What is a Rate Law?
In chemistry, when discussing kinetics, we know that when the concentration of a reactant rises, the rate of the reaction similarly increases. This makes sense, since it logically follows that if we have more reactant in the same volume compared to a lesser amount of reactant, there will be a quicker reaction. However, how do we quantitatively determine how much faster the rate will be? Well, this is where a rate law comes into play. A rate law is defined by saying: R = k[A]^n[B]^m... where R is the rate of the reaction (sometimes also notated as Δ/Δt, which we will delve more in depth into in the next section), k is the rate constant, and n and m are reaction orders for each reactant ([A, B, etc] means the concentration of A, B, etc). That's why there's a ..., in the generalized form a reaction could hypothetically have 3, 4, 5 reactants, though for the AP exam you often won't see more than 2. It's actually quite rare for a reaction with 3+ reactants since it would require three atoms/molecules to bump into each other just right for a reaction to take place. It occurs, but not often and not quickly.
Animation Courtesy of GIPHY
What Does Reaction Order Mean?
n and m define what are called each reactant's reaction order. Reactant order brings us back to the initial question we asked: how do we determine quantitatively how concentration changes the rate of reaction? The reaction order is the answer! It describes how when you increase the concentration, by what factor the rate will increase. For example, let's say the imaginary rate law for the reaction A + B --> C is R = k[A]^2[B]^1. This can tell us that as we increase the concentration of A (assuming a constant [B]), the rate will increase quadratically. For example, if we double the concentration of A, the rate will increase by 4 times. Similarly, if we double the concentration of B, the rate will double, since the order of B is 1. The same applies for orders of 3, 4, etc. (if we double , R goes up by 8 times and 16 times respectively). The total reaction order for the full reaction is the sum of the orders for each reactant.
Quadratic vs. Linear Relationships (x^2 vs x)
Using Experiments to Determine a Rate Law
There's one important thing to note about rate laws. This is that they can only be determined experimentally. What a chemist will do is that they will run a ton of tests at different concentrations and then find the rates for each test. Let's take a look at an example to find out how we mathematically figure this out:
Example Courtesy of Khan Academy
Here we have a reaction: 2NO + 2H2 --> N2 + 2H2O. We can see three experiments
done with different concentrations for each reactant. Let's take a look at the first two reactions where [NO] changes. We see the concentration of [NO] double,
and then we see the rate increase from 1.25 * 10^-5 to 5.00 * 10^-5. This is a change by a factor of 4 (5 * 10^-5/1.25 * 10^-5 = 4). Therefore, since doubling concentration makes rate quadruple, the reaction is 2nd order in NO. Let's take a look at H2 now. For reactions 2 and 3, concentration doubles like before, but the rate increases from 5 * 10^-5 to 1 * 10^-4, an increase by a factor of 1 * 10^-4/5 * 10^-5 = 2. Therefore, the reaction is first order in H2. Now we can put together the rate law! R = k[NO]^2[H2]. As an exercise, pick one of the experiments and plug in the correct numbers to figure out the value of k and then read the next section and figure out the right units for k. (you should get k = 250 M^-2s^-1)
Understanding k, the Rate Constant
The rate constant, k, is a tricky thing to understand. Essentially, it serves as a proportionality constant for the reaction to take place. It makes a bit more sense if you understand the calculus behind kinetics (which we will describe in the next section, though it is by no means required for the AP exam), but essentially all you need to know is that k is a constant that quantifies the rate of each reaction and that it is temperature specific. This means that for the same reaction at different temperatures, the rate constant is different! Another important aspect of the rate constant is that its units change depending on the reaction order. Let's see if we can figure some of them out. Rate is always in M/s, and concentration is in M (M = mol/L). Thus it follows that for certain reactions:
R = k[A]^0 --> R = k, meaning that k is in units M/s.
R = k[A]^1 --> M/s = k * M --> k is in s^-1 (perseconds)
R = k[A]^2 --> M/s = k * M^2 --> k is in M^-1*s^-1 (1/Ms)