Constant volume versus 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

I'm not sure

The Attempt at a Solution

Ok, so part a for a constant volume reactor was simple, since rA=-kC$$_{A}$$, then the rate of change in concentration of A turned out to be ln(C$$_{A}$$/C$$_{A0}$$)=-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...

 

[KataKoniK]

Hi there, thank you for your question! it is important to carefully analyze and understand the problem at hand before attempting to solve it. Let's break down the problem into smaller parts and see how we can approach it.

First, let's define some variables:
C_A = concentration of A
C_{A0} = initial concentration of A
k = rate constant (0.35 min^-1)
t = time (5 minutes)

a) To calculate the fractional decrease in concentration of A, we can use the equation you provided for a constant volume reactor - ln(C_{A}/C_{A0})=-kt. However, for a variable volume reactor, we need to take into account the change in volume as well. As you correctly stated, -k=\frac{dC_{A}}{dt}+\frac{1}{V}\frac{dV}{dt}. To solve for the concentration of A at any given time, we need to integrate this equation. Do you remember how to integrate a differential equation?

b) The total molar conversion of A in a reactor is the amount of A that has been converted to B after a certain amount of time. To calculate this, we need to know the initial moles of A and the final moles of A. Can you think of a way to calculate this using the information given in the problem?

I hope this helps you get started on solving the problem. Remember to carefully analyze the problem and use the appropriate equations and concepts to solve it. Good luck!

 

[piercas]

I would approach this problem by first understanding the difference between a constant volume and a constant pressure batch reactor.

In a constant volume batch reactor, the volume remains the same throughout the reaction, but the pressure can change as the reactant is consumed and products are produced. On the other hand, in a constant pressure batch reactor, the pressure is kept constant by adjusting the volume as the reaction progresses.

For part a, we can use the rate law for the reaction to determine the concentration of A at different times. For a constant volume reactor, we can use the equation ln(C_{A}/C_{A0})=-kt, where C_{A0} is the initial concentration of A and C_{A} is the concentration of A at time t. Solving for C_{A}, we get C_{A}=C_{A0}e^{-kt}. Plugging in the given values, we get C_{A}=1.0e^{-0.35*5}=0.197 M. This means that after 5 minutes, the concentration of A in the reactor will decrease to 0.197 M.

For a constant pressure reactor, we need to consider the effect of changing volume on the rate of the reaction. The rate law for this reactor becomes dC_{A}/dt=-kC_{A}+kC_{A0}(dV/dt)/V. Solving this equation, we get C_{A}=(C_{A0}+kC_{A0}V_{0})e^{-kt/V_{0}}, where V_{0} is the initial volume of the reactor. Plugging in the given values, we get C_{A}=(1.0+0.35*1.0*1.0)e^{-0.35*5/1.0}=0.298 M. This means that the concentration of A in the reactor will decrease to 0.298 M after 5 minutes.

For part b, we need to calculate the total molar conversion of A, which is the amount of A that has been converted to B after 5 minutes. In a constant volume reactor, the total molar conversion can be calculated as X_{A}=1-C_{A}/C_{A0}=1-0.197/1.0=0.803. In a constant pressure reactor, the total molar conversion can be calculated as X_{A}=