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## Homework Statement

I need some help with the following problem:

Consider a thermally insulated enclosure containing an ideal gas initially separated by a thermally insulated partition into two separate regions. One region is at pressure p

_{1}=100 Pa, temprature T

_{1}=200 K, and volume V

_{1}= 10 m

^{3}. The other is at p

_{2}=50 Pa, T

_{2}=300 K, and V

_{2}=20 m

^{3}.

If the partition is removed and the gas from each region is allowed to mix freely, what is the final temprature of the mixture?

## Homework Equations

Ideal gas law:

PV=NKT

p=ρRT

## The Attempt at a Solution

I've first found the final pressure as follows

[itex]U_i \propto P_1 V_1 + P_2 V_2[/itex]

[itex]U_f \propto P_f (V_1+V_2)[/itex]

[itex]P_1 V_1 + P_2 V_2 = P_f (V_1 + V_2)[/itex]

[itex]P_f = \frac{V_1}{V_1 + V_2} P_1 + \frac{V_2}{V_1 + V_2} P_2 = 66.66 \ Pa[/itex]

Now to find the temprature, I wrote

[itex](N_1 + N_2) k T_f = (N_1T_1 + N_2T_2) k = P_f(V_1+V_2) = 66.66[/itex]

But how can I solve for "T

_{f}"? The problem is that I do not know the number of moles of gas in each partition. Is there a way to calculate the number of moles, or should I use a different method to find T

_{f}? Any helps is appreciated.

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