## Thermo Thursday: Compression of an Ideal Gas

Thermo Thursday returns. But on a Tuesday! WHO COULD HAVE SUSPECTED?

This section introduces isothermal and adiabatic compression. Isothermal compression is compression that occurs so slowly that no heat is added to the gas. For isothermal compression of an ideal gas, the temperature remains constant, so you can use the ideal gas law, $PV=NkT$, and the integral equation for work done during compression, $W=-\int _{{V}_{i}}^{{V}_{f}}{P(V)dV}$, to derive that ${W}_{isothermal}=NkT\ln {\frac {{V}_{i}}{{V}_{f}}}$. Since work is being done but the temperature is not changing, heat must be flowing out of the gas, in an amount equal to the work being done. On a PV graph, an isothermal compression takes the shape of a concave-up hyperbola.

Adiabatic compression, in contrast, happens so fast that no heat escapes from the gas during the process. Thus, for adiabatic compression, $\Delta U = W$.  The PV curve for this starts on a lower temperature isotherm and ends on a higher temperature isotherm.  To find an equation for the shape of that curve, we look at the equipartition of energy theorem, $U = \frac {f}{2}NkT$, where f is the number of degrees of freedom per molecule. The infinitesimal change in energy along the curve is then given by $dU=\frac {f}{2}NkdT$. If we assume the compression is quasistatic, then from the equation for work we know that $dU=-PdV$. (We can say this because we previously established that in adiabatic compression the entire change in energy comes from work.) This gives $\frac {f}{2}NkdT=-PdV$. Now you can plug in the ideal gas law for P and do some canceling to get $\frac {f}{2}\frac {dT}{T}=-\frac {dV}{V} \rightarrow \frac {f}{2}\ln {\frac {{T}_{f}}{{T}_{i}}}=-\ln{\frac{{V}_{f}}{{V}_{i}}}$.  This ends up simplifying down to $V{T}^{{f}/{2}}=C$ for some constant C. If you are looking for pressure instead of temperature, you can use the ideal gas law to rewrite this ${V}^{\gamma}P=D$ for some constant D, where $\gamma = {(f+2)}/{f}$ and is called the adiabatic constant.

Problem 1.35: Derive ${V}^{\gamma}P=constant$ from $V{T}^{{f}/{2}}=constant$.

I’m going to use a subscript on the constant term to show when that side of the equations changes. Let’s start with $V{T}^{{f}/{2}}={C}_{0}$. I want to get this equation in terms of pressure, so we use the ideal gas law to say that $T=\frac {PV}{Nk}$. This gives us
$V{(\frac{PV}{Nk})}^{{f}/{2}}={C}_{0}$.
Raising both sides of the equation to the power 2/f  and moving the constant terms to the right side gives
${V}^{{2}/{f}}VP={C}_{1}$.
I can then combine the volume terms,
${V}^{1+\frac{2}{f}}P={V}^{{(f+2)}/{f}}P={V}^{\gamma}P={C}_{1}$.

Problem 1.36: In the course of pumping up a bicycle tire, a liter of air at atmospheric pressure is compressed adiabatically to a pressure of 7 atm. (Air is mostly diatomic nitrogen and oxygen.)

(a) What is the final volume of this air after compression?

I will treat the air as a diatomic ideal gas for these calculations, which means the air molecules have five degrees of freedom. This gives me $\gamma =7/5$. For adiabatic compression, ${V}_{i}^{\gamma}{P}_{i}={V}_{f}^{\gamma}{P}_{f}$, so
${V}_{f}={(\frac{{V}_{i}^{7/5}{P}_{i}}{{P}_{f}})}^{5/7}$
which for an initial pressure of 1 atmosphere and volume of 1 liter, and final pressure of 7 atmospheres, gives a final volume of 0.25 atmospheres.

(b) How much work is done in compressing the air?

The total change in energy is the heat added or lost plus the work done, $\Delta U = Q + W$. Since this is adiabatic compression, there is no heat lost to the environment, so the entire change in energy is due to work. From the equipartition of energy theorem and the ideal gas law I can write $\Delta U= \frac {f}{2} \Delta (PV)$, which for this system gives
$\Delta U = \frac {5}{2} ({P}_{f}{V}_{f}-{P}_{i}{V}_{i})$
Plugging in ${V}_{f}$=0.25 L, ${P}_{f}$=7 atm, ${V}_{i}$=1.0 L, ${P}_{i}$=1 atm, converting to SI units, and calculating gives an energy added due to work of 189.5 J.

(c) If the temperature of the air is initially 300 K, what is the temperature after compression?

I can use the equation $V{T}^{{f}/{2}}=constant$ to say that
${V}_{i}{T}_{i}^{\frac{5}{2}}={V}_{f}{T}_{f}^{\frac{5}{2}}$
and so
${T}_{f}={(\frac {{V}_{i}{T}_{i}^{\frac{5}{2}}}{{V}_{f}})}^{\frac {2}{5}}$.
Plugging in the values and calculating this out gives a final temperature of 522 K.

Problem 1.37: In a Diesel engine, atmospheric air is quickly compressed to about 1/20 of its original volume. Estimate the temperature of the air after compression, and explain why a Diesel engine does not require spark plugs.

Since the air is compressed “quickly,” I will assume adiabatic compression, and since it’s air that’s being compressed it has the same degrees of freedom as the previous problem. So I can use the last equation from part (c) above, plug in ${V}_{f}=\frac {1}{20}{V}_{i}$, and simplify to get
${T}_{f}={(20{T}_{i}^{\frac {5}{2}})}^{\frac {2}{5}} \approx 3.3{T}_{i}$.
So if the initial temperature of the air is 300 K, then after compression the temperature will rise to around 990 K. Since the ignition temperature of Diesel fuel is 483 K, the air after compression is hot enough to ignite it without a spark.

Problem 1.38: Two identical bubbles of gas form at the bottom of a lake, then rise to the surface. Because the pressure is much lower at the surface than at the bottom, both bubbles expand as they rise. However, bubble A rises very quickly so not heat is exchanged between it and the water. Meanwhile, bubble B rises slowly (impeded by a tangle of seaweed), so that it always remains in thermal equilibrium with the water (which has the same temperature everywhere). Which of the two bubbles is larger by the time they reach the surface? Explain your reasoning fully.

Bubble A undergoes adiabatic compression (in this case, expansion), while bubble B undergoes isothermal compression. Initially, the bubbles are identical, so their pressures and volumes are equal. For isothermal compression, ${P}_{B}{V}_{B}=C$ where C is a constant, and for adiabatic compression ${P}_{A}{V}_{A}^{\gamma}=C$. The constants C must be the same, because the bubbles are initially identical. Since $\gamma > 1$, ${V}_{A}<{V}_{B}$. Thus, the bubble that undergoes isothermal compression, bubble B, is larger when it reaches the surface. This can be seen by visual inspection of a graph depicting an isotherm and an adiabat. Notice that for the same initial conditions, the volume of the isotherm rises faster than that of the adiabat.

## Recent Writing by Friends of Mine

My friends are talented and prolific! Look at their things!

NONFICTION:

• Luxury Shopping, From The Other Side of the Register” – In December Carmen Machado wrote for the New Yorker about what it’s like to be working retail during the busiest shopping days of the year.
• Bummed Out and Ugly” – Alice Sola Kim with a beautiful and personal remembrance of Philip K. Dick.
• Crossroads and Coins: A Review of Naomi Mitchison’s Travel Light” – Amal El-Mohtar writing for NPR, another quite personal piece. Having recently read the book myself, I can confirm that it is as delightful as Amal claims.
• Scattered Leaves” – Ben Mauk writing for the New Yorker about the current practice of dismembering ancient manuscripts and selling individual pieces of them on eBay, as well as the long history of “book breaking.”
• To Flip a Flop” – Also in the New Yorker, Elizabeth Weiss writes about the economics of the broadway show, using the example of the largely unsuccessful Spiderman: Turn Off the Dark.

FICTION:

• The Glitch” – Rebekah Frumkin shows off her talent for claustrophobic interiority and familiarity with 64-bit Zelda games in this story at Granta.
• The Engineers” (pdf link) – Rebecca Rukeyser with a story in the Massachusetts Review of expatriate courtship in South Korea.

## The Next Video Game I Can’t Wait To Play

The last time I saw a gameplay video that made me this excited, it was the first teaser video for Portal. A Carnegie Mellon team calling themselves Pillow Castle Games are developing a first person puzzler built around a mechanic that uses projection phenomena for object manipulation. I bought Portal when it came out even though I didn’t have a machine to play it on, and I’d likely go to similar lengths to mess around with this.

## “Yay, Soup!” – Michael J. Pollard as Mr. Mxyzptlk

Michael J. Pollard, who was nominated for an Academy Award for his role as C. W. Moss in Bonnie and Clyde, had a guest star turn on the 1988 television show SuperboySuperboy was notable for recasting the parts of both Clark Kent and Lex Luthor between seasons 1 and 2, and also for being the only entry in the Superman mythos set entirely in Florida. But judging from his Wikipedia page, which currently contains no mention of it, Michael J. Pollard’s involvement has been largely forgotten. That’s a historical oversight which needed correcting, so for posterity I’ve put together a seven minute cut of this treasure. Enjoy.

EDIT: Well that was fun while it lasted.

## My Favorite Books of 2013

It’s astonishing how few books I read as a graduate student. I did a tremendous amount of reading, but it was mostly unpublished fiction by classmates, students, and applicants to the MFA program. I’ve read about 30 books this year, 2/3 of them since I graduated in May. While not that many for me historically, that’s a three year high. Here are my favorites.

Arcadia by Tom Stoppard. Every once in a long while you read a book that immediately becomes a part of your personal canon, something you know from the first encounter that you’ll be returning to and finding new depths in for the rest of your life. Borges was like that for me, and Catch-22, Octavia Butler, Kelly Link, Ted Chiang, and now Arcadia. I was already a fan of Stoppard’s play Rosencrantz and Guildenstern are Dead, which I read in high school. I’d been meaning to read Arcadia for years. I even bought a copy once, but it disappeared. (I think an ex stole it.) Over and over it was recommended by people as something I would like, and I finally got around this year to buying a new copy.

It’s incredible. It has all of my favorite things: clever formalism; patterns that repeat across different scales; nonlinear narrative; fractal mathematics; intellectual humor; and critique of gender roles in social, scientific, and literary regimes. It’s funny, suspenseful, heartbreaking. The best play I’ve ever read. The very first time I have an opportunity to see it produced, I will. In the meantime, the day I read it I got online and bought enough copies to throw an Arcadia reading party. I can’t give a higher recommendation than that.

We Are All Completely Beside Ourselves by Karen Joy Fowler. I was already a fan of Karen Joy Fowler’s work, from her short stories and her novel The Jane Austen Book Club. But her latest novel is in a different league. It’s utterly gorgeous, full of brilliant sentences that add up to an equally brilliant whole. While reading it I was frequently moved to read passages aloud to myself, just to feel the music in the prose. I’ve sold several people the book just by reciting the preface and letting the beauty of the language win them over. It’s convenient that that works, because there’s not really any way to talk about the plot without spoilers that will dramatically change the reading experience. But if that isn’t a concern to you, then you could check out this glowing review by Barbara Kingsolver in the New York Times. For my part, I’ll just say that this year I read Pulitzer Prize winners, Nobel prize winners, bestsellers, and cult classics, but this was my favorite novel that I read in 2013.

Delusions of Gender by Cordelia Fine. The previous was my favorite novel of the year, but this was my favorite work of nonfiction. (So it was a good year for books with bright yellow covers.) If this were just a thorough takedown of biological essentialism, whether historical or modern, it would probably be enough to earn a place on this list. But Cordelia Fine has done more than that. She’s not just taken on the heroic task of going through all the recent books claiming inherent neurological differences between men and women, and tracked down all of the references to assess their legitimacy, but she’s done it with humor. The book is written in delightfully dry tones of academic snark. So, for example, while critiquing the way that Barbara and Allan Pease use scientific studies in their execrably-titled book Why Men Don’t Listen and Women Can’t Read Maps, she observes that of the studies referenced in the Pease’s claim that their “emotion maps” are based on fMRI research, only one of them was a brain study conducted after the academic use of fMRI. And of that she writes, “It might also be worth mentioning that it was a postmortem study. Possibly Sandra Witelson really did present her samples of dead brain tissue with emotionally charged images–but if she did, it’s not mentioned in the published report.” As they say in the ivory tower, oh SNAP!

Nausicaä of the Valley of the Wind by Hayao Miyazaki (deluxe edition box set). This was a graduation gift to myself, something I’d been meaning to read for years. I’m a great fan of Miyazaki’s movies, of which the film adaptation of this manga was the first for which he served as both writer and director. The movie version is glorious, and you should watch it if you haven’t, but the manga is a much larger and more intricate story. This is partly because he had only written/drawn the first two years of the manga when he made the movie, and wouldn’t finish it for another decade. The politics, world building, and characterizations are rich, and the artwork predictably incredible. (This oversized edition is worth it for the greater detail in the artwork alone.) The story does at times have a bit of a formless, sprawling feel to it. That could be because it was Miyazaki’s first (and last) long form manga work. But that hardly matters, as the expansiveness of the world is one of the distinct pleasures on offer here.

The Man Who Fell To Earth by Walter Tevis. (My copy had a different cover that I can’t find good image of. This seems to be the edition in print right now.) For a while this year I was running a science fiction movie club, and picking movies for it was an excellent excuse to watch some classic films that I’d never managed to get around to. One of those was Nick Roeg’s adaptation of The Man Who Fell To Earth starring David Bowie as an alien, which I’d been putting off until after I read the novel. Now that I’ve read/seen both, it’s the book I think I might be going back to. That’s not a knock against the movie, but Tevis’s novel was a startling work of bleak loveliness. If there is such a thing as a page turner consisting entirely of chilly, elegiac portraits of loneliness, this is it. (If you’ve seen the movie but not read the book, which seems likely to be the case for many, know that the book has a lot more tipsy rumination on the impossibility of ever really connecting with other people, and a lot less of David Bowie’s penis.)

Code Name Verity by Elizabeth Wein. (This one isn’t the cover that my copy had either, but I wish it was, because this cover is way better. Mine was a couple of bicycles leaning up against a stone wall.) This is a novel that had been recommended by many people, and the recommendations were often things like, “This book is amazing but also it made me break down crying in public.” So, naturally, I waited until it was dark and cold and miserable outside to read it. The book is made up of a pair of linked epistolary narratives, with an unreliability-powered plot that’s so ostentatiously clever that, in my edition, the cover text touts its cleverness. That alone would make it worth reading. But this book is also that rare creature: a rollicking wartime adventure that is centered on a friendship between two women. It’s set primarily in Nazi-occupied France, full of espionage, aeronautics, and harrowing scenes of painful bravery. Even prepared as I was for an emotionally wrenching experience, the climax was shocking and the denouement deeply affecting. Read it, but not at a time when you’re feeling fragile.

## No Thermo Thursday this week

When Thermo Thursday returns it will be all about adiabatic and isothermal compression, but this was the last week of classes and I got busy. And then I did a podcast this afternoon rather than physics. I’ll try to link to that podcast soon. I also might do a makeup Thermo post this weekend. But for now I’m tired, so no derivations tonight.

## The Traditional Ceremony of Robots

Another semester done, another class of fearless Science Fictionauts heading out into the future with their robot companions.

Writing and Reading Science Fiction, University of Iowa, Fall 2013

## Thermo Thursday: Finishing Compression Work

It’s the end of the semester and I’ve got other projects I need to be working on, so this week I’m just going to do the last problem from the compression work section and leave the adiabats and isotherms for next time.

Problem 1.34: An ideal diatomic gas, in a cylinder with a movable piston, undergoes the rectangular cyclic process shown in the figure to the right. Assume that the temperature is always such that rotational degrees of freedom are active, but vibrational modes are “frozen out.” Also assume that the only type of work done on the gas is quasistatic compression-expansion work. (a) For each of the four steps A through D, compute the work done on the gas, the heat added to the gas, and the change in the energy content of the gas. Express all answers in terms of ${P}_{1}$, ${P}_{2}$, ${V}_{1}$, and ${V}_{2}$. (Hint: Compute $\Delta U$ before Q, using the ideal gas law and the equipartition theorem.)

Step A: This is a diatomic gas with no vibration modes, so for this system the equipartition theorem says that $U = \frac {5}{2}PV$. For this step, $\Delta U = \frac{5}{2}({P}_{2}-{P}_{1}){V}_{1}$. Since the volume of the gas doesn’t change in this step, there is no work being done on the gas. Recalling that $\Delta U = W + Q$, that means that $Q = \frac{5}{2}({P}_{2}-{P}_{1}){V}_{1}$.

Step B: here $\Delta U = \frac{5}{2}{P}_{2}({V}_{2}-{V}_{1})$. The volume is increasing, so the work done on the gas is negative, $W=-{P}_{2}({V}_{2}-{V}_{1})$. Since W is negative, that means that the heat added must be the total energy change plus the amount of energy subtracted by negative work. $Q=\Delta U - W = \frac{7}{2}{P}_{2}({V}_{2}-{V}_{1})$.

Step C: this time $\Delta U = \frac{5}{2}({P}_{1}-{P}_{2}){V}_{2}$. Note that this is a negative quantity, the system has lost energy. Again, there is no volume change, so no work is being done on the gas. Thus the entire energy change is due to the system losing heat energy, so $Q=\frac{5}{2}({P}_{1}-{P}_{2}){V}_{2}$.

Step D: in the last step, $\Delta U = \frac{5}{2}{P}_{1}({V}_{1}-{V}_{2})$. This is another negative quantity. But this time, the work being done on the gas is positive, $W=-{P}_{1}({V}_{1}-{V}_{2})$. Since the work done on the gas is positive, and the energy change is negative, it must be losing even more heat energy than it is gaining in work energy. Here $Q= \Delta U - W = \frac{7}{2}{P}_{1}({V}_{1}-{V}_{2})$.

(b) Describe in words what is physically being done during each of the four steps; for example, during step A, heat is added to the gas (from an external flame or something) while the piston is held fixed.

The author went ahead and did step A for me. In step B, the volume of the gas increases, so the piston is being drawn out (or, more likely, pushed out by internal pressure). But this doesn’t result in any decrease in pressure, which means that the gas is also being heated as the volume increases. In step C, the system just loses heat energy, so the system is being cooled while the piston is held fixed. In step D the piston is compressed, but the cylinder is still being cooled so the pressure doesn’t change.

(c) Compute the net work done on the gas, the net heat added to the gas, and the net change in energy of the gas during the entire cycle. Are the results as you expected? Explain briefly.

The net change in energy is zero, that’s what makes it a cycle. For the work over the whole cycle, that’s ${W}_{cycle}=-({P}_{2}-{P}_{1})({V}_{2}-{V}_{1})$. That’s a negative quantity, meaning that over the course of the whole cycle work was done on the environment. Since the energy change is zero, the heat added perfectly balances the work, so ${Q}_{cycle}=({P}_{2}-{P}_{1})({V}_{2}-{V}_{1})$. That’s a positive quantity. As mentioned at the end of the last Thermo Thursday, this cycle takes in heat energy and converts it to work done on the environment. So these results are as I expected.

## The Planet Money T-Shirt Project

Photo by Alice Gribbin

Here’s me, sitting in the Frank Conroy Reading Room in my NPR Planet Money t-shirt. I started listening to the Planet Money podcast, an economics podcast started by some of the producers of This American Life, right when it started in 2008. For years they’ve wanted to produce a t-shirt and track how it moves through the global economy, from cotton to yarn to cloth to clothes to consumers. Now, people who backed their Kickstarter are starting to get their shirts, and everyone can learn the details about how these objects came to be on a page where they’ve aggregated their reporting. It’s been a fascinating story to follow, with everything from large-scale investigations of the history of international trade to personal stories about individual factory workers.

## Review: The Tom Bihn Synapse 19 backpack

My single strongest brand loyalty is to Tom Bihn bags. They’re brilliantly designed, attractive, near-indestructible, and made in the USA. I describe them as the Apple of bags. My first was an Empire Builder briefcase that I got when I went to Trinity, in an attempt to save my spine from my high school habit of carrying every textbook and binder around on my back in a bag that weighed half as much as I did. I carried it all through college, but it turned out to be so spacious that it didn’t really solve the problem; I still carried around more weight than my shoulders could really support. But as Tom Bihn bags have a modular design, I was able to take out the Brain Cell insert, attach a strap to it, and use it as a minimal MacBook case. Once the 11″ MacBook Air came out I went fully minimal and bought a Ristretto (original style, it’s since been updated) and for the last three years have never carried more than it could hold. 90% of the time that’s all I need, but very occasionally I’ve wanted something more capacious. Then, a Hanukkah miracle: I now own a Synapse 19.

It’s a small, six pocket backpack with shaped shoulder straps and removable sternum and waist straps that, once adjusted, hugs the body better than any pack I’ve ever owned. Mine is the navy blue nylon with Iberian red Dyneema interior, as pictured in the front panel shots here.1 (Though there are many other options. You can even get the whole thing in Dyneema, which is thinner than the nylon but reduces the empty weight by 7%.) The back panel is padded with a breathable mesh overlay to keep your back from sweating, and all of the zippers are rubberized for water resistance. The main compartment is an open space with a 2/3-height elastic pocket along the front. There are o-ring anchor points for attaching keychain lanyards or modular organizers, and it has two pairs of webbing loops to which you can attach the cache with rails to turn it into a checkpoint friendly laptop bag. Traveling to and from Texas for Thanksgiving, I found it very convenient to not have to take out my computer when going through airport security. Instead you just slide the cache out the top of the bag and let the whole thing go through the X-ray machine.

Directly in front of the main compartment is a tall, narrow water bottle pocket, centered on the bag so that it doesn’t throw off the balance when it’s on your back. I don’t carry a water bottle, but this pocket is also perfectly sized for a small book, e-reader, or tablet. I’ve been using it to hold my iPad Mini. The two side pockets are curved and positioned such that they can be easily access while the pack is being worn by dropping one shoulder strap and pulling it around under your arm. One side has sewn-in pen sleeves and the other has a soft, sueded pocket for holding something you don’t want scratched. The website suggest a cell phone, but I’ve been keeping my backup hard drive in mine. Both side pockets have o-ring anchors, as does the small front top pocket behind the logo. The bottom pocket is full width and deeper than it looks. I’ve been keeping gloves, a wool cap, and my unused straps in there.

So far I’ve used it as my only bag on an overnight to Madison, and as my under-seat carryon for my trip back to Texas, and its been perfect for both. Even when I was packing clothes and toiletries along with my computer for the overnight trip, I didn’t quite max out its capacity. That said, it’s still small enough that I’m not at risk for hurting my back again. It easily sits near my feet in a full car, and can hook across the back of a restaurant chair without tipping it over when I stand up. The weather hasn’t been conducive yet to wearing it while riding my bike, but it has been perfect for every other task I’ve thrown at it. It’s the best backpack I’ve ever used. (If this design is attractive but the small size isn’t a plus for you, there’s also the Synapse 25, which is 30% larger but has the same layout.)

1. Photos from Tom Bihn’s site.