- [Instructor] Let's say we have a hypothetical reaction
where reactant A turns into products
and that the reaction is first-order with respect to A.
If the reaction is first-order with respect to reactant A,
for the rate law we can write the rate of the reaction
is equal to the rate constant K
times the concentration of A to the first power.
We can also write that the rate of the reaction
is equal to the negative of the change in the concentration
of A over the change in time.
By setting both of these equal to each other,
and by doing some calculus,
including the concept of integration,
we arrive at the integrated rate law
for a first-order reaction,
which says that the natural log of the concentration of A
at some time T, is equal to negative KT,
where K is the rate constant
plus the natural log of the initial concentration of A.
Notice how the integrated rate law
has the form of Y is equal to mx plus b,
which is the equation for a straight line.
So if we were to graph the natural log of the concentration
of A on the Y axis, so let's go ahead and put that in here,
the natural log of the concentration of A,
and on the X axis we put the time,
we would get a straight line
and the slope of that straight line
would be equal to negative K.
So the slope of this line,
the slope would be equal to the negative
of the rate constant K,
and the Y intercept would be equal to the natural log
of the initial concentration of A.
So right where this line meets the Y axis,
that point is equal to the natural log
of the initial concentration of A.
The conversion of methyl isonitrile to acetonitrile
is a first-order reaction.
And these two molecules are isomers of each other.
Let's use the data that's provided to us in this data table
to show that this conversion is a first-order reaction.
Since the coefficient in front of methyl isonitrile
is a one, we can use this form of the integrated rate law
where the slope is equal to the negative
of the rate constant K.
If our balanced equation had a two as a coefficient
in front of our reactant, we would have had to include 1/2
as a stoichiometric coefficient.
And when we set our two rates equal to each other now
and go through the calculus, instead of getting negative KT,
we have gotten negative two KT.
However for our reaction we don't have a coefficient of two.
We have a coefficient of one and therefore
we can use this form of the integrated rate law.
Also notice that this form of the integrated rate law
is in terms of the concentration of A
but we don't have the concentration of methyl isonitrile
in our data table,
we have the pressure of methyl isonitrile.
But pressure is related to concentration
from the ideal gas law, so PV is equal to nRT.
If we divide both sides by V,
then we can see that pressure is equal to,
n is moles and V is volumes,
so moles divided by volume would be molarity,
so molarity times R times T.
And therefore pressure is directly proportional
to concentration, and for a gas it's easier to measure
the pressure than to get the concentration.
And so you'll often see data
for gases in terms of the pressure.
Therefore, we can imagine this form of the
integrated rate law as the natural log of the pressure
of our gas at time T is equal to negative KT
plus the natural log of the initial pressure of the gas.
Therefore, to show that this reaction
is a first-order reaction we need to graph the natural log
of the pressure of methyl isonitrile on the Y axis
and time on the X axis.
So we need a new column in our data table.
We need to put in the natural log
of the pressure of methyl isonitrile.
So for example, when time is equal to zero
the pressure of methyl isonitrile is 502 torrs.
So we need to take the natural log of 502.
And the natural log of 502 is equal to 6.219.
To save time, I've gone ahead
and filled in this last column here,
the natural log of the pressure methyl isonitrile.
Notice what happens as time increases, right,
as time increases the pressure of methyl isonitrile
decreases since it's being turned into acetonitrile.
So for our graph, we're gonna have the natural log
of the pressure of methyl isonitrile on the y-axis.
And we're gonna have time on the X axis.
So notice our first point here
when time is equal to zero seconds,
the natural log of the pressure as equal to 6.219.
So let's go down and let's look at the graph.
All right, so I've already graphed it here.
And we just saw when time is equal to zero seconds,
the first point is equal to 6.219.
And here I have the other data points already on the graph.
Here's the integrated rate law for a first-order reaction
and I put pressures in there instead of concentrations.
And so we have the natural log of the pressure
of methyl isonitrile on the y-axis
and we have time on the X axis,
and the slope of this line should be equal
to the negative of the rate constant K.
So there are many ways to find the slope of this line,
one way would be to use a graphing calculator.
So I used a graphing calculator
and I put in the data from the data table
and I found that the slope of this line
is equal to negative 2.08
times 10 to the negative fourth.
And since if I go ahead and write y is equal to mx plus b,
I need to remember to take the negative of that slope
to find the rate constant K.
Therefore K is equal to positive 2.08
times 10 to the negative fourth.
To get the units for the rate constant,
we can remember that slope is equal to
change in Y over change in X.
So change in Y would be the natural log of the pressure,
which has no unit, and X the unit is in seconds.
So we would have one over seconds for the units for K.
And finally, since we got a straight line
when we graphed the natural log of the pressure versus time,
we know that this data is for a first-order reaction.
And therefore we've proved that the transformation
of methyl isonitrile to acetonitrile
is a first-order reaction.