Hello. This is my first post in this forum, so bear with me on any flaw or mistake please.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Imagine a gas confined within an insulated container as shown in the figure below. The gas is initially confined to a volume Vi at pressure Pi and temperature Ti. The gas then is allowed to expand into another insulated chamber with volume V2 that is initially evacuated. Show that, for small differences in volume [tex]\Delta[/tex]V= Vf - Vi, the difference of temperature [tex]\Delta[/tex]T before and after the expansion is given by:

[tex]\Delta T = (\frac{P}{Nc_{v}} - \frac{T\alpha}{Nc_{v}K_{t}})\Delta V[/tex]

P - pressure

T - temperature

[tex]\alpha[/tex] - coefficient of thermal expansion

[tex]c_{v}[/tex] - molar heat capacity at constant volume

[tex]K_{t}[/tex] - coefficient of isothermal compressibility

2. Relevant equations

None.

3. The attempt at a solution

It's known that there is no work done on the system, so [tex]\Delta[/tex]U = 0. Since it's an adiabatic process, there's no exchange of heat with the exterior, so [tex]\Delta[/tex]Q = 0.

With the help of my "Concepts in Thermal Physics" and of course, Wikipedia, I managed to write down some promising equations that looked like they were gonna solve the problem on their own, but I just wound up with a huge headache. Here they are:

[tex]\alpha = \frac{1}{V}(\frac{\delta V}{\delta T})_{P}[/tex]

[tex]K_{t} = - \frac{1}{V} (\frac{\delta V}{\delta P})_{T}[/tex]

[tex]dU = 0 = (\frac{\delta U}{\delta T})_{V} dT + (\frac{\delta U}{\delta V})_{T} dV[/tex]

[tex]dU = T dS - P dV[/tex]

[tex]\frac{P}{T} = (\frac{\delta S}{\delta V})_{U}[/tex]

I kinda suck at differential equations, but I'm sure I am missing something, because I just couldn't relate all the variables the way they wanted me to. I've searched the internet, but I just end up reading a problem about Joule's Expansion (which differs on the fact that in this case we are not told that it's an ideal gas, and the gas doesn't expand to twice it's original volume - otherwise it would be fine) or a free gas expansion with constant temperature, which is clearly not the case.

I know that the change of entropy will be a very useful tool in this problem, but I still can't see how. I am going crazy with this.

EDIT: With the equations in this site http://scienceworld.wolfram.com/physics/Heat.html I managed to get an interesting equation: [tex]dT = (\frac{-\alpha K_{T} T}{C_{V}}) dV[/tex], but it still isn't good enough... =/

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# Homework Help: Free expansion of a gas problem

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