## Bottle in a Bottle 2

• heat entropy ideal gas
• assignment Bottle in a Bottle

assignment Homework

##### Bottle in a Bottle
irreversible helium internal energy work first law Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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?

• assignment Free Expansion

assignment Homework

##### Free Expansion
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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

2. What is the change in temperature of the gas?

60 min.

##### Free expansion
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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.
• assignment Icecream Mass

assignment Homework

##### Icecream Mass
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Use integration to find the total mass of the icecream in a packed cone (both the cone and the hemisphere of icecream on top).

• assignment Center of Mass for Two Uncoupled Particles

assignment Homework

##### Center of Mass for Two Uncoupled Particles
Central Forces Spring 2021

Consider two particles of equal mass $m$. The forces on the particles are $\vec F_1=0$ and $\vec F_2=F_0\hat{x}$. If the particles are initially at rest at the origin, find the position, velocity, and acceleration of the center of mass as functions of time. Solve this problem in two ways, with or without theorems about the center of mass motion. Write a short description comparing the two solutions.

• group Outer Product of a Vector on Itself

group Small Group Activity

30 min.

##### Outer Product of a Vector on Itself

Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).
• face Energy and heat and entropy

face Lecture

30 min.

##### Energy and heat and entropy
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

This short lecture introduces the ideas required for Ice Calorimetry Lab or Microwave oven Ice Calorimetry Lab.
• assignment Potential vs. Potential Energy

assignment Homework

##### Potential vs. Potential Energy
AIMS Maxwell AIMS 21 Static Fields Winter 2021

In this course, two of the primary examples we will be using are the potential due to gravity and the potential due to an electric charge. Both of these forces vary like $\frac{1}{r}$, so they will have many, many similarities. Most of the calculations we do for the one case will be true for the other. But there are some extremely important differences:

1. Find the value of the electrostatic potential energy of a system consisting of a hydrogen nucleus and an electron separated by the Bohr radius. Find the value of the gravitational potential energy of the same two particles at the same radius. Use the same system of units in both cases. Compare and the contrast the two answers.
2. Find the value of the electrostatic potential due to the nucleus of a hydrogen atom at the Bohr radius. Find the gravitational potential due to the nucleus at the same radius. Use the same system of units in both cases. Compare and contrast the two answers.
3. Briefly discuss at least one other fundamental difference between electromagnetic and gravitational systems. Hint: Why are we bound to the earth gravitationally, but not electromagnetically?

• group Electrostatic Potential Due to a Ring of Charge

group Small Group Activity

30 min.

##### Electrostatic Potential Due to a Ring of Charge
AIMS Maxwell AIMS 21 AIMS Maxwell AIMS 21 Static Fields Winter 2021

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use the superposition principle $V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}$ to find an integral expression for the electrostatic potential, $V(\vec{r})$, everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for $V(\vec{r})$ either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• computer Approximating Functions with Power Series

computer Computer Simulation

30 min.

##### Approximating Functions with Power Series
AIMS Maxwell AIMS 21 Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021 Central Forces Spring 2021 Static Fields Winter 2021

Power Series Sequence (E&M)

Students use a prepared Mathematica notebook to plot $\sin\theta$ simultaneously with several terms of a power series expansion to judge how well the approximation fits. Students can alter the worksheet to change the number of terms in the expansion and even to change the function that is being considered. Students should have already calculated the coefficients for a power series expansion in a previous activity, Calculating Coefficients for a Power Series.
• Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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 $U=\frac32 Nk_BT$.

1. How many molecules of gas does the large bottle contain? What is the final temperature of the gas?

2. 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).