Checking if a reaction is first-order

Question

I saw this question in my book. Some data about a reaction is given and basically I have to demonstrate that the reaction is indeed first order, in order to "justify"the given data.

In the solution given, the integrated rate law for a first order reaction has been used to find the rate constant for different time intervals and it is stated that since the value comes out to be largely constant, the reaction is first order.

Is this enough proof? And am I correct in assuming that this approach was followed because had the values been placed in the integrated rate law for any other order, the values of the rate constant would not have the constancy as shown?

Or is there a better, more concrete approach towards questions of this type?

This is the solution given:

Answer 1

The integral method is a very good to determine rate constants given that the order of the reaction is known.

For a reaction of unknown order, the differential method is better suited (in my opinion) to determine the order - and then the rate constant can be determined via the integral method (as this gives more reliable results for $k$ than the diff. method).

The generic expression for a n-th order reaction is

$$r = \frac{\mathrm{d}c}{\mathrm{d}t} = k\ce{[A]}^n$$

where n is the order of reaction. Now bringing the equation into a linear form using logarithms:

$$\ln(r) = \ln(k) + n \ln([A])$$

Now approximating $\frac{\mathrm{d}c}{\mathrm{d}t}$ as $\frac{\Delta c}{\Delta t}$ and $\ce{[A]}$ as the average between the measurements, one can tabulate the data in the form: $\ln\left(\frac{\Delta c}{\Delta t}\right)$ vs. $\ln ([\ce{A}_{\text{avg}}])$

Plotting this data will give you a linear curve, whose slope will correspond to the overall order of the reaction.

It is important to keep in mind that the approximation $\frac{\mathrm{d}c}{\mathrm{d}t}$ as $\frac{\Delta c}{\Delta t}$ is only valid for a small time interval $\Delta t$ and when the reaction is far from completion.

If you end up with a fractional order of reaction, look at the magnitude of the change of concentration and perform the linear regression only on the data pairs from the beginning of the reaction.