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CAF123

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## Homework Statement

A rigid thick walled insulating chamber containing a gas at a high pressure ##P_i## is connected to a large insulating empty gas holder where the pressure is held constant at ##P_A## with a piston. A small valve between the two chambers is opened and the gas flows adiabatically into the cylinder. Prove ##n_i u_i - n_f u_f = h' (n_i - n_f)##, where ##n_i ##= no. of moles of gas initially in chamber, ##n_f## = no. of moles of gas left in chamber, ##u_i## = molar internal energy of gas initially in chamber, ##u_f## = molar internal energy of gas left in chamber and ##h'## = final molar enthalpy of gas in cylinder.

## Homework Equations

First Law of thermodynamics,

State function H = U + PV

## The Attempt at a Solution

The LHS side of the show that is the change in internal energy of the gas left in the chamber. It decreases because the gas does work expanding against the constant pressure ##P_A##. Hence $$\Delta U = W = P_A \int_{V_i}^{V_f} dV,$$ where ##V_i## is the volume occupied by gas after expansion and ##V_f## = volume occupied afterwards.

I can write this as ##P_AV_f - P_A V_i = P_A(V_i + V') - P_AV_i## since ##V_f## is composed of both chamber and cylinder. Terms cancel to give $$ W = P_AV' = P_A \left(\frac{(n_i - n_f)RT}{P_A}\right)$$, but this does not seem close to the show that.

Many thanks.

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