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?
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?
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.
What is the change in entropy of the gas? How do you know
this?
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.
Students consider projectile motion of an object that experiences drag force that in linear with the velocity. Students consider the horizontal motion and the vertical motion separately. Students solve Newton's 2nd law as a differential equation.
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.
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.
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 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 \(U=\frac32 Nk_BT\).
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).
