What are the units of k in the following rate law? Rate = k[X][Y]1/2

Question

A. M^-1s^-1

^B. M^-1s^-1/2

^C. M^1/2s^-1

^D. M^-1/2s^-1

^E. M/s

Answer 1

Hi Elva!

The unit of k depends on the overall order of the reaction. This is because all of the units in our rate law must cancel out to give us the units of our rate. Units of rate are typically M/time. In this case, our unit of time is seconds, so our rate is in M/s.

There are two ways to solve this problem.

Method 1: Plugging in units and solving for k.

In the rate law, Rate = k[X][Y]^1/2

1. Our unit of rate is M/s, and
2. [X] and [Y] are concentrations with the unit of molarity (M)

1) Plug in the units for rate and concentration into our rate law:

Rate = k[X][Y]^1/2

M/s = k[M][M]^1/2

2) Solve for k:

M/s = k[M][M]^1/2

(M)/([M][M]^1/2s) = k (we can see that molarity cancels out as follows):

(M)/([M][M]^1/2s) = k

k = 1/(M^1/2s)

This can be rewritten as: M^-1/2s^-1

Method 2: Shortcut method using the overall order of the reaction.

k will usually have two types of units: time and concentration (the exception is first order reactions, in which the unit of k is always time^-1).

1. Unit for time: For k, the unit of time will always be time^-1. Since the unit of time we are using for our rate is seconds, the time component of k will be s^-1. (If the rate were in M/min, then we would be using minutes as our time unit, and the time component of k would be min^-1)
2. Unit for concentration: Except for first order reactions, k will also have a concentration unit (usually molarity) raised to an exponent. This exponent can be found using the overall order of the reaction as follows: Exponent for molarity = 1 - overall order of the reaction.

The overall order can be found by adding up the orders of each reactant in the rate law (i.e. adding up the exponents). If there is no exponent for a particular reactant, we assume the exponent is 1.

So, for the rate law, Rate = k[X][Y]^1/2,

The order with respect to X is 1.

The order with respect to Y is 1/2.

The overall order = 1 + 1/2 = 1.5

Thus the exponent for molarity for this rate law is = 1 - 1.5 = -0.5 (or -1/2)

Putting both the time and concentration units together, the units for k = M^-1/2s^-1

Hope this helps! Please comment if you have any follow-up questions!

Answer 2

Rate is in units Molarity/ second or M/s

[X] and [Y] are concentrations, so they are in units of M

Plug in the units of each of these variables into the equation

(M/s) = k(M)(M)^1/2

To get the unit for k, solve for k