Constant Volume and Constant Pressure Batch Reactor

[missmaria]

Homework Statement

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?

Homework Equations

Since reaction is first order: r$$_{A}$$=-kC$$_{A}$$

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