# 18.2: Rate Laws

Difficulty Level: At Grade Created by: CK-12

## Lesson Objectives

• Use a rate law to describe the concentration dependence of a reaction rate.
• Determine the rate law for a reaction by analysis of experimental data.
• Calculate the specific rate constant from experimental data.

## Lesson Vocabulary

• first-order reaction
• rate law
• specific rate constant

### Recalling Prior Knowledge

• What is a reaction rate?
• Why does the rate of a reaction depend on the concentrations of the reactants?

The rate of a chemical reaction increases as the concentrations of the reactants increase. The specific relationships between each concentration and the reaction rate are given by a rate law. In this lesson, you will learn how to write and analyze rate laws for various reactions.

## Rate Law Concept

Consider a simple chemical reaction in which reactant A is converted into product B according to the equation below.

\begin{align*}\mathrm{A \rightarrow B}\end{align*}

As seen in the last lesson, the rate of reaction is given by the change in concentration of A as a function of time. If this reaction proceeds by a single-step mechanism, which we will explain more completely later in this chapter, the rate of disappearance of A is also proportional to the concentration of A.

\begin{align*}\mathrm{- \dfrac{\Delta [A]}{\Delta t} \propto [A]}\end{align*}

Since the rate of a reaction generally depends upon collision frequency, it stands to reason that as the concentration of A increases, the reaction rate increases. Likewise, as the concentration of A decreases, the reaction rate decreases. The expression for the rate of the reaction can be shown as follows:

\begin{align*}\mathrm{rate=- \dfrac{\Delta [A]}{\Delta t}}\end{align*} or rate = k[A]

The proportionality between the rate and [A] becomes an equal sign by the insertion of a constant (k). A rate law is an expression showing the relationship of the reaction rate to the concentrations of each reactant. The specific rate constant (k) is the proportionality constant relating the rate of the reaction to the concentrations of the reactants. The rate law and the specific rate constant for any chemical reaction must be determined experimentally. The value of the rate constant is temperature dependent. A large value for the rate constant means that the reaction is relatively fast, while a small value means that the reaction is relatively slow.

In the reaction described above, the rate of the reaction is directly proportional to the concentration of A, which can also be written as A raised to the first power. That is to say, [A] = [A]1. A first-order reaction is a reaction in which the rate is directly proportional to the concentration a single reactant. As a first-order reaction proceeds, the rate of the reaction decreases because the concentration of the reactant also decreases (Figure below). Thus, the graph of concentration versus time is curved. The reaction rate (Δ[A]/Δt) at any given time can be determined graphically by the slope of a line that is tangent to the curve at that point. For example, the rate of the reaction at the time where the red line intersects with the black curve is given by:

\begin{align*}\mathrm{rate=- \dfrac{[A]_{final}-[A]_{initial}}{\Delta t}=- \dfrac{0.35 \ M-0.63 \ M}{3.0 \ s-1.0 \ s}=0.14\:M/s}\end{align*}

This graph shows how the concentration of a reactant changes as a reaction proceeds. The rate of the reaction is determined at any point by measuring the slope of a tangent to the curve.

The rates of some reactions depend on the concentrations of more than one reactant. Consider a reaction in which a molecule of A collides with a molecule of B to form product C.

\begin{align*}\mathrm{A+B \rightarrow C}\end{align*}

Doubling the concentration of A alone would double the reaction rate. Likewise, doubling the concentration of B alone would also double the rate. The rate law must reflect this dependence on the concentrations of both reactants.

\begin{align*}\mathrm{rate=k[A][B]}\end{align*}

This reaction is said to be first order with respect to A, and first order with respect to B. Overall, it is a second-order reaction. We have again derived this rate law based on the assumption that the reaction proceeds by a single-step mechanism. With real reactions, where we cannot necessarily make this type of assumption, the rate law and the order of a reaction must be determined experimentally.

## Determining the Rate Law

In order to experimentally determine a rate law, a series of experiments must be performed with various starting concentrations for each reactant. The initial rate is then measured for each of the reactions. Consider the reaction between nitrogen monoxide gas and hydrogen gas to form nitrogen gas and water vapor.

\begin{align*}\mathrm{2NO}{(g)} \mathrm{+2H_{2}}{(g)} \mathrm{\rightarrow N_{2}}{(g)} \mathrm{+2H_2O}{(g)}\end{align*}

The following data (Table below) were collected for this reaction at 1280°C.

Experiment [NO] [H2] Initial Rate (M/s)
1 0.0050 0.0020 1.25 × 10−5
2 0.010 0.0020 5.00 × 10−5
3 0.010 0.0040 1.00 × 10−4

Notice that the starting concentrations of NO and H2 were varied in a specific way. In order to compare the rates of reaction and determine the order with respect to each reactant, the initial concentration of each reactant must be changed while the other is held constant.

Comparing experiments 1 and 2: the concentration of NO was doubled, while the concentration of H2 was held constant. The initial rate of the reaction quadrupled, since 5.00 × 10−5/1.25 × 10−5 = 4. Therefore, the order of the reaction with respect to NO is 2. In other words, rate ∝ [NO]2. Because 22 = 4, doubling the value of [NO] increases the rate by a factor of four.

Comparing experiments 2 and 3: the concentration of H2 was doubled while the concentration of NO was held constant. The initial rate of the reaction doubled, since 1.00 × 10−4/5.00 × 10−5 = 2. Therefore, the order of the reaction with respect to H2 is 1 (rate ∝ [H2]1). Because 21 = 2, doubling the value of [H2] also doubles the reaction rate.

The overall rate law incorporates both of these results into a single equation:

\begin{align*}\mathrm{rate=k[NO]^2[H_2]}\end{align*}

The sum of the exponents is 2 + 1 = 3, making the reaction third-order overall. Once the rate law for a reaction is determined, the specific rate constant can be found by substituting the data for any of the experiments into the rate law and solving for k.

\begin{align*}\mathrm{k=\dfrac{rate}{[NO]^2[H_2]}=\dfrac{1.25 \times 10^{-5} \ M/s}{(0.0050 \ M)^2(0.0020 \ M)}=250 \ M^{-2}s^{-1}}\end{align*}

Notice that the rate law for the reaction does not exactly correspond to the chemical equation for the overall reaction. The coefficients of NO and H2 in the balanced equation are both 2, but the order of the reaction, with respect to H2, is only one. The units for the specific rate constant vary with the order of the reaction.

So far, we have seen reactions that are first- or second-order with respect to a given reactant. Occasionally, the rate of a reaction may not depend on the concentration of one of the reactants at all. In this case, the reaction is said to be zero-order with respect to that reactant. The analysis of reaction mechanisms in the next lesson will illustrate how this is possible.

## Lesson Summary

• The concentration dependence of a reaction rate is shown in an equation called a rate law. The rate of a reaction is equal to a specific rate constant multiplied by the concentration of each reactant raised to some power.
• The specific rate constant is unique for every reaction and is dependent upon temperature. A large rate constant indicates a relatively fast reaction, while a small rate constant indicates a relatively slow reaction.
• Rate laws must be determined experimentally. The order of the reaction with respect to each reactant and the value of the specific rate constant can be determined by a set of experiments in which the concentrations of various reactants are systematically varied.

## Lesson Review Questions

### Reviewing Concepts

1. What is the general relationship between the rate of a reaction and the concentrations of the reactants? Explain.
2. How does the size of the specific rate constant (k) for a reaction relate to the general speed of the reaction?
3. A certain reaction is found to depend only upon the concentration of reactant A. How will the rate of the reaction be affected when the initial concentration of A is doubled, given each of the three possibilities for the order of the reaction?
1. 1st order
2. 2nd order
3. zero order
4. What are the units of the specific rate constant when the reaction is zero order? 1st order? 2nd order? 3rd order?
5. When performing a set of experiments to determine a rate law, why are initial rates of reaction measured and compared?
6. For the reaction A + B → C, the reaction is first-order with respect to A and second-order overall. Write the rate law for the reaction.

### Problems

1. For the decomposition reaction X → Y + Z, the reaction is first-order with respect to X, and the value of the specific rate constant is 0.0296 s−1. Calculate the initial reaction rate when starting with the following concentrations of X.
1. [X] = 0.410 M
2. [X] = 0.0223 M
2. The reaction between the peroxydisulfate ion (S2O82−) and the iodide ion (I) is: \begin{align*}\mathrm{S_{2}O^{2-}_{8}}{(aq)} \mathrm{+3I^-}{(aq)} \mathrm{\rightarrow 2SO^{2-}_{4}}{(aq)} \mathrm{+I^-_{3}}{(aq)}\end{align*}.
1. From the following data (Table below), which were collected at a set temperature, determine the rate law and calculate the specific rate constant.
Experiment [S2O82−] (M) [I] (M) Initial Rate (M/s)
1 0.080 0.034 2.2 × 10−4
2 0.080 0.017 1.1 × 10−4
3 0.16 0.017 2.2 × 10−4
2. What is the initial rate of reaction if the concentrations of both S2O82− and I are 0.5 M?
3. Nitrogen monoxide reacts with hydrogen at elevated temperatures according to the following equation: \begin{align*}\mathrm{2NO}{(g)} \mathrm{+2H_{2}}{(g)} \mathrm{\rightarrow N_{2}}{(g)} \mathrm{+2H_2O}{(g)}\end{align*}. Determine the rate law from the following data (Table below), and calculate the rate constant for the temperature at which these experiments were conducted.
Experiment [NO] (M) [H2] (M) Initial Rate (M/s)
1 5.0 × 10−3 2.0 × 10−3 1.30 × 10−5
2 1.0 × 10−2 2.0 × 10−3 5.20 × 10-5
3 1.0 × 10−2 4.0 × 10−3 1.04 × 10−4

## Points to Consider

Chemical reactions can generally be broken down into a series of simple steps that convert the reactants into the products. This set of steps is referred to as the reaction mechanism.

• What is the relationship between a reaction mechanism and the rate law for that reaction?
• What is an intermediate in a chemical reaction?

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