A valuable model for figuring out how we're going to save the Earth
Let's start by visualizing the energy flow associated with driving a gasoline-powered car. We will use a box and arrow diagram, where boxes represent where energy can accumulate, and arrows show energy flow.
The energy clearly starts in the form of gasoline in the tank. Where does it go?
Actually ask this of students.
Visualize the energy as an indestructable, incompressible liquid.
“Energy is conserved”
The heat can look like
Hot exhaust gas
The radiator (its job is to dissipate heat)
Friction heating in the drive train
The work contribute to
Rubber tires heated by deformation
Wind, which ultimately ends up as heating the atmosphere
The most important factors for a coarse-grain model of highway driving:
The 75:25 split between “heat” and “work”
The trail of wind behind a car
What might we have missed? Where else might energy have gone?
We ignored the kinetic energy of the car, and the energy dissipated as heat in the brakes. On the interstate this is appropriate, but for city driving the dominant “work” may be in accelerating the car to 30 mph, and with that energy then converted into heat by the brakes.
Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.
Consider a column of atoms each of mass \(M\) at temperature \(T\) in
a uniform gravitational field \(g\). Find the thermal average
potential energy per atom. The thermal average kinetic energy is
independent of height. Find the total heat capacity per atom. The
total heat capacity is the sum of contributions from the kinetic
energy and from the potential energy. Take the zero of the
gravitational energy at the bottom \(h=0\) of the column. Integrate
from \(h=0\) to \(h=\infty\). You may assume the gas is ideal.
Consider a system of
fixed volume in thermal contact with a resevoir. Show that the mean
square fluctuations in the energy of the system is \begin{equation}
\left<\left(\varepsilon-\langle\varepsilon\rangle\right)^2\right>
= k_BT^2\left(\frac{\partial U}{\partial T}\right)_{V}
\end{equation} Here \(U\) is the conventional symbol for
\(\langle\varepsilon\rangle\). Hint: Use the partition function
\(Z\) to relate \(\left(\frac{\partial U}{\partial T}\right)_V\) to
the mean square fluctuation. Also, multiply out the term
\((\cdots)^2\).
A one-dimensional
harmonic oscillator has an infinite series of equally spaced energy
states, with \(\varepsilon_n = n\hbar\omega\), where \(n\) is an
integer \(\ge 0\), and \(\omega\) is the classical frequency of the
oscillator. We have chosen the zero of energy at the state \(n=0\)
which we can get away with here, but is not actually the zero of
energy! To find the true energy we would have to add a
\(\frac12\hbar\omega\) for each oscillator.
Show that for a harmonic oscillator the free energy is
\begin{equation}
F = k_BT\log\left(1 - e^{-\frac{\hbar\omega}{k_BT}}\right)
\end{equation} Note that at high temperatures such that
\(k_BT\gg\hbar\omega\) we may expand the argument of the logarithm
to obtain \(F\approx k_BT\log\left(\frac{\hbar\omega}{kT}\right)\).
From the free energy above, show that the entropy is
\begin{equation}
\frac{S}{k_B} =
\frac{\frac{\hbar\omega}{kT}}{e^{\frac{\hbar\omega}{kT}}-1}
- \log\left(1-e^{-\frac{\hbar\omega}{kT}}\right)
\end{equation}
Entropy of a simple harmonic oscillatorHeat capacity of a simple harmonic oscillator
This entropy is shown in the nearby figure, as well
as the heat capacity.
The goal of this problem is
to show that once we have maximized the entropy and found the
microstate probabilities in terms of a Lagrange multiplier \(\beta\),
we can prove that \(\beta=\frac1{kT}\) based on the statistical
definitions of energy and entropy and the thermodynamic definition
of temperature embodied in the thermodynamic identity.
The internal energy and
entropy are each defined as a weighted average over microstates:
\begin{align}
U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i
\end{align}:
We saw in clase that the probability of each microstate can be given
in terms of a Lagrange multiplier \(\beta\) as
\begin{align}
P_i &= \frac{e^{-\beta E_i}}{Z}
&
Z &= \sum_i e^{-\beta E_i}
\end{align}
Put these probabilities into the above weighted averages in
order to relate \(U\) and \(S\) to \(\beta\). Then make use of the
thermodynamic identity
\begin{align}
dU = TdS - pdV
\end{align}
to show that \(\beta = \frac1{kT}\).
You are given the following Gibbs free energy:
\begin{equation*}
G=-k T N \ln \left(\frac{a T^{5 / 2}}{p}\right)
\end{equation*}
where \(a\) is a constant (whose dimensions make the argument of the logarithm dimensionless).
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 (Hint: find the volume as a partial derivative of the Gibbs free energy).
A student is invited to “act out” motion corresponding to a plot of effective potential vs. distance. The student plays the role of the “Earth” while the instructor plays the “Sun”.
Students consider the change in internal energy during three different processes involving a container of water vapor on a stove. Using the 1st Law of Thermodynamics, students reason about how the internal energy would change and then compare this prediction with data from NIST presented as a contour plot.
Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.
Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.
Find the equilibrium value at temperature \(T\)
of the fractional magnetization \begin{equation}
\frac{\mu_{tot}}{Nm} \equiv \frac{2\langle s\rangle}{N}
\end{equation} of a system of \(N\) spins each of magnetic moment
\(m\) in a magnetic field \(B\). The spin excess is \(2s\). The energy
of this system is given by \begin{align}
U &= -\mu_{tot}B
\end{align} where \(\mu_{tot}\) is the total magnetization. Take the
entropy as the logarithm of the multiplicity \(g(N,s)\) as given in
(1.35 in the text): \begin{equation}
S(s) \approx k_B\log g(N,0) - k_B\frac{2s^2}{N}
\end{equation} for \(|s|\ll N\), where \(s\) is the spin excess, which
is related to the magnetization by \(\mu_{tot} = 2sm\). Hint:
Show that in this approximation \begin{equation}
S(U) = S_0 - k_B\frac{U^2}{2m^2B^2N},
\end{equation} with \(S_0=k_B\log g(N,0)\). Further, show that
\(\frac1{kT} = -\frac{U}{m^2B^2N}\), where \(U\) denotes
\(\langle U\rangle\), the thermal average energy.
Find the entropy of a set of \(N\) oscillators of frequency
\(\omega\) as a function of the total quantum number \(n\). Use the
multiplicity function: \begin{equation}
g(N,n) = \frac{(N+n-1)!}{n!(N-1)!}
\end{equation} and assume that \(N\gg 1\). This means you can
make the Sitrling approximation that
\(\log N! \approx N\log N - N\). It also means that
\(N-1 \approx N\).
Let \(U\) denote the total energy \(n\hbar\omega\) of the
oscillators. Express the entropy as \(S(U,N)\). Show that the total
energy at temperature \(T\) is \begin{equation}
U = \frac{N\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1}
\end{equation} This is the Planck result found the hard
way. We will get to the easy way soon, and you will never again need
to work with a multiplicity function like this.
The goal of this problem
is to show that once we have maximized the entropy and found the
microstate probabilities in terms of a Lagrange multiplier \(\beta\),
we can prove that \(\beta=\frac1{kT}\) based on the statistical
definitions of energy and entropy and the thermodynamic definition of
temperature embodied in the thermodynamic identity.
The internal energy and entropy are each defined as a weighted average
over microstates: \begin{align}
U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i
\end{align} We saw in clase that the probability of each microstate
can be given in terms of a Lagrange multiplier \(\beta\) as
\begin{align}
P_i &= \frac{e^{-\beta E_i}}{Z}
&
Z &= \sum_i e^{-\beta E_i}
\end{align} Put these probabilities into the above weighted averages
in order to relate \(U\) and \(S\) to \(\beta\). Then make use of the
thermodynamic identity \begin{align}
dU = TdS - pdV
\end{align} to show that \(\beta = \frac1{kT}\).
Found in: Thermal and Statistical Physics course(s)
Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
The Gibbs free energy,
\(G\), is given by
\begin{align*}
G = U + pV - TS.
\end{align*}
Find the total differential of \(G\). As always, show your work.
Interpret the coefficients of the total differential \(dG\) in
order to find a derivative expression for the entropy \(S\).
From the total differential \(dG\), obtain a different
thermodynamic derivative that is equal to
\[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]
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?
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.
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)?
Consider a system which has an internal energy \(U\) 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?
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:
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.
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.
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?
Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s)
(Messy algebra) Convince yourself that the expressions for kinetic energy in original and center of mass coordinates are equivalent. The same for angular momentum.
Consider a system of two particles of mass \(m_1\) and \(m_2\).
Show that the total kinetic energy of the system is the same as that of two
“fictitious” particles: one of mass \(M=m_1+m_2\) moving with the velocity of the
center of mass and one of mass \(\mu\) (the reduced mass) moving with the
velocity of the relative position.
Show that the total angular momentum of the system can similarly be decomposed
into the angular momenta of these two fictitious particles.
Groups are asked to analyze the following standard problem:
Two identical lumps of clay of (rest) mass m collide head on, with each
moving at 3/5 the speed of light. What is the mass of the resulting lump of
clay?
As discussed in
class, we can consider a black body as a large box with a small hole
in it. If we treat the large box a metal cube with side length \(L\)
and metal walls, the frequency of each normal mode will be given by:
\begin{align}
\omega_{n_xn_yn_z} &= \frac{\pi c}{L}\sqrt{n_x^2 + n_y^2 + n_z^2}
\end{align} where each of \(n_x\), \(n_y\), and \(n_z\) will have
positive integer values. This simply comes from the fact that a half
wavelength must fit in the box. There is an additional quantum number
for polarization, which has two possible values, but does not affect
the frequency. Note that in this problem I'm using different
boundary conditions from what I use in class. It is worth learning to
work with either set of quantum numbers. Each normal mode is a
harmonic oscillator, with energy eigenstates \(E_n = n\hbar\omega\)
where we will not include the zero-point energy
\(\frac12\hbar\omega\), since that energy cannot be extracted from the
box. (See the
Casimir effect
for an example where the zero point energy of photon modes does have
an effect.)
Note
This is a slight approximation, as the boundary conditions for light
are a bit more complicated. However, for large \(n\) values this gives
the correct result.
Show that the free energy is given by \begin{align}
F &= 8\pi \frac{V(kT)^4}{h^3c^3}
\int_0^\infty \ln\left(1-e^{-\xi}\right)\xi^2d\xi
\\
&= -\frac{8\pi^5}{45} \frac{V(kT)^4}{h^3c^3}
\\
&= -\frac{\pi^2}{45} \frac{V(kT)^4}{\hbar^3c^3}
\end{align} provided the box is big enough that
\(\frac{\hbar c}{LkT}\ll 1\). Note that you may end up with a
slightly different dimensionless integral that numerically evaluates
to the same result, which would be fine. I also do not expect you to
solve this definite integral analytically, a numerical confirmation
is fine. However, you must manipulate your integral until it
is dimensionless and has all the dimensionful quantities removed
from it!
Show that the entropy of this box full of photons at temperature
\(T\) is \begin{align}
S &= \frac{32\pi^5}{45} k V \left(\frac{kT}{hc}\right)^3
\\
&= \frac{4\pi^2}{45} k V \left(\frac{kT}{\hbar c}\right)^3
\end{align}
Show that the internal energy of this box full of photons at
temperature \(T\) is \begin{align}
\frac{U}{V} &= \frac{8\pi^5}{15}\frac{(kT)^4}{h^3c^3}
\\
&= \frac{\pi^2}{15}\frac{(kT)^4}{\hbar^3c^3}
\end{align}
For electrons with an energy \(\varepsilon\gg mc^2\), where
\(m\) is the mass of the electron, the energy is given by
\(\varepsilon\approx pc\) where \(p\) is the momentum. For electrons
in a cube of volume \(V=L^3\) the momentum takes the same values as
for a non-relativistic particle in a box.
Show that in this extreme relativistic limit the Fermi energy of a
gas of \(N\) electrons is given by \begin{align}
\varepsilon_F &= \hbar\pi c\left(\frac{3n}{\pi}\right)^{\frac13}
\end{align} where \(n\equiv \frac{N}{V}\) is the number density.
Show that the total energy of the ground state of the gas is
\begin{align}
U_0 &= \frac34 N\varepsilon_F
\end{align}
In our week on radiation, we saw that the Helmholtz free energy of a
box of radiation at temperature \(T\) is \begin{align}
F &= -8\pi \frac{V(kT)^4}{h^3c^3}\frac{\pi^4}{45}
\end{align} From this we also found the internal energy and entropy
\begin{align}
U &= 24\pi \frac{(kT)^4}{h^3c^3}\frac{\pi^4}{45} V \\
S &= 32\pi kV\left(\frac{kT}{hc}\right)^3 \frac{\pi^4}{45}
\end{align} Given these results, let us consider a Carnot engine that
uses an empty metalic piston (i.e. a photon gas).
Given \(T_H\) and \(T_C\), as well as \(V_1\) and \(V_2\) (the two
volumes at \(T_H\)), determine \(V_3\) and \(V_4\) (the two volumes
at \(T_C\)).
What is the heat \(Q_H\) taken up and the work done by the gas
during the first isothermal expansion? Are they equal to each other,
as for the ideal gas?
Does the work done on the two isentropic stages cancel each other,
as for the ideal gas?
Calculate the total work done by the gas during one cycle. Compare
it with the heat taken up at \(T_H\) and show that the energy
conversion efficiency is the Carnot efficiency.
These notes, from the third week of https://paradigms.oregonstate.edu/courses/ph441 cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.
Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.
Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.
Consider a system that may be unoccupied with energy zero, or
occupied by one particle in either of two states, one of energy zero
and one of energy \(\varepsilon\). Find the Gibbs sum for this
system is in terms of the activity \(\lambda\equiv e^{\beta\mu}\).
Note that the system can hold a maximum of one particle.
Solve for the thermal average occupancy of the system in terms of
\(\lambda\).
Show that the thermal average occupancy of the state at energy
\(\varepsilon\) is \begin{align}
\langle N(\varepsilon)\rangle =
\frac{\lambda e^{-\frac{\varepsilon}{kT}}}{\mathcal{Z}}
\end{align}
Find an expression for the thermal average energy of the system.
Allow the possibility that the orbitals at \(0\) and at
\(\varepsilon\) may each be occupied each by one particle at the
same time; Show that \begin{align}
\mathcal{Z} &= 1 + \lambda + \lambda e^{-\frac{\varepsilon}{kT}} +
\lambda^2 e^{-\frac{\varepsilon}{kT}}
\\
&= (1+\lambda)\left(1+e^{-\frac{\varepsilon}{kT}}\right)
\end{align} Because \(\mathcal{Z}\) can be factored as shown, we
have in effect two independent systems.
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.
Consider a white dwarf of mass \(M\) and radius \(R\). The dwarf
consists of ionized hydrogen, thus a bunch of free electrons and
protons, each of which are fermions. Let the electrons be degenerate
but nonrelativistic; the protons are nondegenerate.
Show that the order of magnitude of the gravitational self-energy is
\(-\frac{GM^2}{R}\), where \(G\) is the gravitational constant. (If
the mass density is constant within the sphere of radius \(R\), the
exact potential energy is \(-\frac53\frac{GM^2}{R}\)).
Show that the order of magnitude of the kinetic energy of the
electrons in the ground state is \begin{align}
\frac{\hbar^2N^{\frac53}}{mR^2}
\approx \frac{\hbar^2M^{\frac53}}{mM_H^{\frac53}R^2}
\end{align} where \(m\) is the mass of an electron and \(M_H\) is
the mas of a proton.
Show that if the gravitational and kinetic energies are of the same
order of magnitude (as required by the virial theorem of mechanics),
\(M^{\frac13}R \approx 10^{20} \text{g}^{\frac13}\text{cm}\).
If the mass is equal to that of the Sun (\(2\times 10^{33}g\)), what
is the density of the white dwarf?
It is believed that pulsars are stars composed of a cold degenerate
gas of neutrons (i.e. neutron stars). Show that for a neutron star
\(M^{\frac13}R \approx 10^{17}\text{g}^{\frac13}\text{cm}\). What is
the value of the radius for a neutron star with a mass equal to that
of the Sun? Express the result in \(\text{km}\).
Students observe the motion of a puck tethered to the center of the airtable. Then they plot the potential energy for the puck on their small whiteboards. A class discussion follows based on what students have written on their whiteboards.
This lab gives students a chance to take data on the first day of class (or later, but I prefer to do it the first day of class). It provides an immediate context for thermodynamics, and also gives them a chance to experimentally measure a change in entropy. Students are required to measure the energy required to melt ice and raise the temperature of water, and measure the change in entropy by integrating the heat capacity.
In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.
In this activity students combine energy conservation with the relationship between the de Broglie wavelength and momentum to find the wavelength of atoms that have been dropped a given distance.
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.
Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.
