Experimental Determination of Rate Law
We’ve stated this a few times now, but it bears repeating: The values of k, x, and y in the rate law equation (rate = k[A]x[B]y) must usually be determined experimentally for a given reaction at a given temperature. Although rate laws can be quite complex and the orders of the reaction difficult to discern, the MCAT limits its coverage of this topic to fairly straightforward reaction mechanisms, experimental data, and rate laws.
The first step in determining a rate law for a specific reaction is to write out the generic rate law on the scratch material provided for you at the testing center. Then look for the necessary data. Typically, you’ll recognize a chart that includes initial concentrations of the reactants and the initial rates of product formation as a function of the reactant concentrations. Usually, the data for three or four trials are included in the chart.
Once you’ve located the data, the next step is to identify a pair of trials in which the concentration of one of the reactants is changed while the concentration of all other reactants remains constant. Under these conditions, any change in rate of product formation (if there is any) from the one trial to the other is due solely to the change in concentration of one reactant. Let’s imagine that compound A’s concentration is constant, while the concentration of B has been doubled. If the rate of the formation of product C has subsequently quadrupled, then you can say to yourself (using your in-the-head-voice, because nobody wants to hear you talk to yourself on Test Day!), “Doubling the concentration of B has resulted in a quadrupling of the production rate of C, so to determine the order of the reaction, y, with respect to reactant B, I need to calculate the power by which the number 2 must be raised to equal 4. Because 2y = 4, y = 2.” (Kaplan wants you to think exactly this way. When you read this sentence, exactly as we’ve written it, you will be analyzing the data in the correct manner and working through the correct process for generating the correct rate law.)
MCAT Expertise
Traditionally, the MCAT has loved rate problems. With practice, you’ll be able to do these quickly in your head with minimal paper-and-pencil calculations. Remember to look for pairs of reaction trials in which the concentration of only one species changes while the other(s) remain constant.
The next step is to repeat this process for the other reactant, using different data from a different pair of trials, always making sure that the concentration of only the reactant whose order you are trying to determine is changed from one trial to the other while the concentration of any other reactant remains the same. Once you’ve determined the orders of the reaction with respect to each reactant, you then can write the complete rate law, replacing the x and the y (and sometimes z) with actual numbers. To determine the value of the rate constant k, you will need to plug in actual values—you can use the data from any one of the trials; pick whichever trial has the most arithmetically convenient numbers—for the reactant concentrations and the product formation rate, once you know the values for the exponent for each reactant.
Example: Given the data below, find the rate law for the following reaction at 300 K.
Solution: First, look for two trials in which the concentrations of all but one of the substances are held constant.
a) In Trials 1 and 2, the concentration of A is kept constant, while the concentration of B is doubled. The rate increases by a factor of 8.1/2.0, approximately 4. Write down the rate expression of the two trials.
Trial 1: r1 = k[A]x [B]y = k(1.00)x (1.00)y
Trial 2: r2 = k[A]x [B]y = k(1.00)x (2.00)y
Divide the second equation by the first:
b) In Trials 2 and 3, the concentration of B is kept constant, while the concentration of A is doubled; the rate is increased by a factor of 15.9/8.1, approximately 2. The rate expressions of the two trials are as follows:
Trial 2: r2 = k(1.00)x (2.00)y Trial 3: r3 = k(2.00)x (2.00)y
Divide the second equation by the first,
The order of the reaction with respect to A is 1 and with respect to B is 2; the overall reaction order is 1 + 2 = 3.
To calculate k, substitute the values from any one of the above trials into the rate law; e.g.,
2.0 M/sec = k × 1.00 M × (1.00 M)2 k = 2.0 M-2 sec-1
Therefore, the rate law is r = 2.0 M-2 sec-1 [A][B]2.
REACTION ORDERS
We classify chemical reactions on the basis of kinetics into classes of reactions called zero-order, first-order, second-order, mixed-order, or higher-order reactions. We will continue to consider the generic reaction aA + bB
Zero-Order Reactions
A zero-order reaction is one whose rate of formation of product C is independent of changes in concentrations of any of the reactants, A and B. These reactions have a constant reaction rate equal to the rate coefficient (rate constant) k. The rate law for a zero-order reaction is
rate = k[A]0[B]0 = k
where k has units of M·s-1. (Remember that any number raised to the zero power equals 1.) We will remind you that the rate constant itself is dependent upon temperature; thus, it is possible to change the rate for a zero-order reaction by changing the temperature. The only other way to change the rate of a zero-order reaction is by the addition of a catalyst, which lowers the energy of activation, thereby increasing the value of k.
MCAT Expertise
Temperature is the only factor that can change the rate of a zero-order reaction.
First-Order Reactions
A first-order reaction (order = 1) has a rate that is directly proportional to only one reactant, such that doubling the concentration of, say, reactant A results in a doubling of the rate of formation of product C. The rate law for a first-order reaction is
rate = k[A]1 or rate = k[B]1
where k has units of s-1. A classic example of a first-order reaction is the process of radioactive decay. From the rate law, in which the rate of decrease of the amount of a radioactive isotope A is proportional to the amount of A,
The concentration of radioactive substance A at any time t can expressed mathematically as
[At] = [Ao]e-kt
where [Ao] is the initial concentration of A, [At] is the concentration of A at time t, k is the rate constant, and t is time. It is important to recognize that a first-order rate law with a single reactant suggests that the reaction begins when the molecule undergoes a chemical change all by itself, without a chemical interaction and, usually, without a physical interaction with any other molecule.