- irreversible helium internal energy work first law
*assignment*Bottle in a Bottle 2*assignment*Homework##### Bottle in a Bottle 2

heat entropy ideal gas Energy and Entropy 2021 (2 years)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 m

^{23}and the volume of the big bottle is 0.01 m^{3}. 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).

- Discuss your results.

*assignment*Directional Derivative*assignment*Homework##### Directional Derivative

Static Fields 2022 (6 years)You are on a hike. The altitude nearby is described by the function \(f(x, y)= k x^{2}y\), where \(k=20 \mathrm{\frac{m}{km^3}}\) is a constant, \(x\) and \(y\) are east and north coordinates, respectively, with units of kilometers. You're standing at the spot \((3~\mathrm{km},2~\mathrm{km})\) and there is a cottage located at \((1~\mathrm{km}, 2~\mathrm{km})\). You drop your water bottle and the water spills out.

- Plot the function \(f(x, y)\) and also its level curves in your favorite plotting software. Include images of these graphs. Special note: If you use a computer program written by someone else, you must reference that appropriately.
- In which direction in space does the water flow?
- At the spot you're standing, what is the slope of the ground in the direction of the cottage?
- Does your result to part (c) make sense from the graph?

*grading*Free expansion*grading*Quiz60 min.

##### Free expansion

Energy and Entropy 2021 (2 years) Students will determine the change in entropy (positive, negative, or none) for both the system and surroundings in three different cases. This is followed by an active whole-class discussion about where the entropy comes from during an irreversible process.*group*Quantifying Change*group*Small Group Activity30 min.

##### Quantifying Change

In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.*face*Phase transformations*face*Lecture120 min.

##### Phase transformations

Thermal and Statistical Physics 2020phase transformation Clausius-Clapeyron mean field theory thermodynamics

These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.*assignment*Symmetry Arguments for Gauss's Law*assignment*Homework##### Symmetry Arguments for Gauss's Law

Static Fields 2022 (5 years)Instructions for 2022: You will need to complete this assignment in a 15 minute appointment on Zoom or in person with one of the members of the teaching team between 1/21 and 10 pm on 1/26. Here is a link to a sign-up page.

You are required to watch a sample video for how to make symmetry arguments here. As demonstrated in the video you should bring with you to the meeting a cylinder, an observer, and a vector.

Use good symmetry arguments to find the possible direction for the electric field due to a charged wire. Also, use good symmetry arguments to find the possible functional dependence of the electric field due to a charged wire. Rather than writing this up to turn in, you should find a member of the teaching team and make the arguments to them verbally.

*group*A glass of water*group*Small Group Activity30 min.

##### A glass of water

Energy and Entropy 2021 (2 years)thermodynamics intensive extensive temperature volume energy entropy

Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.*face*Thermal radiation and Planck distribution*face*Lecture120 min.

##### Thermal radiation and Planck distribution

Thermal and Statistical Physics 2020Planck distribution blackbody radiation photon statistical mechanics

These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.-
Energy and Entropy 2021 (2 years)
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? -
###### Media & Figures