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.)