## 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.

## Thermo Thursday: Chapter 1.5: Compression Work, continued

I never got around to finishing the problem set last week, so instead of editing the last entry, I’ll just finish it for this week.

Problem 1.32: By applying a pressure of 200 atm, you can compress water to 99% of its usual volume. Sketch this process (not necessarily to scale) on a PV diagram, and estimate the work required to compress a liter of water by this amount. Does the result surprise you?

I’m just going to assume a linear function here and say that the work done is the area under the PV curve. Here that ends up being about 100 J to compress water by 1%.

Problem 1.33: An ideal gas is made to undergo a cyclic process shown in the figure to the right. For each of the steps A, B, C, determine whether each of the following is positive, negative, or zero: (a) the work done on the gas; (b) the change in the energy content added to the gas; (c) the heat added to the gas. Then determine the sign of each of these three quantities for the whole cycle. What does this process accomplish?

For A, the work being done on the gas is (a) negative, by the equation $W = -P\Delta V$. The energy added to the gas is (b) positive, which I could demonstrate with the equipartition of energy principle, but would rather show by observing that if the gas increases in volume but maintains the same pressure, that means that the frequency of collisions with the container must remain the same as when the gas was at lower volume. The only way for that to happen is if the molecules are moving faster, so energy must have been added to the system. Since the energy of the system is the heat plus the work, and the work is negative and the energy is positive, then the heat added must also be (c) positive (and greater than the absolute value of the work).

For B, the work done on the gas is (a) zero, since there is no change in volume. By equipartition of energy for an ideal gas, $U = \frac{3}{2}PV$, the energy increase is (b) positive. You can also get this from observing that increased pressure at the same volume means more collisions with the container per unit time, which means that the molecules of the gas must be moving faster, and so must have more energy. Since there is more energy and no work has been done, the heat added is (c) positive.

For C, the work done on the gas must be (a) positive, because the volume decreases. The energy added to the gas is (b) negative, because even though the volume is decreasing, the pressure is going down. Since the work done on the gas is positive and the overall energy change is negative, the heat added to the gas is also (c) negative.

For the whole cycle, the net work must be (a) positive, because the average pressure is higher during step C than during step A. Since the pressure and volume at the end of the cycle are the same as at the start, the net energy change must be (b) zero. And since the work is positive and the energy change is zero, the heat added must be (c) negative. So this cycle takes in energy as work and emits the energy as heat.

There’s another problem here, but it’s basically the same, except it’s a four-step rectangular cycle that goes clockwise, and turns heat added into work done by the gas. It’s late, so I’m not going to go through the details.

## Thermo Thursday: chapter 1.5: Compression Work

This section introduces the equations for the work done on a system by compression, such as when a piston is used to compress a volume of gas. Assuming nearly quasistatic compression (that is, compression slow enough that all of the gas can be said to be at a single pressure; in practice, usually any compression slower than the speed of sound in the gas), the work done per infinitesimal change in volume is $W = -P\Delta V$. When the pressure changes significantly during compression, you have to approximate the process as a series of small compressions, unless you have an equation for the pressure as a function of volume. If you do, then you can integrate, and the equation for work becomes $W=-\int _{{V}_{i}}^{{V}_{f}}{P(V)dV}$.

Problem 1.31: Imagine some helium in a cylinder with an initial volume of 1 liter and an initial pressure of 1 atm. Somehow the helium is made to expand to a final volume of 3 liters, in such a way that its pressure rises in direct proportion to its volume. (a) Sketch a graph of the pressure vs. volume for this process.

(b) Calculate the work done on the gas during this process, assuming that there are no “other” types of work being done.

Here I actually do have an equation for pressure with respect to volume. Since the relationship is linear, I can say that $P = V (\frac {atm}{L})$. Plugging into the equation for work gives $W = -\frac {atm}{L} \int _{1 (L)}^{3 (L)}{V dV}$. This ends up giving –4 atmosphere-liters, which converts to approximately –400 J. The negative sign indicates that, rather than work being done on the gas, it is the gas that is doing work on the environment (presumably by displacing whatever was in the volume it expands to occupy).

(c) Calculate the change in the helium’s energy content during this process.

Helium is a monatomic gas, so it has three quadratic degrees of freedom. By the equipartition of energy theorem, $U = \frac {3}{2}NkT$, which by the ideal gas law is equivalent to $U = \frac {3}{2}PV$. The change in energy, then, is $\Delta U = \frac {3}{2} (P_{final}V_{final} - P_{initial}V_{initial})$. This gives 12 atmosphere-liters, or approximately 1200 J.

(d) Calculate the amount of heat added to or removed from the helium during this process.

The first law of thermodynamics is $\Delta U = Q + W$, which I can rewrite $Q = \Delta U - W$. In this case, the work done is -400 J, and the change in energy is 1200 J, so the heat added to the system is 1600 J.

(e) Describe what you might do to cause the pressure to rise as the helium expands.

Normally as volume increased pressure would decrease. (Solve the ideal gas law for pressure and volume ends up in the denominator). To counter that, you would have to increase the temperature. So you could cause the pressure to rise as the helium expands by heating it.

I’ll have to finish these problems another time, as I need to go to the reception for Marina Warner, who is receiving the Truman Capote award. I’ll edit this post later. (Tumblr, you won’t see the edit, you’ll have to click through.)

## Review: The Air-O-Swiss 7135 Humidifier

Let’s get this decade of my life started off right. Let me tell you about my high-tech humidifier.

The unit in question is the Air-O-Swiss 7135. It’s an ultrasonic model with a replaceable demineralization cartridge impregnated with silver ions to impede bacterial growth. It has programmable controls and a built-in humidistat, so you can set it to either run for a given duration, or to turn itself on and off to maintain a desired percent humidity. It also has an optional preheater so that the mist doesn’t lower the ambient temperature of the room.

I’ve been loving it. Once the weather here changed and I had to turn on the heat in my house I was waking up with sore throats, aching sinuses, nostrils that felt like they’d been packed with sand while I slept. I got nosebleeds, an infection, lost my voice. Things got better when I went out and bought a hot mist (boiling) humidifier as a stopgap measure, but that raised the humidity in my room so high that it got musty, and on very cold nights water would condense on the windows and exterior walls. With the Air-O-Swiss, though, I can watch the hygrometer display and see it adjusting its output to maintain the humidity where I want it. It oscillates, but my experience is that it manages to keep things stable plus or minus around three percent. Since it’s cool/warm mist, I can set it up near where I sleep and have the occasional lovely and ominous curl of mist roll silently over the bed and disappear in front of me like a ghost. I’m sleeping better, and waking up better, than I have in a long time.

Ultrasonic humidifiers work by using a transducer to physically separate water molecules into a mist. This makes it quieter than models that boil water, or evaporative humidifiers that use a fan to blow air through a wick. The downside is that since the mist is being mechanically created rather than produced through a chemical process like evaporation or boiling, the mechanism is indiscriminate about what it aerosolizes. The transducer is happy to vibrate minerals, microbes, whatever happens to be in the water into the air for me to breathe. There are varieties of pneumonitis that are actually known as humidifier lung. As I’m on immunosuppressive drugs, that makes keeping the thing clean of particular importance. Fortunately, this model makes it easy. It comes with a solvent and has an indicator light for monthly cleanings, but I do it more often than that. About every three days, actually, as recommended by the Mayo clinic. The mouth of the tank is wide enough to allow quick and easy water changes, and the base mostly easily-scrubbable flat surfaces. It even comes with a little brush for getting scale, which can harbor bacteria, off the transducer plate. So far, though, the filtration and demineralization features are good enough that in a week of operation I haven’t noticed any of the white dust that’s typical of ultrasonic humidifiers. Assuming I don’t ironically die of Legionnaires’ disease in the next couple of months, I’m very pleased with everything about the device.

The bad news: it costs \$180. That’s on the very pricey side for a humidifier, and if that matters you could probably buy a less expensive one, a hygrometer, and a timer switch from a hardware store for less than the Air-O-Swiss. But if the all-in-one convenience is valuable to you, or you’re having a birthday and willing to ask for the kind of thing that you want but would never buy for yourself, then the Air-O-Swiss is great.

## I turned 30 two days ago…

and still no Sandmen to take me to the Carrousel. Looks like I’m safe.

## More Remidios Varo

Because why not?

(Couldn’t find the title of this one)

The Debutante

Woman Leaving the Psychoanalyst

Transit in a Spiral

(couldn’t find the title to this one either)

## Another Glorious Moment in Texas Political History

One nice thing about having an entire ascendant political class utterly divorced from pragmatism or reality is that, since the people involved do still have to live in the real world, their own efforts occasionally backfire in hilarious ways. My favorite one of these is probably still the time that Texas, in its zeal to ban gay marriage, overshot the mark and accidentally amended the state constitution in a way that techincally banned all marriage. But now there’s a new contender.

Texas, like many other conservative-controlled states, has passed strict Voter ID laws in an effort to disenfranchise the poor, minorities, and women. In Texas you must now be able to show a current government ID whose name “substantially matches” that on the voter registry to be able to vote. This is actually somewhat weaker than the version of the law pushed for by Texas attorney general Greg Abbott, who wanted an exact match to be required. But State Senator Wendy Davis, Democrat and noted superhero, pointed out that women who had gotten married and changed their name would frequently be unable to vote, and got the law changed so that if the names were similar but not identical you could still vote providing you signed an affidavit.

In an ironic twist, Greg Abbott turned out to be registered to vote under that name, but arrived at his polling place to discover that his driver’s license reads “Greg Wayne Abbott.” So under his preferred legislation, he would have been unable to vote at all. Which is especially interesting given that he’s the presumptive Republican candidate to replace Rick Perry as Governor. And his presumptive opponent? Wendy Davis, the woman who preserved his franchise in 2013.

You’ve got a year to improve the optics on that one, Greg.

## Introducing Senhor Testiculo, Your Friendly Neighborhood Megascrotum

A Brazilian testicular cancer awareness group has introduced its new mascot Senhor Testiculo, or “Mr. Balls.”

Give Uncle Scrotor a hug” made real. Happy Friday, everyone.