In many environmental problems it's necessary to get to high conversion to eliminate a
pollutant. A pollutant may be in low concentration, but we still need to react 90 or 99% of it.
So I just want to use kinetics here to look at the reaction time because we'll see
that it can take longer to go from 90 to 99% conversion than it would take from 0 to 90%
conversion and actually can take dramatically longer in some cases. So what we're going
to do is look at a simple batch reactor to do the comparison. We're going to first look
at a first order reaction. The reaction is just A goes to B where the rate constant is
0.0115 and this rate constant was just chosen to make the numbers come out even. If we do
a mass balance on a batch reactor for component A, it's the change in the number of moles with
time equals, minus sign because A is reacting away, the rate constant times the concentration
times the volume so I've written in A concentration times the reactor volume which we then can
write and this is easy to integrate where we rearrange and then integrate from 0 to time,
t, from our initial number of moles to our number of moles at time, t. So the integration
and the substitution of the limits in the integral, gives us a number of moles of A
at any given time divided by the initial number of moles is equal to e^-kt. And then we're
going to calculate conversion. So x is conversion, fractional conversion, and compare the different
values. So I'm going to multiply conversion by 100 just to use percents. If I plug the numbers
in for this rate constant we find it takes 200 s for the batch reactor to get to
90% conversion. If instead we want to start, convert 99% initially, it takes 400 s
and if you want to convert 99.9% is takes 600 s. So notice 200 s to convert
the first 90%, another 200 s to convert the next 9% and another 200 s to convert
the next 0.9%. So it takes longer and longer to convert that last percent, the last partial
percent, so it can be very difficult to get to very high conversions, but the situation
is actually much worse, it takes much longer if we look at a second order reaction because
as we lower the concentration for a second order reaction, the rate is dropping a lot more.
So what I want to do is do the same calculation now for a second order reaction. So I have
written down a rate constant times initial concentration. I've picked a value so that
we'll get the same time for a 90% conversion to make the comparison a little clearer. Now
we write the rate expression. In order to simplify this we're going to assume liquid phase or what
that really is assuming, it's constant volume. So that means I can write a mass balance.
So volume cancels and then I can solve this equation for concentration by again separating variables,
integrating from 0 to time, t, integrating from our initial concentration of batch reactor
to concentration at a later time. Keep in mind that we're still looking at a batch reactor,
it's isothermal and we just want to compare reaction time. So I've written down the calculations
where we've integrated 1 over C_A squared to get us minus 1 over C_A. Substituted the
limits, where here the limits were 0 to t, and just rearranged and then wrote the conversion.
So it's important to note that conversion is only defined in terms of concentrations
like this when it's constant volume. So I can do this because it's constant volume.
I end up with an expression for x as CA_0 kt over 1 + CA_0 kt. And then what I want
to do is just do the calculation of conversion. So for 90% as I mentioned I picked the rate
constant so it's 200 seconds so it's the same as for first order. For 99% now here's where
the difference is rather dramatic, it takes 10 times as long and for 99.9% it takes 10
times as long again. So 10 times longer, if I convert 90%, to convert that next 9% it
takes 10 times as long. To convert the next 9.9% it takes 100 times as long so it's a
dramatic difference for a second order reaction. It can be very difficult to get to very high
conversion, increases dramatically to get the last percent converted to reach higher
conversions. We're already at high conversion we want to go to the next fraction of percent,
it takes much longer. This is something you need to keep in mind then in systems that
compared to first order, where indeed it takes longer, we're looking at really dramatic differences
when we look at a second order process and this as we mentioned, can be important when
we look at environmental problems.