The decomposition of dinitrogen pentoxide is a first order reaction .
2N2O5(g) ⇾ 4NO2(g) + O2(g)
The rate of the reaction can be expressed in terms of the rate at which the concentration of the reactant decreases.
Let’s take the first order reaction in which reactant A is converted into product B.
Once again, the rate of reaction can be expressed in terms of the rate at which the concentration of the reactant, A, decreases but it could also be expressed in terms of the rate at which the concentration of the product, B, increases.
If we were to plot a graph of [A] against time, we could draw a tangent to the curve at a particular time and the gradient would be the differential of [A] with respect to time.
The rate equation tells us that the rate of change in the concentration of A is proportional to the concentration of A, and it can be used to determine the rate of reaction at a particular instant during the reaction.
rate = k [A]
If we now combine our two equations for rate of reaction we have the differential rate equation for this reaction.
Differentiating a rate equation tells us the rate at which the concentration of the reactant (y variable) changes with respect to time (𝑥 variable).
Since the rate of the reaction is constantly changing (just look at the gradient of the curve above), it would be useful to have an expression that describes how the concentration of A changes with respect to time. We can’t determine this directly since we can’t measure the rate of the reaction directly – experiments tend to measure the change in concentration of a reactant or product over a specific time period.
The solution is to solve the differential rate equation as this will give us the concentration of the reactant (y variable) as a function of time (the 𝑥 variable). Think of it as starting with an expression that describes the rate of change in concentration (i.e. the gradient) and ending up with an expression for the curve or straight line itself. We start with a differential rate equation and end up with an integrated rate equation.
Integrating a rate equation solves the differential rate equation so that we know the exact nature of the relationship between the concentration of the reactant (y variable) and time (𝑥 variable).
What this means is that if we know the concentration of the reactant, A, at the start of the reaction (the initial [A]) and a value for the rate constant, k, we can use the integrated rate equation to find the concentration of a reactant or product at any time during the reaction.
We can also rearrange our integrated rate equation for a first order reaction into the form y = m𝑥 + c and plot our experimental results in such a way that the negative gradient of the straight line is k, the rate constant.
And finally, we can use it to show that all first order reactions have the form of an exponential decay, y = e-𝑥 i.e. the concentration of the reactant, A, decays exponentially with time.
Starting with our differential rate equation, our first job is to separate the two variables, [A] and time.
Step 2 requires us to integrate both sides of the equation. Our limit will be from [A] = [A]0 to [A] = [A]t where [A]0 is the initial concentration of the reactant at time = 0 and [A]t is the concentration of the reactant at time = t.
And then using standard integrals on both sides (note the use of the second law of logarithms here):
We can now put the two sides of the equation back together and then rearrange to give us the integrated rate equation:
You should be able to see that using this form of the rate equation, if we know the initial concentration of the reactant, [A]0, and the value of k, the rate constant, we can work out the concentration of reactant (or product) at any time during the experiment.
And with a quick shuffle around, the integrated rate equation takes the form y = m𝑥 + c.
And now for the grand finale!
The important thing here is not that you are able to differentiate and integrate rate equations from first principles, but that you can appreciate where all the different forms of the rate equation for a first order reaction have come from and that you can use these equations to plot graphs and find the value for k with confidence.
If you are interested in the maths and in understanding differentiation and integration, then I highly recommend an introduction to differential equations and an introduction to integration at www.mathsisfun.com 😎.
Your can find a practice question on first order reactions using the integrated rate equation here.