First-order reactions | Kinetics | AP Chemistry | Khan Academy

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 that 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. This law states 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 = mx + 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, let's go ahead and put that in here: the natural log of the concentration of A. On the x-axis, we would put the time. We would get a straight line, and the slope of that straight line would be equal to negative k. Therefore, the slope of this line 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. 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 1, 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 2 as a coefficient in front of our reactant, we would have had to include a one-half as a stoichiometric coefficient. When we set our two rates equal to each other now and go through the calculus, instead of getting negative kt, we've got negative two kt. However, for our reaction, we don't have a coefficient of 2; we have a coefficient of 1, 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 = nRT. If we divide both sides by V, then we can see that pressure is equal to n over V.

Moles divided by volume would be molarity, so molarity times R times T. Therefore, pressure is directly proportional to concentration. For a gas, it's easier to measure the pressure than to get the concentration. 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. 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 of 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 going to have the natural log of the pressure of methyl isonitrile on the y-axis, and we're going to have time on the x-axis. Notice our first point here; when time is equal to zero seconds, the natural log of the pressure is 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've put pressures in there instead of concentrations. So we have the natural log of the pressure of methyl isonitrile on the y-axis, and we have time on the x-axis. The slope of this line should be equal to the negative of the rate constant k.

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. If I go ahead and write y = mx + 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 units, and x has a unit in seconds. So we would have one over seconds for the units for k. 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. Therefore, we've proved that the transformation of methyl isonitrile to acetonitrile is a first order reaction.