Constant Volume and Constant Pressure Batch Reactor

[missmaria]

1. The problem statement, all variables and given/known data
Consider the following two well-mixed, isothermal batch reactors for the elementary and irreversible decomposition of A to B, A\stackrel{k}{\rightarrow}2B
reactor1: The reactor volume is constant (Pressure is variable)
reactor2: The reactor pressure is constant (Volume is variable)
Both reactors are charged with pure A at 1.0 atm and k=0.35min^{-1}

a) what is the fractional decrease in the concentration of A in reactors 1 and 2 after 5 minutes?
b)what is the total molar conversion of A in reactors 1 and 2 after 5 minutes?



2. Relevant equations

Since reaction is first order: r_{A}=-kC_{A}

3. The attempt at a solution

Ok, so part a for a constant volume reactor was simple, since the ODE was easy and it turned out that \frac{C_{A}}{C_{A0}}=exp(-kt)
but variable volume has me stumped. I figured out that -k=\frac{dC_{A}}{dt}+(\frac{1}{V})(\frac{dV}{dt} but i don't know where to go from here.

I don't even know where to start for part b, please help...

[NotMaria]

This problem is a bit complicated, but I remember it from my days as an undergraduate. You have to use a Laplacian transform of the Ideal Gas equation (assuming that T is high enough at 1atm to support this assumption). This should give you an equation for concentration in the form of a Gaussian, which, when integrated, will give an error function. You should work this out for yourself, but your answer should be of the form:

Ca / Cao = erf(-Vr * k * t)

Hope this helps.

[missmaria]

I'm not sure i understand what exactly you mean, could you spell it out stepwise?

[NotMaria]

just kidding. You actually use the Ideal gas equation and (think about the assumptions made for the ideal gas law) calculate delta(V). Then, think about what Cj means