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 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).
Quantum Fundamentals 2023 (2 years)
The following two problems ask you to make Fermi estimates. In a good Fermi estimate, you start from basic scientific facts you already know or quantities that you can reasonably estimate based on your life experiences and then reason your way to estimate a quantity that you would not be able guess. You may look up useful conversion factors or constants. Use words, pictures, and equations to explain your reasoning:
Imagine that you send a pea-sized bead of silver through a Stern-Gerlach device oriented to measure the z-component of intrinsic spin. Estimate the total z-component of the intrinsic spin of the ball you would measure in the HIGHLY improbable case that every atom is spin up.
Protons, neutrons, and electrons are all spin-1/2 particles. Give a (very crude) order of magnitude estimate of the number of these particles in your body.
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.
VaporizationHeatThermal and Statistical Physics 2020
The pressure
of water vapor over ice is 518 Pa at \(-2^\circ\text{C}\). The vapor
pressure of water at its triple point is 611 Pa, at
0.01\(^\circ\text{C}\) (see
Estimate in \(\text{J
mol}^{-1}\) the heat of vaporization of ice just under freezing. How
does this compare with the heat of vaporization of water?
Clausius-ClapeyronThermal and Statistical Physics 2020
Calculate based on the Clausius-Clapeyron equation the value of
\(\frac{dT}{dp}\) near \(p=1\text{atm}\) for the liquid-vapor
equilibrium of water. The heat of vaporization at
\(100^\circ\text{C}\) is \(2260\text{ J g}^{-1}\). Express the result
in kelvin/atm.
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.
Consider one mole of an
ideal monatomic gas at 300K and 1 atm. First, let the gas expand
isothermally and reversibly to twice the initial volume; second, let
this be followed by an isentropic expansion from twice to four times
the original volume.
How much heat (in joules) is added to the gas in each of these two
processes?
What is the temperature at the end of the second process?
Suppose the first process is replaced by an irreversible expansion
into a vacuum, to a total volume twice the initial volume. What is
the increase of entropy in the irreversible expansion, in J/K?
Energy and Entropy 2021 (2 years)
The spring constant \(k\) for a one-dimensional spring is defined by:
\[F=k(x-x_0).\]
Discuss briefly whether each of the variables in this equation is extensive or intensive.
Energy and Entropy 2021 (2 years)
Use the NIST web site “Thermophysical Properties of Fluid Systems” to answer the following questions. This site is an excellent resource for finding experimentally measured properties of fluids.
Find the partial derivatives
\[\left(\frac{\partial {S}}{\partial {T}}\right)_{p}\]
\[\left(\frac{\partial {S}}{\partial {T}}\right)_{V}\]
where \(p\) is the pressure, \(V\) is the volume, \(S\) is the entropy, and \(T\) is the temperature. Please find these derivatives for one gram of methanol at one atmosphere of pressure and at room temperature.
Why does it take only two variables to define the state?
Why are the derivatives above different?
What do the words isobaric, isothermal, and isochoric mean?
