We're going to look at a batch reactor that has a reversible reaction, isothermal reactor,
and all we want to do in this screencast is set up the mass balance.
So I don't actually have any numerical values to substitute in to get the numerical final
answer, and other screencasts go through in detail how to obtain numerical answers for
this type of problem.
First thing is to write down the rate of reaction from the information given.
The forward reaction is second order in reactant A and the reverse reaction is first order,
so k_f and k_r are rate constants, and this rate of reaction, one way of defining this
is it's the rate of reaction of component A divided by the stoichiometric coefficient
of component A, because A is reactant stoichiometric coefficient is negative, stoichiometric coefficient
of B is positive, so rate of reaction of B divided by plus 1.
That's the rate of reaction.
If I write a mass balance, and I'll write it in terms of the total number of moles of
A, how this changes with time in the batch reactor, that's the rate of reaction times
the volume.
And the rate of reaction is minus 2 times r times the volume, so this is minus 2 times
k_f, C_A squared, and k_r, concentration of B, times the volume.
Because this is is liquid phase, we can assume the density is constant, and what that allows
us to do is, if I write the number of moles as a concentration of A times the volume,
means the volume of the reactor is not going to change.
That means I can write the left side as the volume times the concentration of A with respect
to time.
The right side hasn't changed, and therefore the volume cancels and the time for a given
conversion is independent of volume.
So now I have an equation where our unknowns are concentration of A, concentration of B,
and time, so I need to relate concentration of B to the concentration of A, and I can
use stoichiometry.
Initial concentration of B minus concentration of B at a later time divided by the stoichiometric
coefficient, same expression for A divided by its stoichiometric coefficient.
So I can substitute in here this concentration of B, and then I have an equation with 1 dependent
and 1 independent variable.
So let me make that substitution.
So notice the substitution I've made is to solve this equation here for the concentration
of B, and it's related to initial concentration of B and the change in the concentration of
A divided by 2.
Okay, now I can separate variables.
Of course, I could solve this numerically, but I'm going to write down the equation for
separated variables.
So here's the equation with variables separated.
Integrate from time equals zero to some time.
Time equals zero, the concentration of A is C_A0, and then C_A at some later time.
So here's the mass balance, batch reactor, reversible reaction, and this we can solve
numerically or we can solve analytically.
The whole objective here is to show how to set up the mass balance where we have a reversible
reaction.