Animated Solution for Chemistry - Chemical Kinetics: Consider following two reactions, $A⟶\text{Product};-\frac{d[A]}{dt}=k_1[A{]}^0$ $B⟶\text{Product};-\frac{d[B]}{dt}=k_2[B]$ $k_1$ and $k_2$ are expressed in terms of molarity (${\text{mol L}}^{-1}$) and time (${\text{s}}^{-1}$) as

The study of chemical kinetics is not just about how fast a reaction goes, but also about understanding the mathematical rules that govern that speed. One of the most fundamental concepts in kinetics is the order of a reaction, which tells us how the rate depends on the concentration of the reactants.

In this problem, we are given two different reactions and asked to find the units of their rate constants, $k_1$ and $k_2$. Let's break down the thought process step-by-step.

We are presented with two rate laws: For reaction A: $-\frac{d[A]}{dt}=k_1[A{]}^0$ For reaction B: $-\frac{d[B]}{dt}=k_2[B{]}^1$

The power to which the concentration term is raised in the rate law is the order of the reaction. Looking closely at the first equation, the concentration of A is raised to the power of 0. This immediately tells us that the first reaction is a zero-order reaction ($n=0$).

Similarly, in the second equation, the concentration of B is raised to the power of 1 (implied, since there is no exponent written). This means the second reaction is a first-order reaction ($n=1$).

Now, how do we determine the units of the rate constant $k$? The rate of any reaction always has the units of concentration per unit time, which is typically ${\text{M s}}^{-1}$ (Molarity per second).

To find the unit of $k$ for any order $n$, we can use a very handy general formula:

This formula is a lifesaver! It works because the rate (${\text{M s}}^{-1}$) must equal $k$ multiplied by the concentration ($\text{M}$) raised to the power of $n$. Rearranging this gives us the master formula above.

Let's apply our master formula to both reactions.

For the first reaction (zero-order, $n=0$): Substitute $n=0$ into the formula: This makes perfect sense. In a zero-order reaction, the rate is equal to the rate constant, so they must have the exact same units.

For the second reaction (first-order, $n=1$): Substitute $n=1$ into the formula: Here, the concentration units cancel out completely, leaving only the inverse of time.

Comparing our results with the given options, we see that $k_1$ has units of ${\text{M s}}^{-1}$ and $k_2$ has units of ${\text{s}}^{-1}$. This perfectly matches option (d).

Remembering the general formula for the units of the rate constant makes these types of questions incredibly straightforward and quick to solve!