Pressure and Volume of a Neutral Hydrogen Cloud

[Parzeevahl]

Homework Statement

A diffuse cloud of neutral hydrogen atoms in space has a temperature of 50 K. Calculate the pressure (in Pa) and the volume (in cubic light-years) occupied by the cloud if its mass is 100 times the mass of the sun.

Relevant Equations

PV = NkT (k = Boltzmann constant)

Here's how I approached it. We know the total mass of the cloud, it is given. Let's call it 'M'. We can also find out the mass of a single hydrogen atom from its atomic weight. Let's call this 'm'. Then

N = M / m

is the total number of hydrogen atoms in the cloud. The temperature (T) is given - it is 50 K.

Then the gas equation becomes: PV = (M / m) * k * (50 K)

So, here are two unknown variables here that I have to find out - P and V. But I have only one equation (the gas equation above). How can I find out the other equation, since for two unknown variables we need two equations?

 

[scottdave]

I don't know how you can solve this without more information. Such as the density of the cloud. This may be given as a certain number of atoms per cubic centimeter. Did they tell you any information such as this?
I was doing some searching when reading this problem, because I did not know if these atoms obey the Ideal Gas Law.

I didn't really find a definite answer to that, but it seems that they may. I did find this following article interesting, though:

https://www.britannica.com/science/hydrogen-cloud

 

[Parzeevahl]

Yes, I also thought of the density and checked again, but the question didn't supply any such information.

Also, thanks for the article, I'll take a look :) !

 

[DaveE]

I suppose gravity would be the force that keeps the volume finite? Then the temperature would have to be an average value, I guess?

 

[Chestermiller]

I suppose gravity would be the force that keeps the volume finite? Then the temperature would have to be an average value, I guess?

I agree that it makes sense to include gravity. But the temperature is given, so it doesn't need to be considered an average value. However, the pressure and density should vary with radial position.

 

[jbriggs444]

I agree that it makes sense to include gravity. But the temperature is given, so it doesn't need to be considered an average value. However, the pressure and density should vary with radial position.

I am not convinced that this is the intended interpretation. But...

Suppose that we arbitrarily fix the density at the center of the [spherically symmetric] cloud. Now we should be able to obtain a differential equation for the density gradient at radius r in terms of r and the cumulative mass at or beneath radius r.

We solve the differential equation. If the limit as $r \to \infty$ is finite, we re-scale the initial density to obtain the desired 100 solar mass total and hope that the resulting density still qualifies as "sparse".

Unfortunately, even if this approach succeeds, we will not have obtained a volume. We might, I suppose, consider the pressure at the center as the "pressure" part of the answer and the hypothetical volume that would be required for 100 solar masses at that uniform pressure as the "volume".

 

[DaveE]

Unfortunately, even if this approach succeeds, we will not have obtained a volume.

Yes I find the volume issue problematic. I guess you have to pick a statistical definition, like "containing 99% of the atoms". For any volume chosen there could be an atom that has enough energy to leave that space.