Thermodynamics of Hell Question

[thechunk]

Hey all. I was reading that story about the physics student who is asked to show whether hell is endothermic or exothermic (here's the http://www.people.virginia.edu/~rjh9u/hellthrm.html to the story) when I came upon the following statements:

1. If hell is expanding at a slower rate than the rate at which souls enter hell, then the temperature and pressure in hell will increase until all hell breaks loose.

2. If hell is expanding at a rate faster than the increase of souls in hell, than the temperature and pressure will drop until hell freezes over.

From what I learned in my high school physics class, these are valid statements, however how can one prove the above statements mathematically using the ideal gas laws.

What I have done as of right now, was taking the derivative with respect to time of PV=nRT(nothing is constant except R) to get

$$\frac{dP}{dt} V + \frac{dV}{dt} P = \frac{dn}{dt} T R + \frac{dT}{dt} n R$$

Then I said the system is at STP for convenience with one mol of gas (or souls whatever you want to call them) and got the following

$$\frac{dV}{dt} - 22.4 \frac{dn}{dt} = .0821 \frac{dT}{dt} - 22.4 \frac{dP}{dt}$$

Now assuming that the change in volume as well as the change in number of moles is positive, what conclusions can I make from the above statement. Maybe I took this in the wrong direction or very probably overcomplicated the situation but any help would be much appreciated.

 

[Berislav]

V, P and T are not constant so you can't assume STP for simplicity.

The first statement about hell is one about pressure. That means that you have to solve your ODE for pressure and take the limit of time to infinity. It should go to infinity.

The second statement is about temperature. So solve the ODE for T and take the same limit. It should go to zero.

You could study the simple case where $\frac{dV}{dt}$ and $\frac{dn}{dt}$ are constants. That way it is easier to solve the ODE and possible to take the limit.

EDIT:
You should also need to remove T and P from the first and second equation, respectfully, via the original law PV=nRT. So you need only solve for either P or T.

 

[thechunk]

Ah, I kind of see, thanks for the help