• ideal gas internal energy engine
• assignment Bottle in a Bottle 2

assignment Homework

##### Bottle in a Bottle 2
heat entropy ideal gas 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).

• 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?

assignment Homework

Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

The isothermal compressibility is defined as $$K_{T}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{T}$$ $K_T$ 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 $$K_{S}=-\frac{1}{V} \left(\frac{\partial V}{\partial p}\right)_{S}$$ 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 $$\frac{K_{T}}{K_{S}} = \frac{C_{p}}{C_{V}}$$ 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}

• group Heat and Temperature of Water Vapor (Remote)

group Small Group Activity

5 min.

##### Heat and Temperature of Water Vapor (Remote)

In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.
• assignment Nucleus in a Magnetic Field

assignment Homework

##### Nucleus in a Magnetic Field
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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 $E=0$.

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

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

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

• assignment Entropy and Temperature

assignment Homework

##### Entropy and Temperature
Energy Temperature Ideal gas Entropy Thermal and Statistical Physics Spring 2020

Suppose $g(U) = CU^{3N/2}$, where $C$ is a constant and $N$ is the number of particles.

1. Show that $U=\frac32 N k_BT$.

2. Show that $\left(\frac{\partial^2S}{\partial U^2}\right)_N$ is negative. This form of $g(U)$ actually applies to a monatomic ideal gas.

• group Using $pV$ and $TS$ Plots

group Small Group Activity

30 min.

##### Using $pV$ and $TS$ Plots
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

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.
• 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).
• assignment Paramagnet (multiple solutions)

assignment Homework

##### Paramagnet (multiple solutions)
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021 We have the following equations of state for the total magnetization $M$, and the entropy $S$ of a paramagnetic system: \begin{align} M&=N\mu\, \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}\\ S&=Nk_B\left\{\ln 2 + \ln \left(e^{\frac{\mu B}{k_B T}}+e^{-\frac{\mu B}{k_B T}}\right) +\frac{\mu B}{k_B T} \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}} \right\} \end{align}
1. List variables in their proper positions in the middle columns of the charts below.

2. Solve for the magnetic susceptibility, which is defined as: $\chi_B=\left(\frac{\partial M}{\partial B}\right)_T$

3. Using both the differentials (zapping with d) and chain rule diagram methods, find a chain rule for:

$\left(\frac{\partial M}{\partial B}\right)_S$

4. Evaluate your chain rule. Sense-making: Why does this come out to zero?

• Energy and Entropy Fall 2020

A diesel engine requires no spark plug. Rather, the air in the cylinder is compressed so highly that the fuel ignites spontaneously when sprayed into the cylinder.

In this problem, you may treat air as an ideal gas, which satisfies the equation $pV = Nk_BT$. You may also use the property of an ideal gas that the internal energy depends only on the temperature $T$, i.e. the internal energy does not change for an isothermal process. For air at the relevant range of temperatures the heat capacity at fixed volume is given by $C_V=\frac52Nk_B$, which means the internal energy is given by $U=\frac52Nk_BT$.

Note: in this problem you are expected to use only the equations given and fundamental physics laws. Looking up the formula in a textbook is not considered a solution at this level.

1. If the air is initially at room temperature (taken as $20^{o}C$) and is then compressed adiabatically to $\frac1{15}$ of the original volume, what final temperature is attained (before fuel injection)?

2. By what factor does the pressure increase?