Query in Chemical Kinetics

Question

I have been recently studying chemical kinetics. I've understood what the rate law and the rate constant mean, and also that, kinetics depends completely on experimental observations, due to which we have zero order reactions as well as fractional order reactions.

Take for example, this: $$\text{Rate of a certain chemical reaction} = k [A]^p [B]^q [C]^r $$

Here, $[A]$, $[B]$ and $[C]$ are the molar concentrations of the three reactants involved, while $p$, $q$ and $r$ are the number of molecules of each respectively taking part in the reaction.

So, $\text{the order of the reaction} =p+q+r $.

Now, this $p$, $q$ and $r$ are completely determined experimentally, and cannot be determined theoretically.

How are these values determined experimentally? How does one find out how many molecules of each reactant are taking part in the reaction? Is there any special instrument to do this? If not, then how do scientists find the order of a reaction?

As an attempt to shorten out the answering area, consider this reaction: $$\ce {NO2 +CO -> NO +CO2} $$ Now, experimentally it has been determined that $\text {the rate of this reaction} = k[\ce{NO2}]^2$. I want to know how have scientists determined that the rate depends on the square of the concentration of $\ce {NO2} $ and not on the other reactant.

N.B.: My question is how it can be experimentally determined, and not why it has to be experimentally determined.

Answer 1

Order of a reaction can be found by the following methods:

1. Plotting a graph of concentration of reactant versus time

For example, let us take decomposition of $\ce{N2O}$ to $\ce{N2}$ and $\ce{O2}$ on $\ce{Pt}$ surface. $$\ce{2N2O ->[Pt] 2N2 + O2}$$

The graph will look similar to the following:

The slope of the graph is constant. That is, it does not vary with time.

This is possible when the reaction is zeroth order. Hence, we can conclude that the reaction is zero-order.

For a $\mathrm{1^{st}}$ order reaction, the slope of the graph drawn between $\ln[\mathrm{Reactant}]$ with time is constant, for a $\mathrm{2^{nd}}$ order reaction, the slope of the graph drawn between $\mathrm{\frac{1}{[Reactant]}}$ with time is constant.

In general, for an $n^{\text{th}}$ order reaction, the slope of the graph drawn between $\mathrm{\frac{1}{[Reactant]^{n-1}}}$ with time is constant (where $n \neq 1$).

2. Half life method

This is useful only when a single reactant is present.

In this method, half life of a reaction is measured, varying initial concentration of the reactant. $$t_{\frac{1}{2}} \propto \frac{1}{\mathrm{[A]^{n-1}}}$$ where $n$ is the order of the reaction and $\mathrm{[A]}$ is the initial concentration of the reactant.

3. Initial rate method

In this method, the initial rate of the reaction is measured at different initial concentrations of the reactant.

Let the reaction be $n^{th}$ order w.r.t. a reactant $\mathrm{A}$, then $$\mathrm{r_0 = k[A]^n}$$ at different initial concentration of $\mathrm{A}$, say $a_1$ and $a_2$, the observed initial rates be $r_1$ and $r_2$, then $$n = \frac{\log(r_1)-\log(r_2)}{\log(a_1)-\log(a_2)}$$

In case of multiple reactants, say for a hypothetical reaction $$\ce{11A + 20 B -> 5 C + 9 D },$$ you want to find the order of it.

Let the rate equation be $$ r = k[\ce{A}]^n[\ce{B}]^m$$ First take a reactant in excess and find the order of reaction with respect to other reactant by comparing initial rates or half lives.

Answer 2

You know that rate of a reaction is proportional to molar concentrations of each of the reactant with each term raised to the number of molecules participating in the simultaneous collision i.e reaction. This law is called Law of Mass Action. This is true only for elementary reactions, where all reactant molecules simultaneously collide to form products.

For example:

$$\ce{H2(g) + I2(g) -> 2HI(g)}$$
is a elementary reaction and its

$$\text{rate}=k[\ce{H2}]^1[\ce{I2}]^1$$

which matches with experimentally determined rate.

But the complex reaction:

$$\ce{NO2 + CO-> NO + CO2}$$

is a two step reaction with the formation of an intermediate; $\ce{NO3}$. The mechanism of the reaction is as follows:

Step 1(Rate determining step):

$$\ce{2NO2\xrightarrow[ ]{\text{slow}}NO3 + NO}\label{eqn:1}\tag{1}$$

This step is slower compared to $(\ref{eqn:2})$. Hence the rate of the overall reaction depends on this step.

Therefore overall,

$$\text{rate} =k[\ce{NO2}]^2$$

Step 2:

$$\ce{NO3 + CO\xrightarrow[ ]{\text{fast}} NO + CO2}\label{eqn:2}\tag{2}$$

These are the steps through which the reactants have to go in order to form products. The rate determining step(RDS) will be always the slowest because it has the highest activation energy (forms an unstable product).

Note that each step is an elementary step and law of mass action is separately applicable to each step.

In the case of elementary reactions reactant molecules collide simultaneously giving product molecules with no formation of an intermediate. Hence, law of mass action can predict rates of only such reactions.

Mechanisms can be verified experimentally, with the help of intermediates. There are various method of determining rate of a reaction:

1. Initial rate method: Maintaining concentration of one reactant constant and varying the other.
2. Optical rotation method (if reactants are optically active): Measuring the net optical activity of reaction mixture.
3. Titration: Titrating the reaction mixture.
4. Barometry: Measuring pressure of container of gaseous reactants.

Thus the exponents in the rate law can be determined.