[Dzoni]

Right, so I've got this Thermo assignment, and the question I'm having problems with looks like this:



I can easily work out all the conditions in both compartments before the heating, using PV=nRT. There are two things I'm having problems with, however.

Firstly, to find the conditions AFTER the heating, I get 3 unknowns and 2 equations, which is obviously a problem.

What I have so far is this:

P_A = P_B, since I'm assuming energy was added to the system slowly enough such that we can take the process to be reversible.

Using the ideal gas equation for both compartments, we get

n_AT_A/V_A = n_BT_B/V_B

and we also know that V_A + V_B = V_tot

The unknowns are V_A, V_B and T_B.

Anyone have any ideas on where I can find a third equation?

Also, once I have worked out the conditions, I'll have to work out the amount of energy added by the heater. Since dU = 0 for an ideal gas, the energy balance reduces to Q = -W where W=-?PdV. Am I right in saying I'll need to work out separate W terms for each of the two compartments? If so, how would I do this? My initial thought is simple enough, just use P=nRT/V in the integral, but the obvious problem is that the expansion of compartment A and the compression of B are not isothermal, so T cannot be taken as constant in the integration... How do I get around this?

Any help will be much appreciated! Thanks in advance!

[Dzoni]

OK, so I feel like a bit of an idiot... dU ? 0 in this case.

Nevertheless, I still need to work out the conditions before I can even think about the energy balance... Is there any way I can relate the pressure in A before the heating to the pressure in A after the heating?

[Yggdrasil]

If the piston is a perfect insulator then the initial temp in chamber B should be the final temp in chamber B.

[Dzoni]

If the piston is a perfect insulator then the initial temp in chamber B should be the final temp in chamber B.

But, seeing as the piston is free to move, then as the temperature of compartment A increases, it expands, hence compressing the gas in compartment B, which will make B's temperature rise. The fact that the piston is a perfect insulator means energy cannot get in or out of compartment B via the mechanism of heat, but by no means does it imply that B's temperature will stay the same. I would agree with you if the piston was fixed, but then there would be no point in even including compartment B in the discussion!

[Donaldson Tan]

Work Done by A = - Work Done on B

[Dzoni]

Geodome, I agree completely. But how does that relate to this? But is that really necessary for the calculations?

My plan of attack was this: An energy balance on A will reveal dU + PdV = dQ. Integrating that whole lot and using dU = C_vdT, I should get Q, not so?

HOWEVER, the issue arises in trying to calculate the PdV term in B (or A for that matter). W = ?PdV, where P = nRT/V. Great, except for the fact that T isn't constant... How do I account for this in the integration?

[Yggdrasil]

Ok, here's how to derive a thrid equation:

Because the piston is a perfect insulator, the compression in chamber A is adiabatic.  This means that:

dU = dw
nC_vdT = -PdV
nC_vdT = -(nRT/V)dV

After separation of variables, we get:

(dT/T) = -(R/C_v)(dV/V)

After integrating:

ln (T_f/T_i) = (-R/C_v) ln (V_f/V_i)
(T_f/T_i) = (V_i/V_f) ^ (R/C_v)

[Dzoni]

Yggdrasil, yes, I should have done that myself, but at first I figured that since the C_v of the gas isn't given, and we don't know what the gas is, there must be a way to do it without that information. Since then, afters hours of various attempts and head-scratching, I've come to realise that that's impossible, so I'm going to give my answer in terms of C_v.

Would there be anything wrong with using PV^? = constant, where ? ? C_p/C_v? I suspect it would lead to the same answer, numerically...

[Kunisch]

Depending on how picky your teachers are you might wanna write that the volume of the heater is assumed to be  = 0 (else it would cause a rise in pressure)


Good luck..

Sorry, dont have any time for more help.. finishing a math project worth 4,375 ECTS points

[Dzoni]

Myeah, don't think my lecturer would be that picky, but I'll add it in for fun. Thanks...