A rate law is a mathematical expression that relates the rate of a reaction to another factor, often concentration. Rate laws take on the general form
`"rate"=k[C]^n`
`k` is the rate constant of the reaction
`[C]` is the concentration of the species
`n` is the "order" of the reaction
The rate of a reaction will always have units of `M/s` or `"mol"/(L*s)` . Thus, the rate constant will have different units depending on the order of the reaction.
Both the order of the reaction and the rate constant can only be determined experimentally. In other words, we cannot determine the order of a reaction and/or the rate constant outside of directly performing the reaction in a lab and collecting data.
This specific kind of rate law described above is called a differential rate law. A differential rate law expresses the rate as a function of concentration. This explains why the rate of a reaction will change as the reaction proceeds; as the concentration of the species increases or decreases, the rate increases/decreases alongside it.
We briefly touched upon this above, but the order of a reaction is just the sum of the exponents. If we have a rate law of
`"rate"= k[A]^2`
the order of the reaction would be 2. Similarly, if we had a reaction with the rate law
`"rate"=k[A][B]`
the order would also be 2, since each of the concentration terms is raised to the power of 1 i.e `"rate" = k[A]^1[B]^1` .
Usually the orders of reactions don't exceed 2 in real life, though in practice problems they're sometimes higher.
A first order reaction is one in which the rate of reaction depends linearly on the concentration of the single reactant:
`"rate"=k[C]`
The higher the concentration of `C` , the higher the rate. Conversely, the lower the concentration, the lower the rate. To describe this reaction, we say that the reaction is dependent on `C` to the first order.
For first order rate laws, the rate constant has units of `s^(-1)` , where `s` is seconds.
There are two possible forms of second order reactions. The first depends solely on the concentration of one reactant and is written as:
`"rate" = k[A]^2`
If we increase the concentration of `A` two times, the rate increases by 4. Thus, the rate is proportional to the squared concentration.
The second involves two reactants and is first order in both of them. This rate law is written as:
`"rate" = k[A][B]`
If we increase the concentration of `A` by 2 while holding `[B]` constant, the rate increases by a factor of 2. The same thing would happen if we increased the concentration of `B` while holding `[A]` constant. If we increased both `[A]` and `[B]` by 2, the rate increases by a factor of 4. We can therefore say that the reaction is first order in `A` , first order in `B` , and second order overall.
The unit of the rate constant in second order reactions is `1/(M*s)` or `L/("mol"*s)` .
The concept of a zeroeth order reaction is somewhat odd. In a zeroeth reaction, the rate of the reaction is constant and independent of concentration. The rate law is written as:
`"rate"=k`
This is interesting as a zeroeth order reaction will have the same rate regardless of the value of the concentration of the species. In a zeroeth order reaction, it wouldn't matter is `[A]` was `100M` or `0.01M` , the reaction would proceed at the same rate.
A psuedo-first order reaction is a reaction that can be simplified to be approximated as a first order reaction. Imagine that you had a second order reaction following the form
`"rate"=k[A][B]`
but had the problem that both reactants `A` and `B` cost a lot of money. To reduce the cost of experiment, we could use a concentration of one reactant significantly higher than the other. For demonstration purposes, let's say that the concentration of `A` was made to be significantly higher than the concentration of `B` .
Since `[A]`>>`[B]` , we know that `[A]` is essentially a constant because even after reacting with all of `[B]` , `[A]` will be hardly affected. Thus, we can rewrite the rate law as
`"rate"=k[A]_0[B]`
and since `[A]_0` , the initial concentration of `[A]` , is essentially constant, we can define a new rate constant `k'=k[A]_0` .
`"rate"=k'[B]`
Now the reaction looks like a first order reaction! If we tested this experiment out and determined the rate at different concentrations of `B` , we can determine `k'` . From there, we just have to plug back in `k'=k[A]_0` to find the original `k` of the experiment!
An analogy can be made to pouring a gallon of acid into the ocean. Even if the acid was strong and in high concentration, the concentration of the ocean is just so much higher that even after the acid reacts with the ocean, the change in the concentration of the ocean is negligible. For most purposes, psuedo-first order reactions are great approximations.
Psuedo-first order reactions are not limited to approximating second order reactions, by the way. The same process can be done for a third order reaction by making two of the reactants in excess. If we wanted to model
`"rate"=k[A][B][C]`
as a psuedo-first order reaction, we would just have to take two of the reactants in excess. In this case, let's make `[A]` and `[B]` in excess (it doesn't matter which ones we pick). Since the concentrations of `A` and `B` will be practically untouched, we can say that
`"rate"=k[A]_0[B]_0[C]`
If we define a new rate constant `k'=k[A]_0[B]_0` , we get
`"rate"=k'[C]`
and we have a psuedo-first order reaction!
1. A rate law gives a relationship between the rate of a reaction and one of its variables.
2. A differential rate law gives a relationship between the concentration of a species and the rate of a reaction.
3. The rate constant and order of a reaction can only be determined experimentally.
4. A first order reaction has a rate that's proportional to the concentration of the species.
5. A second order reaction has a rate that's proportional to the square of the species.
6. A zeroeth order reaction has a rate that's independent of the concentration.
7. A psuedo-first order reaction is a reaction that is modeled as being first-order by having the concentrations of all but one reactant in excess.