How Does Gas Expansion Relate to Changes in Molar Internal Energy and Enthalpy?

[CAF123]

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.

 

[Chestermiller]

Let u' be the final molar internal energy of the gas in the cylinder. In terms of u_i and n_i, what is the initial internal energy of the gas in the combined system? In terms n_i, u_f, n_f, and u', what is the final internal energy of the combined system? In terms of P_A and V (where V is the final volume of gas within the cylinder), how much work was done? Write the first law expression involving the change in internal energy of the combined system and the work which was done.

Chet

 

[CAF123]

Let u' be the final molar internal energy of the gas in the cylinder. In terms of u_i and n_i, what is the initial internal energy of the gas in the combined system?

The initial state is just composed of the chamber so that is $n_i u_i$.

In terms n_i, u_f, n_f, and u', what is the final internal energy of the combined system?

That is $u_f n_f + (n_i -n_f)u'$

In terms of P_A and V (where V is the final volume of gas within the cylinder), how much work was done?

I think it is what I derived above: $P_A V$

Write the first law expression involving the change in internal energy of the combined system and the work which was done.

So, $$u_f n_f + (n_i - n_f)u' - n_iu_i = P_AV \Rightarrow u_f n_f - u_i n_i = P_A V - (n_i - n_f)u'$$
I can relate $h' = u' + P_AV$ and sub for $P_AV$ but this is not quite the result either.

Thanks.

 

[Chestermiller]

The initial state is just composed of the chamber so that is $n_i u_i$.


That is $u_f n_f + (n_i -n_f)u'$


I think it is what I derived above: $P_A V$



So, $$u_f n_f + (n_i - n_f)u' - n_iu_i = P_AV \Rightarrow u_f n_f - u_i n_i = P_A V - (n_i - n_f)u'$$
I can relate $h' = u' + P_AV$ and sub for $P_AV$ but this is not quite the result either.

Thanks.

Very nice job. Now, only one more step. Recognize that V is the total volume of the cylinder, not the final volume per mole. Call the final volume per mole v'. What is v' in terms of the n's and V?

Chet