- [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.