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Sequence
Sequence
Warm-Up (Welcome Activity Reviewing Material from Undergraduate Physics)
This content is used in the Physics Department at OSU with incoming graduate students to remind them of undergraduate content before classes start and to help them to decide whether or not to take some Bridge Courses. This sequence is intended to run in two blocks of three hours each. The sessions should be run by someone with a deep knowledge of all of the relevant courses, the specific activities, and active engagement in general.
This session may be the first opportunity for the incoming graduate students to meet each other as well as some faculty and other graduate students. So start with a 1/2 hour dedicated to introductons.
Consider inviting some or all of the following people to participate:
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.”
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.
These notes from week 6 of https://paradigms.oregonstate.edu/courses/ph441 cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.
This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between changes in entropy and heat.
In this remote-friendly activity, students use a microwave oven (and optionally a thermometer) to measure the latent heat of melting for water (and optionally the heat capacity). From these they compute changes in entropy. See also Ice Calorimetry Lab.
- how to distinguish the two different uses of the word “linear” in a linear charge density that varies linearly;
- some of the words for describing functional variation: linear, quadratic, exponential, falls off like ..., proportional to the square, etc.
- how to “name the thing you don't know” with an algebraic symbol so that it can appear in an equation.
- Partial derivatives
- Physical representation
- Thermodynamic variables
- Practicing changing certain variables while holding others constant
Students are placed into small groups and asked to create an experimental setup they can use to measure the partial derivative they are given, in which entropy changes.
In this activity students estimate the optical depth of the atmosphere at the infrared wavelength where carbon dioxide has peak absorption.
Students calculate two different (thermodynamic) partial derivatives of the form \(\left(\frac{\partial A}{\partial B}\right)_C\) from information given on the same contour map.
These lecture notes from the ninth week of https://paradigms.oregonstate.edu/courses/ph441 cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.
In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.
These are notes, essentially the equation sheet, from the final review session for https://paradigms.oregonstate.edu/courses/ph441.
This small group activity using surfaces introduces a geometric interpretation of partial derivatives in terms of measured ratios of small changes. Students work in small groups to identify locations on their surface with particular properties. The whole class wrap-up discussion emphasizes the equivalence of multiple representations of partial derivatives.
A pretzel is to be dipped in chocolate. The pretzel is in the shape of a quarter circle, consisting of a straight segment from the origin to the point (2,0), a circular arc from there to (0,2), followed by a straight segment back to the origin; all distances are in centimeters. The (linear) density of chocolate on the pretzel is given by \(\lambda = 3(x^ 2 + y^2 )\) in grams per centimeter. Find the total amount of chocolate on the pretzel.Main ideas
- Calculating (scalar) line integrals.
- Use what you know!
Prerequisites
- Familiarity with \(d\boldsymbol{\vec{r}}\).
- Familiarity with “Use what you know” strategy.
Warmup
It is not necessary to explicitly introduce scalar line integrals, before this lab; figuring out that the (scalar) line element must be \(|d\boldsymbol{\vec{r}}|\) can be made part of the activity (if time permits).
Props
- whiteboards and pens
- “linear” chocolate covered candy (e.g. Pocky)
Wrapup
Emphasize that students must express each integrand in terms of a single variable prior to integration.
Emphasize that each integral must be positive!
Discuss several different ways of doing this problem (see below).
Details
In the Classroom
- Make sure the shape of the pretzel is clear! It might be worth drawing it on the board.
- Some students will work geometrically, determining \(ds\) on each piece by inspection. This is fine, but encourage such students to try using \(d\vec{r}\) afterwards.
- Polar coordinates are natural for all three parts of this problem, not just the circular arc.
- Many students will think that the integral “down” the \(y\)-axis should be negative. They will argue that \(ds=dy\), but the limits are from \(2\) to \(0\). The resolution is that \(ds = |dy\,\boldsymbol{\hat x}|=|dy|=-dy\) when integrating in this direction.
- Unlike work or circulation, the amount of chocolate does not depend on which way one integrates, so there is in fact no need to integrate “down” the \(y\)-axis at all.
- Some students may argue that \(d\boldsymbol{\vec{r}}=\boldsymbol{\hat T}\,ds\Longrightarrow ds=d\boldsymbol{\vec{r}}\cdot\boldsymbol{\hat T}\), and use this to get the signs right. This is fine if it comes up, but the unit tangent vector \(\boldsymbol{\hat T}\) is not a fundamental part of our approach.
- There is of course a symmetry argument which says that the two “legs” along the axes must have the same amount of chocolate --- although some students will put a minus sign into this argument!
Subsidiary ideas
- \(ds=|d\boldsymbol{\vec{r}}|\)
These notes from the fourth week of https://paradigms.oregonstate.edu/courses/ph441 cover blackbody radiation and the Planck distribution. They include a number of small group activities.
This small group activity has students reasoning about how the Planck distribution shifts when the temperature is doubled. This leads to a qualitative argument for the Stefan-Boltzmann law.
- Work as area under curve in a \(pV\) plot
- Heat transfer as area under a curve in a \(TS\) plot
- Reminder that internal energy is a state function
- Reminder of First Law
This very short lecture introduces Wein's displacement law.
These lecture notes covering week 8 of https://paradigms.oregonstate.edu/courses/ph441 include a small group activity in which students derive the Carnot efficiency.