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# Rate Constant

If you're reading this, you're probably diving into reaction rates, rate laws, and rate constants right now in your chemistry studies. A key skill in chemical kinetics is the ability to calculate the rate constant for chemical reactions mathematically. So let's talk about rate constants now!

• First, we will review reaction rates and look at the definition of rate constant.
• Then, we will look at the units for the rate constant and the equation for rate constant.
• After, we will solve some problems involving rate constant calculations.

## Rate Constant Definition

Before diving into the rate constant, let's review reaction rates and rate laws.

The reaction rate is referred to as the speed at which a specific reaction proceeds from reactants to products.

The reaction rate is directly proportional to temperature, so when temperature increases, the reaction rate becomes faster than before! This is because the more energy the reaction mixture has, the quicker the particles move around, successfully colliding with others more frequently.

Two other important factors that affect reaction rates are concentration and pressure. Similar to the effects of temperature, an increase in concentration or pressure will also lead to an increase in the rate of the reaction.

To get the instantaneous rate of a reaction we monitor the change in the concentration of a component over a series of very short periods which span over a short interval of time. If the plot of the concentration of a reaction component, over a given short time interval, yields a linear curve, then the slope of the graph is equal to the instantaneous reaction rate.

The rate law for a reaction is a mathematical expression that relates the rate of reaction to changes in the concentrations of either reactants or products.

The equation for the instantaneous reaction rate can be expressed as a change in product concentration over a series of very short time intervals, for example over 10 seconds. Since the concentrations of products increase with time, the reaction rate in terms of products will be positive. On the other hand, if the instantaneous reaction rate is expressed in terms of reactants, because the concentrations of reactants decrease with time, the reaction rate will be negative.

$$\text{aA + bB}\longrightarrow \text{cC + dD}$$

$$\text{Reaction rate} = \text{ }\color {red}- \color {black}\frac{1}{a}\frac{\Delta[\text{A}]}{\Delta \text{t}} = \text{ } \color {red} - \color {black}\frac{1}{b}\frac{\Delta[\text{B}]}{\Delta \text{t}} = \text{ } \frac{1}{c}\frac{\Delta[\text{C}]}{\Delta \text{t}} = \text{ } \frac{1}{d}\frac{\Delta[\text{D}]}{\Delta \text{t}}$$

Let's look at an example. Suppose that you are dealing with the chemical reaction below. What would be the reaction rate of N2?

$$2\text{ NH}_{3}(\text{g})\text{ }\rightleftharpoons \text{N}_{2} (\text{g})\text{ + 3 H}_{2}\text{(g)}$$

This is fairly simple to answer. All we need to do is look at the reaction and apply the equation for the instantaneous reaction rate! So, for N2, the instantaneous reaction rate would be $$\frac{1}{1}\frac{\Delta[\text{N}_{2}]}{\Delta \text{t}}$$, where, Δ[N2], is the change in concentration (Final concentration - Initial concentration), and Δt is a very short time interval.

Now, what if you were given the same exact chemical reaction and were told that the instantaneous reaction rate of N2 is equal to 0.1 M/s? Well, we could use this instantaneous reaction rate to find the instantaneous reaction rate of H2! Since 3 moles of H2 are produced for every 1 mole of N2, then the reaction rate for H2 will be three times that of N2!

For an in-depth explanation of rates of reaction and rate laws, check out "Reaction Rates" and "Rate Law"!

The second topic we need to review is rate law. Rate laws must also be determined experimentally, and its general equation for a power rate law is as follows:

$$\text {Rate} = \color {#1478c8}k \color {black}[\text{A}]^{\text{X}}[\text{B}]^{\text{Y}}...$$

Where,

• A and B are reactants.

• X and Y are the reaction orders of the reactants.

• k is the rate constant

When it comes to reaction orders, the greater the value, the more that a change in the concentration of that reactant will affect the overall reaction rate.

• Reactants whose exponents (reaction orders) equal zero will not have an effect on reaction rates when their concentration is changed.

• When the reaction order is 1, doubling the concentration of the reactant will double the reaction rate.

• Now, if the reaction order is 2, if the concentration of that reactant gets doubled, the rate of reaction will be quadrupled.

For example, the experimentally determined rate law for a reaction between NO and H2 is $$\text{Rate = }k[\text{NO}]^{2}[\text{H}_{2}]^{1}$$. By adding the reaction orders, we can determine the overall reaction order of the rate law expression, which is 3 in this case! Therefore, this reaction is third-order overall.

$$2\text{ NO (g) + 2 H}_{2}\text{ (g)}\longrightarrow\text{ N}_{2}\text{ (g) + 2 H}_{2}\text{O (g)}$$

Now, take another look at the rate law equation above. Notice that there is a rate constant (k) present in its formula! But what exactly does it mean? Let's take a look at the definition of rate constant.

The rate constant k is used by chemists to compare the speed of different reactions, as it gives the relation between the rate of reaction and the reactant concentration in the reaction.

Just like rate laws and reaction orders, rate constants are also determined experimentally!

## Rate Constant Units

Rate constant units vary based on the order of reactions. In zero-order reactions, the rate law equation is Rate = k and the unit of rate constant in this case is, $$\text{mol L}^{-1}\text{s}^{-1}$$.

For first-order reactions, Rate = k[A]. The constant rate unit, in this case, is $$\text {s}^{-1}$$. On the other hand, second-order reactions have a rate law of, Rate = k[A][B], and rate constant unit of. $$\text{mol}^{-1}\text{L}\text{ s}^{-1}$$.

 Reaction Order Rate Law Rate Constant Units 0 $$\text{Rate = }k$$ $$\text{mol L}^{-1}\text{s}^{-1} \textbf{ or }\text{M s}^{-1}$$ 1 $$\text{Rate = }k[\text{A}]$$ $$\text {s}^{-1}$$ 2 $$\text{Rate = }k[\text{A}][\text{B}]$$ $$\text{mol}^{-1}\text{L}\text{ s}^{-1} \textbf{ or }\text{M}^{-1} \text { s}^{-1}$$ 3 $$\text{Rate = }k[\text{A}]^{2} \text{[B]}$$ $$\text{mol}^{-2}\text{L}^{2}\text{ s}^{-1} \textbf{ or }\text{M}^{-2} \text { s}^{-1}$$

## Rate Constant Equation

Depending on the reaction order we are dealing with, the equation to calculate the rate constant differs. Zero-order reactions are by far the easiest ones to solve for the rate constant because k is equal to the rate of the reaction (r).

$$k = r$$

In the case of a first-order reaction, k will be equal to the rate of the reaction divided by the reactant concentration.

$$k = \frac{r}{[A]}$$

Now, for second and third-order reactions, we would have the rate constant equations $$k = \frac{r}{[A][B]}$$ and $$k = \frac{r}{[A]^{2}[B]}$$, respectively.

## First Order Rate Constant

To better understand the rate constant, let's talk about the first-order reactions and first-order rate constant.

Reactions whose rate depends solely on the concentration of a single reactant are called first-order reactions. Hence, $$\text{rate = }-\frac{1}{a}\frac{\Delta[\text{A}]}{\Delta \text{t}} = k[\text{A}]^{1}$$.

When a kinetic plot is done for a first order reaction, the kinetic graph of ln[A]t versus t yields a straight line with a slope of negative k.

Figure 2. ln [A] vs. time graph for a first-order reaction, Isadora Santos - StudySmarter Originals.

## Rate Constant Calculations

Lastly, let's walk through how to do calculations involving rate constant, similar to what you will most likely encounter during the AP chemistry exam.

### Solving a Multi-Step Problem

Sometimes analyzing a chemical equation doesn't tell the full story. As you should be aware, final chemical equations are usually the overall chemical equations. This means that there may be more than one step that produces the overall equation. For example, take the following overall chemical equation, where each step is fully written out, including how fast each step relatively occurs.

$$1. \text{ NO}_{2}\text{ + NO}_{2}\longrightarrow \text{NO}_{3}\text{ + NO } (slow)$$

$$2. \text{ NO}_{3}\text{ + CO}\longrightarrow \text{NO}_{2}\text{ + CO}_{2}\text{ } (fast)$$

$$\rule{8cm}{0.4pt}$$

$$\text{ NO}_{2}\text{ + CO}_{2}\longrightarrow \text{NO}\text{ + CO}_{2}\text{ }$$

As you can see, the overall chemical equation is found by canceling the common reactants and products. This applies to the entire system of chemical equations. (For example, the NO2 in the reactants of step 1 cancels the NO2 in the products of step 2, which is why NO2 doesn't appear in the products of the overall reaction.) But how would you figure out what the rate law is for a problem like this? Take a second to think about what determines how fast this reaction occurs.

Intuitively, the overall reaction is only as fast as its slowest step. This means that the overall rate law for this reaction would be its slowest step, which would be Step 1. This also means that Step 1 would be the rate-Determining step. As for solving the rate constant, we now just follow the same process we have before. We need to set up a rate law equation using the rate-determining step, and then solve for k.

$$\text{Rate = }k[\text{NO}_{2}][\text{CO}_{2}]$$

$$k = \frac{\text{Rate}}{[\text{NO}_{2}][\text{CO}_{2}]}$$

### Solving an Experimental Problem

As mentioned earlier in this lesson, chemists have to experimentally determine a chemical equation's unique rate law. But how do they do this? As it turns out, the AP test has problems that are just like this.

For example, let's say that we have chlorine gas reacting with nitric oxide, and we want to determine the rate law and rate constant from the following experimental data. How would we do this? Let's take a look!

$$2 \text{ NO (g) + Cl}_{2}\text{ (g)} \rightleftharpoons \text{2 NOCl (g)}$$

 Experiment Initial concentration of NO (M) Initial concentration of Cl2 (M) Initial Rate (M/s) 1 0.10 0.10 0.18 2 0.10 0.20 0.36 3 0.20 0.20 1.44

In this type of calculation, the first step is to find the rate law. The basic rate law expression, in this case, can be written as:

$$\text{Rate = }k [\text{NO}]^{X}[\text{Cl}_{2}]^{Y}$$

However, we don't know the reaction orders of the reactions, so we need to use the experimental data collected from three different experimental trials to find out what type of reaction order we are dealing with!

First, choose two trials where only one concentration changes. In this case, let's compare experiments 2 and 3. Experiment 2 used 0.10 M of NO and 0.20 M of Cl2, whereas experiment 3 used 0.20 M of NO and 0.20 M of Cl2. When comparing them, notice that doubling the NO concentration (from 0.10 M to 0.20 M) and keeping the concentration of Cl2 constant causes an increase in the initial rate from 0.36 M/s to 1.44 M/s.

So, if you divided 1.44 by 0.36, you will get 4, which means that doubling the concentration of NO, quadrupled the initial rate from experiment 1. So, the rate law equation, in this case, will be:

$$\text{Rate = }k [\text{NO}]^{2}[\text{Cl}_{2}]^{1}$$

Now that we know the rate law expression, we can re-arrange it to solve for the rate constant, $$k$$!

$$k = \frac{\text{Rate}}{[\text{NO}]^{2}[\text{Cl}_{2}]}$$

$$k = \frac{\text{1.44 M/s}}{[\text{0.20 M}]^{2}[\text{0.20 M}]} = \textbf {180} \textbf{ M}^{-2}\textbf{s}^{-1}$$

As a matter of fact, it doesn't matter which experiment trial you choose to use for your rate constant calculation. For example, if I used the data from experiment 1 instead, I would still get the same rate constant value!

$$k = \frac{\text{0.18 M/s}}{[\text{0.10 M}]^{2}[\text{0.10 M}]} = 180 \text{ M}^{-2}\text{s}^{-1}$$

Hopefully, you now feel more confident when approaching problems involving rate constant. Remember: take your time with these sorts of calculations, and always double-check your work!

## Rate Constant - Key takeaways

• The reaction rate is referred to as the speed at which a specific reaction proceeds from left to right.
• The rate constant k is used by chemists to compare the speed of different reactions, as it gives the relation between the rate of reaction and the reactant
• Rate constant units vary based on the order of reactions.

## References

1. Chad’s Videos. (n.d.). Chad’s Prep -- DAT, MCAT, OAT & Science Prep. Retrieved September 28, 2022, from https://courses.chadsprep.com/courses/take/organic-chemistry-1-and-2
2. Jespersen, N. D., & Kerrigan, P. (2021). AP chemistry premium 2022-2023. Kaplan, Inc., D/B/A Barron’s Educational Series.
3. Moore, J. T., & Langley, R. (2021a). McGraw Hill : AP chemistry, 2022. Mcgraw-Hill Education.
4. Theodore Lawrence Brown, Eugene, H., Bursten, B. E., Murphy, C. J., Woodward, P. M., Stoltzfus, M. W., & Lufaso, M. W. (2018). Chemistry : the central science (14th ed.). Pearson.

The rate constant k is used by chemists to compare the speed of different reactions, as it gives the relation between the reaction rate and the concentration of the reactant in the reaction.

To find the rate constant, we first need to find the rate law expression for the reaction, and we re-arrange it to solve for the rate constant, k.

The rate constant k is equal to the reaction's velocity provided that the reactants are in the units of M or mol/L.

The reaction rate is referred to as the speed that a specific reaction proceeds from left to right. The rate constant gives the relation between the reaction rate and the concentration of the reactant in the reaction.

Rate constant is affected by the reaction rate and the concentration of reactants.

## Final Rate Constant Quiz

Question

The ______  is referred to as the speed at which a specific reaction proceeds from left to right (reactants to products) in a given amount of time.

Reaction rate

Show question

Question

What happens to a reaction rate if temperature is increased?

The reaction rate increases

Show question

Question

Reaction rate is ______ to both concentration and pressure.

directly proportional

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Question

True or false: The equation for a reaction rate is expressed as a change in product concentration over a time interval.

True

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Question

Reactants whose exponents (reaction orders) equal _____ will not have an effect on reaction rates when their concentration is changed.

0

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Question

When the reaction order is _____ , doubling the concentration of the reactant will double the reaction rate

1

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Question

If the reaction order is ____ , if the concentration of that reactant gets doubled, the rate of reaction will be quadrupled.

2

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Question

True or false: rate laws and rate constants are determined experimentally.

True

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Question

The ______ is used by chemists to compare the speed of different reactions, as it gives the relation between the rate of reaction and the reactant concentration in the reaction.

Rate constant (k)

Show question

Question

What is the unit of rate constant in first-order reactions?

s-1

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Question

What is the unit of rate constant in second order reactions?

M-1 s-1

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Question

What is the equation used to calculate rate constant (k) is second-order reactions?

Since the rate law expression of a second order reaction is rate = k[A][B], we can re-arrange the equation to solve for k.

Show question

Question

True or false: In zero-order reactions, k = r

True

Show question

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