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We have looked at first order kinetics
and the integrated rate law for first order kinetics.
We are ready now to look at second order kinetics.
We are going to have a very similar process.
We are going to derive the equations that we will use.
The integrated rate law
for second order reactions, we are going to
see how you come up with the half-life equation.
So my goal or you is to be able to use
the second order integrated rate law
to calculate the interdependence between concentration and time,
including the half-life calculations.
We will finish this lesson with a brief
discussion of zeroth order kinetics
and also tie the three together
with some graphical representations
some plot of the three.
So lets look here at the same basic reaction.
A reactant of some sort going to So lets look here at the same basic reaction.
A reactant of some sort going to
products, but this time it is a second order reaction.
We know from the rate expression,
that we have the express for rate.
We also know, since it is second order,
that this is the rate law for the reaction.
Once again if I set these two
portions here equal to each other
this portion equal to portion,
and I employ an integration
I will come up with this expression.
Which relates concentration and time.
This equation is one to commit to memory.
It is a second order reaction,
integrated rate law.
We are not going to, in the lesson time, use this equation
to do calculations, we you will have an
opportunity to view some of those within my
talk hand lessons.
Lets do a little graphical manipulation,
and get it into the slope intercept form of a line again. Lets do a little graphical manipulation,
and get it into the slope intercept form of a line again.
All I have done is a rearrangement.
When you rearrange it you suddenly
see you have it in the form
of a line, a linear relationship,
the slop intercept form, that one can plot
one over a [1/A] on the y-axis
and plot
t along the x-axis.
When I do that, if it is second order
we will get a straight line.
Sometimes that is a way to obtain
the order of the reaction is to
plot, as time goes by, a
relationship of 1 over concentration verse
time, and see if gives a straight line, and if it does
it is a first order reaction.
Now when, you have that linear relationship it is a first order reaction.
Now when, you have that linear relationship
there and y = mx + b
what is the slope going to be equal to?
You think about it and choose your answer.
Did you say 'k', certainly that is correct.
That is the relationship of what m is and that is 'k'
So a plot of
1 over A [1/A] along the y-axis
time [t] along the x-axis
will give a straight line, but the straight line will go off
in this direction, as a positive slope.
Lets work on the half-life information here.
We know that half-life is represented with
t 1/2, so we are going to substitute that in
for t in the equation.
We are going to be at half the original amount
so I am going to replace the A here
with this portion.
With those substitutions, we have this
and once again if I
combine the A naught terms
and solver for half-life
this will give me T 1/2
is equal to 1 over 'k' [1/k]
time the initial concentration.
Let look at this relationship here
this is very different then our half-life
relationship for a
first order reaction.
In a first order reaction we saw that
the amount of time it takes to get to half your start,
whatever that time was will be the same
time it will be to drop in half again.
This is not true for second order kinetics.
Second order kinetics, when we drop it in half
and then we drop it in half again, that second
half-life is going to be longer
because the concentration
is part of that expression.
Concentration is in the denominator
so as we drop it in half
and now we have less amount, we are putting
a smaller number in that denominator
that is going to stretch out the half-life so
it is going to take even longer to drop it in half
a second time, and so forth.
That is the second order half-life equation.
We want to look at zeroth order equations next.
What do we know about zeroth order reactions
they are rated to the zeroth power
of the equation, rate is just a constant.
So if rate is a constant
a plot of concentration verses
time is going to give you a nice straight line.
What would that straight line look like?
Well, if it is concentration
and time plotted like this
concentration is certainly going to drop off
but it is going to drop off in a linear fashion.
So, it is not a common reaction type but it is going to drop off in a linear fashion.
So, it is not a common reaction type
but it is out there.
What about a plot of rate verses time?
If instead of concentration
I put rate as time goes by.
rate is a constant, it does not change so
I would get a nice straight line.
That is a bit of the
graphical representation
of zeroth order reactions.
I would like to finish with just a
summary of those three.
Sometimes you are given in a problem
a list of concentrations
and time.
So they give you some data points.
I will use these line to represent the data points.
They will say, with this information,
tell me what is the order
of the reaction. So the reaction is A
going to products
and you are trying to figure out
what is the order.
I suggest that you start with what is the order.
I suggest that you start with
first order, because that is the most common.
If you were to plot
natural log [ln] of A
verse time
and you get a straight line
then you know it is first order.
So you would just take all these A's and
determine the concentration
the natural log [ln] of those values
so you get new values here
plot these tow portions
on our graph and see if it is a linear relationship.
If it is, it is first order, on our graph and see if it is a linear relationship.
If it is, it is first order,
but if it is not,
if it is not a nice straight line,
then you want to go to the next option. if it is not a nice straight line,
then you want to go to the next option.
Which is typically to plot
one over A [1/A]
verse time.
So you come back up to your data points here
and you figure each of those out in terms of
one over the concentration [1/concentration].
Now if you plot those one over the concentration [1/concentration].
Now if you plot those
and that gives you a straight line
then you know it is second order.
If that does not give you a straight line,
you might want to finish with
the zeroth order. You can do these in any
order you want.
But if you were to plot these
and get a straight line, then it is a zeroth order.
If you were to plot all three of those
and not get a straight line
you do not know the order
but you do know what it is not.
You know it is not zeroth, you know it is not first,
you know it is not second order.
Now, also from each of these you could get
the value of the rate constant.
If you run into a situation where time
and concentration are given
and they want to know the order
then you want to do it and they want to know the order
then you want to do it
using these linear plots.
Now we have looked at
the second order kinetics.
We have looked at its integrated rate law
we have looked at half-life for second order.
We have done a brief mention of zeroth order
and we have given a summary
of first, second, and zeroth
order kinetics, of you were to plot
concentration information verse time.
And this completes this lesson.