- #1

CAF123

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## Homework Statement

Consider an isolated evacuated chamber. A small adiabatic valve is opened so that the air enters the chamber slowly from the atmosphere. What is the final temperature of the air inside the chamber once it has reached atmospheric pressure? Assume air is an ideal gas with ##\gamma = 1.4##. The air outside the chamber forms a reservoir at P = 1atm and constant temperature 298K throughout.

## Homework Equations

Free expansion, Ideal Gas, Adiabatic process.

## The Attempt at a Solution

I think I may consider this problem to consist of a two partitioned system with the vacuum on one side and the air on another. As the air enters the chamber, it does no work since there is no external agent inside the chamber hindering the expansion. I think this is then a free expansion of air, which means the temperature difference is given by ##\Delta T = \int_{V_i}^{V_f} \left(\frac{\partial T}{\partial V}\right)_U dV = \frac{P_o}{nR} \int_{V_i}^{V_f} dV##

My problem is in writing expressions for ##V_i## and ##V_f##. Take the system to be the number of moles, ##n## in the chamber that have expanded adiabatically. I can write ##V_i = nRT_1/P_o##, and ##V_f = nRT_2/P_o## and the expansion of those n moles can be written as ##P_oV_i^{\gamma} = P_oV_f^{\gamma}## for the initial and final states.

I have tried reexpressing the equations above but I always end up with things like T

_{1}=T

_{2}.

Many thanks.