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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

- Thermodynamic Identity
- Name the Experiment I
- Heat Capacity

**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*

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

- (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 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.*