# change in temperature in the Gay-Loussac-Joule experiment

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

Consider the apparatus from the Gay-Loussac-Joule experiment. You have 2 containers of gas, with a valve between them. The volume of each container is V. Each vessel contains n

_{A}and n

_{B}moles of gas. Their temperatures (T

_{1}) are initially the same.

When the valve is opened and the gases are allowed to mix, find an expression that shows the change in temperature.

**2. Relevant equations**

We will assume a Van der Waals gas with an equation of state as follows:

##nRT = \left(P+\frac{a}{V^2}\right)(V-b)##

and the internal energy of both gases can be described by:

##u = c_VT – \frac{a}{V} + u_0##

**3. The attempt at a solution**

OK, I assumed we need the conservation of energy, so that there will be a ##u_i## (initial energy) and a ##u_f## (final).

I figured at first I should just add the energies of the two gases as follows:

gas 1: ##u = c_vT – \frac{a}{V} + u_0##

gas 2: ##u = c_vT – \frac{a}{V} + u_0##

and the total energy has to be the same as the van der Waals gas in a space 2x the volume, so ##u_f = c_vT – \frac{a}{2V} + u_0##.

I also know that the equation of state for the combined gas should be

##(n_A + n_B)RT = \left(P+\frac{a}{V^2}\right)(V-b)##

and fro there I should get a reasonable expression for the change in temperature.

The problem is I am not quite sure how to make the connection. My first attempt at finding a delta T was this:

##2u_i = 2c_vT_1 – 2\frac{a}{V} + 2u_0 = u_f = c_vT_2 – \frac{a}{2V} + u_0##

##2c_vT_1 – c_vT_2 = 2\frac{a}{V} – \frac{a}{2V} – u_0##

##(2T_1 – T_2)c_v = \frac{a}{2V} – u_0##

##(2T_1 – T_2) = \frac{a}{2Vc_v} – u_0##

But i am sensing that isn’t right because I am not incorporating ##n_A## and ##n_B##.

Anyhow, I feel like I am almost getting i but there is some crucial step I am missing.

EDIT: I used twice the initial energy to account for there being two volumes of gas, and wasn’t sure if I should change the a constants (it didn’t seem like it given the parameters of the problem).

ANy help is much appreciated. Thanks.

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