Thermodynamic process when nebula is heated

Question

The basic thermodynamics problem is stated as follows.

The nebula contains a very tenuous gas of a given number density (atoms per volume) that is being heated to a given temperature. What is the gas pressure?

1. What are the basic assumptions that should be taken in solving this problem? There is no sealed container, obviously, but if nebulæ were to allowed to expand indefinitely then it would be an isobaric process, would it not? Seen how that would not allow us to determine the pressure (?), then some sort of constraint is to be placed. If we are to assume that it is, in fact, an isochoric process, would it be a simple matter of finding molar mass from given number density and plugging it in the formula of Ideal Gas Law ($p = nRT / V$) assuming the volume of 1 metre cubed?

2. Given the numbers ($1 × 10^{8}$ atoms per m$^3$, 7500K) what should be realistic order of magnitude for the answer (in $Pa$ or $atm$) for the purposes of assessment as many certainly would not have intuitive grasp of your average nebular pressures?

In general, for questions like this (advanced question from introductory chapter) is it detrimental to overthink the problem, that is: do I look for more difficulty than I should?

Answer 1

0) This sounds like the type of problem that was meant to be just plug-and-chug with the ideal gas formula. Nebulae, being as sparse as they are, are often excellent ideal gases. That said, overthinking problems is not always bad.

1) Nebulae can be self-gravitating, or they can be gravitationally bound to stars, or they could be semi-transient phenomena "contained" by the bulk motion of turbulent eddies. So yes, there are ways to make isochoric changes, more or less. This problem in particular seems like it would have been better worded without implying heating over time, which of course can be very complicated. I read it as "given a temperature and density, find the pressure," rather than "given an initial density and a temperature change, find the new pressure."

If you wanted to consider expansion while heating, you would have to know more about the timescales. For instance, a supernova shockwave could pass through the gas, heating it before it has time to expand. Or a star could turn on nearby, heating it more or less uniformly via radiative transfer. Or maybe it is being heated everywhere simultaneously by the decay of some radioactive element. Since it can only expand at the speed of sound, it could be a while before its density drops.

2) An interesting look at how interstellar densities are actually measured in practice can be found in this article by Jenkins and Tripp. Figure 7 shows the distribution of pressures of atomic hydrogen in their study. Taking the peak to be $$ \frac{P}{k} = 3\times10^3~\mathrm{K}/\mathrm{cm}^3, $$ with Boltzmann's constant $k = 1.4\times10^{-23}~\mathrm{J}/\mathrm{K}$, this corresponds to $$ P = 4\times10^{-20}~\mathrm{J}/\mathrm{cm}^3 = 4\times10^{-14}~\mathrm{Pa}. $$ This can vary by a few orders of magnitude (the numbers you give are for slightly denser region), but it gives you a sense of how thin the stuff out there is.