PDM Derivatives*

- Chain Rule Diagrams

*Surfaces Activity: “Squishability” of Water Vapor*

### Unit: First Law of Thermodynamics

**Hour 8: Heat and Temperature**- Intro to Thermodynamic Properties
*Lab: Ice Lab I*

- Comparing Thermodynamic Properties

**Hour 9: The first law**- QUIZ
- Lecture: Jargon
- Intensive and Extensive
- T, S, V, p, U
- Work, Heat
- The First Law

- Snapping a Rubber Band

- (CoffeeAndBagels)
*What goes here?*\newcommand\juice{\ensuremath{\mathcal{O}}} \newcommand\cocoa{\ensuremath{\mathcal{C}}} \newcommand\priceorange{\ensuremath{\mathcal{P_O}}} \newcommand\pricecocoa{\ensuremath{\mathcal{P_C}}}

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.\footnote{Yes, this is ridiculous. It would be slightly less ridiculous if we were talking about nations and commodities, but also far less humorous.}

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 haveThe 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 $P_C$.

Use the equality of mixed partial derivatives (Clairut's theorem) to find a relationship between $P_B$, $P_C$, $B$ and $C$. Write this relationship mathematically, and also describe in words what it means.

Solve for the total differential of your net worth. Once again use Clairut's theorem considering second derivatives of $W$ to find a different partial derivative relationship between $P_B$, $P_C$, $B$ and $C$.

- (ExtensiveIntensiveChecking)
*What goes here?*For each of the following equations, check whether it could possibly make sense. You will need to check both dimensions and whether the quantities involved are intensive or extensive. For each equation, explain your reasoning.

You may assume that quantities with subscripts such as $V_0$ have the same dimensions and intensiveness/extensiveness as they would have without the subscripts.

\[p = \frac{N^2k_BT}{V}\]

\[p = \frac{Nk_BT}{V}\]

\[U = \frac32 k_BT\]

\[U = - Nk_BT \ln\frac{V}{V_0}\]

\[S = - k_B \ln\frac{V}{V_0}\]

\[S = - k_B \ln\frac{V}{N}\]

- (ExtensiveInternalEnergy)
*What goes here?*Consider a system which has an internal energy defined by: \begin{align} U &= \gamma V^\alpha S^\beta \end{align} where $\alpha$, $\beta$ and $\gamma$ are constants. The internal energy is an extensive quantity. What constraint does this place on the values $\alpha$ and $\beta$ may have?

- Thermodynamic Identity
- Name the Experiment I
- Heat Capacity

- (FirstLaw) PRACTICE
*What goes here?*You heat an insulated piston with a resistor. You run 5 A through the resistor at 10 V for a total of 10 seconds. The pressure is fixed at 1 Pa (which is 1 N/m$^2$). If the system expands by 0.1 m$^3$, what is the change in internal energy of the system?

You heat an insulated piston with a resistor. You run 5 A through the resistor at 5 V for a total of 10 seconds. The pressure is fixed at 1 Pa (which is 1 N/m$^2$). If the system expands by 0.5 m$^3$, what is the change in internal energy of the system?

Consider and insulated cylinder full of an ideal gas, whose internal energy is given by \[U=\frac{3}{2} N k_B T\] What happens to the temperature of the gas when I compress the insulated piston? Why?

- (BottleInBottle)
*What goes here?*The internal energy of helium gas at temperature $T$ is to a very good approximation given by \begin{align} U &= \frac32 Nk_BT \end{align} Consider a very irreversible process in which 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. What is the change in temperature when this process is complete? How much of the helium will remain in the small bottle? \begin{center} \includegraphics[width=3in]{../figs/bottle-in-bottle} \end{center}

: ADD IMAGE

**Hour 11: Heat capacity and latent heat***Lab: Ice Lab II*

- This should go somewhere around here once things get rearranged:
*Heat Capacity of Water Vapor*

- (PDMElevator)
*What goes here?***Elevator**The table of data in the spreadsheet (available on the course website and reproduced in the table below) shows the value of each state variable at different points in an elevator cycle on the Partial Derivative Machine. (Note: You may find it useful to do calculations using the spreadsheet or another computer program. If you do so, please include both a printout of your calculations and any formulas that you used in the spreadsheet.): INSERT TABLE OF DATA

Calculate the work done by each force ($F_L$ and $F_R$) during each of the four processes (loading, raising, unloading, and lowering). Discuss your method for calculating the work: do not assume a functional for for the data. Include relevant graphs of the data and give a physical explanation for the sign of each work.

What is the total energy of the system at the end of each process (loading, raising, unloading, and lowering)? Give a physical explanation for the total energy of the system after the elevator has returned to the initial state (ground floor, empty).

Calculate the following derivatives for at least two different states each (indicate the states you chose!): $\left(\frac{\partial X_R}{\partial F_R}\right)_{F_L}$ (“iso-force”) and $\left(\frac{\partial X_R}{\partial F_R}\right)_{X_L}$ (“iso-position”). Describe your method in detail and include any graphs you used.

The data you have been given was taken from real experiments done using the partial derivative machine — and some of the data wasn't perfect! Discuss how you chose to handle this during your calculations.

**Hour 12: Mechanical cycles**- Elevators
*Small Group Activity: The Elevator Cycle*

**Hour 13: Heat, work, and processes**- Work
- Heat
- Using $p V$ and $T S$ Plots

- (HeatAndWork)
*What goes here?***Heat and work**FOr each of the following processes, solve for the heat or work done.A system expands from initial volume $V_0$ to final volume $V_f$. During the process the pressure is given by \[p=\frac{N k_b T}{V}\] where $k_B T$ and $N$ are constant. How much work does the system do on its environment?

A system is heated from initial entropy $S_0$ to final entropy $S_f$. During this process the temperature is held fixed. How much energy is transferred into the system by heating during this process?

A system is heated from initial entropy $S_0$ to final entropy $S_f$. During this process the temperature is given by \[T=T_0 + \frac{S-S_0}{C_V}\] where $T_0$ and $C_V$ are constants. How much energy is transferred into the system by heating during this process?

A system expands from initial volume $V_0$ to final volume $V_f$. During the process the pressure is given by \[p=p_0 \left( \frac{V_0}{V} \right) ^\gamma \] where $p_0$ is the initial pressure. How much work does the system do on its environment?

**Hour 14: A simple cycle**- Analyzing a Simple Curve

- (SimpleCycleInSTDiagram)
*What goes here?***Simple cycle in S-T diagram**(This problem is identical to the in class group exercise) Consider the cycle below. This cycle represents a heat engine, in which energy is transferred by heating from a hot thermal reservoir, work is done, and energy is transferred by heating to a cold thermal reservoir.: ADD PICTURE, PROMPT IS VERY CONFUSING

What is the heat in each step?

What is the net heat for the cycle?

What is the net work done for the cycle? What can you say about the work done in each step?

What is the efficiency of the system as a function of the temperatures of the hot and the cold reservoirs?

**Hour 15: Holiday**

### Unit: Internal Energy

- Quiz
- Name the Experiment II (Heat and/or First Law)

**Hour 17: The second law**- Lecture: Jargon
- Second Law
- Reversible and Irreversible
- Adiabatic
- Quasistatic

- (EntropyChange)
*What goes here?***Entropy change**The following are processes, in which the system is designated in*italics*:a\) A hot

*potato*is left on the counter top.b\) An

*ice cube in a sealed bag*is placed in a glass of room-temperature water.c\) A

*sealed (no air leaks) steel piston of air*is slowly compressedFor each process, answer the following questions. For each question, give a brief explanation. Answers without explanation will

*not*receive credit.Is the change in entropy of the system positive, negative, zero, or impossible to determine?

Is the change in entropy of the surroundings positive, negative, zero, or impossible to determine?

Is the change in entropy of the system plus surroundings positive, negative, zero, or impossible to determine?

- Free expansion not-quiz
- Discussion of not-quiz

- (FreeExpansion)
*What goes here?*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.\footnote{This relationship happens to be linear at low temperatures, where “low” is defined relative to the energy of the excited states of the molecules or atoms.} 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 temperature of the gas?

What is the change in entropy of the gas? How do you know this?

**Hours 19-20: Legendre Transformations****LINKS THAT WORK:****LINKS THAT DO NOT WORK:**- Legendre Transformations

- (GibbsFreeEnergy)
*What goes here?*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.

Compute the internal energy $U$.

**Hour 21: Maxwell relations**- Maxwell Relations
- Maxwell Relation Activities

- (HardSphereFluid)
*What goes here?***Hard-sphere fluid**A hard-sphere fluid (or gas) is just like and ideal gas except that the gas (fluid) particles are not point particles, but instead have a finite volume. The following equations approximately describe the hard-sphere fluid.\[U=\frac{3}{2} N k_B T\]

\[\eta=\frac{N}{V} \frac{4 \pi}{3} R^3\]

\[\frac{F}{N k_B T}=\ln \left( \frac{N}{V} \left( \frac{3 h^2}{4 \pi m} \frac{N}{U} \right)^\frac32 \right) + \frac{4 \eta - 3 \eta^2}{\left( 1-\eta \right)^2}-\frac12\]

where $k_B$ is a constant, and you may consider the number of spheres $N$ as a constant for this problem. Similarly the mass $m$, the radius $R$, and Plank's constant $h$ are constants. In these equations $\eta$ is called the packing fraction, and represents the fraction of the volume that is occupied by hard spheres (which has no impact whatsoever on how you solve this problem).

Solve for \[S \equiv -\left( \frac{\partial F}{\partial T} \right) _V\]

Solve for \[S \equiv -\left( \frac{\partial F}{\partial V} \right) _T\]

Solve for \[\left( \frac{\partial U}{\partial V} \right) _T\]

Solve for \[\left( \frac{\partial U}{\partial V} \right) _S\]

- (SimpleCycleII)
*What goes here?*

- Name The Experiment III

- (IsothermalAdiabaticCompressibility)
*What goes here?*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}

- (NonIdealGas)
*What goes here?*The equation of state of a gas that departs from ideality can be approximated by \[ p=\frac{NkT}{V}\left(1+\frac{NB_{2}(T)}{V}\right) \] where $B_{2}(T)$ is called the second virial coefficient which increases monotonically with temperature. Find $\left( \frac{\partial U}{\partial V}\right)_{T}$ and determine its sign.

**Hours 23-24: Using a Maxwell relation to make an entropy change**- QUIZ
*Lab: Rubber Band Lab*

- Sometimes Always Never True (1st half)

**Hours 26-28: Holiday**

### Unit: Statistical Mechanics

**Hour 29: Statistical view of entropy**- QUIZ
- Fairness Function and Probability

- (ThermodynamicPotentialsAndMaxwellRelations) PRACTICE
*What goes here?*For the three thermodynamic potentials defined as

\begin{quote} \(F=U-T S\) \hfill Helmholtz free energy

\(H=U+P V\) \hfill Enthalpy

\(U-T S+P V\) \hfill Gibbs free energy \end{quote}

determine the total differentials for each thermodynamic potential: $dF$, $dH$, and $dG$. Use the thermodynamic identity ($dU=T dS-p dV$) to simplify.

Identify a Maxwell relation for each of the three potentials.

- (TwoLevelSystem)
*What goes here?***A two level system**A system consists of $N$ identical non-interacting (independent) objects. Each object has two energy eigenstates labeled $A$ and $B$, with energies $E_A=E_0$ and $E_B=-E_0$ ($E_0=\left|E_0 \right|$).Two students argue that this object can't be real because the low energy state is not zero, i.e. $E_B=0$ and $E_A=2 E_0$. What is the effect of the energy of the lowest energy state on thermal averages?

As a function of temperature $T$, what are the probabilities $p_A$ and $p_B$ of finding an object in this system in eigenstates $A$ and $B$?

Calculate the internal energy $U$ of this system as a function of temperature.

Each object has a property $X$, which takes the values of $X_A=+1$ when an object is in state $A$ and $X_B=-1$ when an object is in state $B$. Solve for the average value $\langle X \rangle$ as a function of temperature of the property $X$ for this system.

What are the limits for $T\rightarrow 0$ and $T\rightarrow\inf$ for $\langle X \rangle$?

**Hour 30: Maximizing Entropy**- Maximizing S
- Lagrange Multipliers

- Weighted Averages
- Derive Boltzman Ratio and connect to Thermo ($U = \sum p_iE_i$)

- (BoltzmannRatio)
*What goes here?*At low temperatures, a diatomic molecule can be well described as a

*rigid rotor*. The Hamiltonian of such a system is simply proportional to the square of the angular momentum \begin{align} H &= \frac{1}{2I}L^2 \end{align} and the energy eigenvalues are \begin{align} E_{lm} &= \hbar^2 \frac{l(l+1)}{2I} \end{align}What is the energy of the ground state and the first and second excited states of the $H_2$ molecule?

At room temperature, what is the relative probability of finding a hydrogen molecule in the $l=0$ state versus finding it in any one of the $l=1$ states?\\ i.e. what is $P_{l=0,m=0}/\left(P_{l=1,m=-1} + P_{l=1,m=0} + P_{l=1,m=1}\right)$

At what temperature is the value of this ratio 1?

At room temperature, what is the probability of finding a hydrogen molecule in any one of the $l=2$ states versus that of finding it in the ground state?\\ i.e. what is $P_{l=0,m=0}/\left(P_{l=2,m=-2} + P_{l=2,m=-1} + \cdots + P_{l=2,m=2}\right)$

- (NucleusInMagneticField)
*What goes here?*Nuclei of a particular isotope species contained in a crystal have spin $I=1$, and thus, $m = \{+1,0,-1\}$. The interaction between the nuclear quadrupole moment and the gradient of the crystalline electric field produces a situation where the nucleus has the same energy, $E=\varepsilon$, in the state $m=+1$ and the state $m=-1$, compared with an energy $E=0$ in the state $m=0$, i.e. each nucleus can be in one of 3 states, two of which have energy $E=\varepsilon$ and one has energy $E=0$.

Find the Helmholtz free energy $F = U-TS$ for a crystal containing $N$ nuclei which do not interact with each other.

Find an expression for the entropy as a function of temperature for this system. (Hint: use results of part a.)

Indicate what your results predict for the entropy at the extremes of very high temperature and very low temperature.

**Hour 32: Free**- Free

**Hours 33-34: Exploring a Diatomic Gas**- Diatomic Gas

**Hour 35: Review**- Review

- (HotMetal)
*No HW problem by this name on archive.*