# 12.3 Rate Laws

### Learning Objectives

- Explain the form and function of a rate law
- Use rate laws to calculate reaction rates
- Use rate and concentration data to identify reaction orders and derive rate laws

As described in the previous module, the rate of a reaction is affected by the concentrations of reactants. **Rate laws** or **rate equations** are mathematical expressions that describe the relationship between the rate of a chemical reaction and the concentration of its reactants. In general, a rate law (or differential rate law, as it is sometimes called) takes this form:

in which [*A*], [*B*], and [*C*] represent the molar concentrations of reactants, and ** k** is the

**rate constant**, which is specific for a particular reaction at a particular temperature. The exponents

*m*,

*n*, and

*p*are usually positive integers (although it is possible for them to be fractions or negative numbers). The rate constant

*k*and the exponents

*m*,

*n*, and

*p*must be determined experimentally by observing how the rate of a reaction changes as the concentrations of the reactants are changed. The rate constant

*k*is independent of the concentration of

*A*,

*B*, or

*C*, but it does vary with temperature and surface area.

The exponents in a rate law describe the effects of the reactant concentrations on the reaction rate and define the **reaction order**. Consider a reaction for which the rate law is:

If the exponent *m* is 1, the reaction is first order with respect to *A*. If *m* is 2, the reaction is second order with respect to *A*. If *n* is 1, the reaction is first order in *B*. If *n* is 2, the reaction is second order in *B*. If *m* or *n* is zero, the reaction is zero order in *A* or *B*, respectively, and the rate of the reaction is not affected by the concentration of that reactant. The **overall reaction order** is the sum of the orders with respect to each reactant. If *m* = 1 and *n* = 1, the overall order of the reaction is second order (*m* + *n* = 1 + 1 = 2).

The rate law:

describes a reaction that is first order in hydrogen peroxide and first order overall. The rate law:

describes a reaction that is second order in C_{4}H_{6} and second order overall. The rate law:

describes a reaction that is first order in H^{+}, first order in OH^{−}, and second order overall.

##### Interactive Video Quiz

### Example 1

**Writing Rate Laws from Reaction Orders**

An experiment shows that the reaction of nitrogen dioxide with carbon monoxide:

is second order in NO_{2} and zero order in CO at 100 °C. What is the rate law for the reaction?

**Solution**

The reaction will have the form:

The reaction is second order in NO_{2}; thus *m* = 2. The reaction is zero order in CO; thus *n* = 0. The rate law is:

Remember that a number raised to the zero power is equal to 1, thus [CO]^{0} = 1, which is why we can simply drop the concentration of CO from the rate equation: the rate of reaction is solely dependent on the concentration of NO_{2}. When we consider rate mechanisms later in this chapter, we will explain how a reactant’s concentration can have no effect on a reaction despite being involved in the reaction.

**Check Your Learning**

In a transesterification reaction, a triglyceride reacts with an alcohol to form an ester and glycerol. Many students learn about the reaction between methanol (CH_{3}OH) and ethyl acetate (CH_{3}CH_{2}OCOCH_{3}) as a sample reaction before studying the chemical reactions that produce biodiesel:

The rate law for the reaction between methanol and ethyl acetate is, under certain conditions, determined to be:

What is the order of reaction with respect to methanol and ethyl acetate, and what is the overall order of reaction?

### Answer:

order in CH_{3}OH = 1; order in CH_{3}CH_{2}OCOCH_{3} = 0; overall order = 1

It is sometimes helpful to use a more explicit algebraic method, often referred to as the **method of initial rates**, to determine the orders in rate laws. To use this method, we select two sets of rate data that differ in the concentration of only one reactant and set up a ratio of the two rates and the two rate laws. After canceling terms that are equal, we are left with an equation that contains only one unknown, the coefficient of the concentration that varies. We then solve this equation for the coefficient.

##### Interactive Video Quiz

### Example 2

**Determining a Rate Law from Initial Rates**

Ozone in the upper atmosphere is depleted when it reacts with nitrogen oxides. The rates of the reactions of nitrogen oxides with ozone are important factors in deciding how significant these reactions are in the formation of the ozone hole over Antarctica (Figure 1). One such reaction is the combination of nitric oxide, NO, with ozone, O_{3}:

This reaction has been studied in the laboratory, and the following rate data were determined at 25 °C.

Trial | [NO] (mol/L) | [O_{3}] (mol/L) |
[latex]\frac{{\Delta}[\text{NO}_2]}{{\Delta}t}\;(\text{mol\;L}^{-1}\text{s}^{-1})[/latex] |
---|---|---|---|

1 | 1.00 × 10^{−6} |
3.00 × 10^{−6} |
6.60 × 10^{−5} |

2 | 1.00 × 10^{−6} |
6.00 × 10^{−6} |
1.32 × 10^{−4} |

3 | 1.00 × 10^{−6} |
9.00 × 10^{−6} |
1.98 × 10^{−4} |

4 | 2.00 × 10^{−6} |
9.00 × 10^{−6} |
3.96 × 10^{−4} |

5 | 3.00 × 10^{−6} |
9.00 × 10^{−6} |
5.94 × 10^{−4} |

Table 3. |

Determine the rate law and the rate constant for the reaction at 25 °C.

**Solution**

The rate law will have the form:

We can determine the values of *m*, *n*, and *k* from the experimental data using the following three-part process:

*Determine the value of*m*from the data in which [NO] varies and [O*In the last three experiments, [NO] varies while [O_{3}] is constant._{3}] remains constant. When [NO] doubles from trial 3 to 4, the rate doubles, and when [NO] triples from trial 3 to 5, the rate also triples. Thus, the rate is also directly proportional to [NO], and*m*in the rate law is equal to 1.*Determine the value of*n*from data in which [O*In the first three experiments, [NO] is constant and [O_{3}] varies and [NO] is constant._{3}] varies. The reaction rate changes in direct proportion to the change in [O_{3}]. When [O_{3}] doubles from trial 1 to 2, the rate doubles; when [O_{3}] triples from trial 1 to 3, the rate increases also triples. Thus, the rate is directly proportional to [O_{3}], and*n*is equal to 1.The rate law is thus:[latex]\text{rate} = k[\text{NO}]^1[\text{O}_3]^1 = k[\text{NO}][\text{O}_3][/latex]*Determine the value of*k*from one set of concentrations and the corresponding rate*.[latex]\begin{array}{r @{{}={}} l} k & \frac{\text{rate}}{[\text{NO}][\text{O}_3]} \\[0.5em] & \frac{6.60\;\times\;10^{-5}\;\rule[0.5ex]{2.5em}{0.1ex}\hspace{-2.5em}\text{mol\;L}^{-1}\text{s}^{-1}}{(1.00\;\times\;10^{-6}\;\rule[0.5ex]{2.75em}{0.1ex}\hspace{-2.75em}\text{mol\;L}^{-1})(3.00\;\times\;10^{-6}\;\text{mol\;L}^{-1})} \\[0.5em] & 2.20\;\times\;10^{7}\;\text{L\;mol}^{-1}\text{s}^{-1} \end{array}[/latex]The large value of

*k*tells us that this is a fast reaction that could play an important role in ozone depletion if [NO] is large enough.

### Example 3

**Determining Rate Laws from Initial Rates**

Using the initial rates method and the experimental data, determine the rate law and the value of the rate constant for this reaction:

Trial | [NO] (mol/L) | [Cl_{2}] (mol/L) |
[latex]-\frac{{\Delta}[\text{NO}]}{{\Delta}t}\;(\text{mol\;L}^{-1}\text{s}^{-1})[/latex] |
---|---|---|---|

1 | 0.10 | 0.10 | 0.00300 |

2 | 0.10 | 0.15 | 0.00450 |

3 | 0.15 | 0.10 | 0.00675 |

Table 5. |

**Solution**

The rate law for this reaction will have the form:

As in Example 2, we can approach this problem in a stepwise fashion, determining the values of *m* and *n* from the experimental data and then using these values to determine the value of *k*. In this example, however, we will use a different approach to determine the values of *m* and *n*:

*Determine the value of*m*from the data in which [NO] varies and [Cl*We can write the ratios with the subscripts_{2}] is constant.*x*and*y*to indicate data from two different trials:[latex]\frac{\text{rate}_x}{\text{rate}_y} = \frac{k[\text{NO}]_x^m[\text{Cl}_2]_x^n}{k[\text{NO}]_y^m[\text{Cl}_2]_y^n}[/latex]Using the third trial and the first trial, in which [Cl

_{2}] does not vary, gives:[latex]\frac{\text{rate}\;3}{\text{rate}\;1} = \frac{0.00675}{0.00300} = \frac{k(0.15)^m(0.10)^n}{k(0.10)^m(0.10)^n}[/latex]After canceling equivalent terms in the numerator and denominator, we are left with:

[latex]\frac{0.00675}{0.00300} = \frac{(0.15)^m}{(0.10)^m}[/latex]which simplifies to:

[latex]2.25 = (1.5)^m[/latex]We can use natural logs to determine the value of the exponent

*m*:[latex]\begin{array}{r @{{}={}} l} \text{ln}(2.25) & m\text{ln}(1.5) \\[0.5em] \frac{\text{ln}(2.25)}{\text{ln}(1.5)} & m \\[0.5em] 2 & m \end{array}[/latex]We can confirm the result easily, since:

[latex]1.5^2 = 2.25[/latex]*Determine the value of*n*from data in which [Cl*_{2}] varies and [NO] is constant.[latex]\frac{\text{rate}\;2}{\text{rate}\;1} = \frac{0.00450}{0.00300} = \frac{k(0.10)^m(0.15)^n}{k(0.10)^m(0.10)^n}[/latex]Cancelation gives:

[latex]\frac{0.0045}{0.0030} = \frac{(0.15)^n}{(0.10)^n}[/latex]which simplifies to:

[latex]1.5 = (1.5)^n[/latex]Thus

*n*must be 1, and the form of the rate law is:[latex]\text{Rate} = k[\text{NO}]^m[\text{Cl}_2]^n = k[\text{NO}]^2[\text{Cl}_2][/latex]*Determine the numerical value of the rate constant*k*with appropriate units.*The units for the rate of a reaction are mol/L/s. The units for*k*are whatever is needed so that substituting into the rate law expression affords the appropriate units for the rate. In this example, the concentration units are mol^{3}/L^{3}. The units for*k*should be mol^{−2}L^{2}/s so that the rate is in terms of mol/L/s.To determine the value of

*k*once the rate law expression has been solved, simply plug in values from the first experimental trial and solve for*k*:[latex]\begin{array}{r @{{}={}} l} 0.00300\;\text{mol\;L}^{-1}\text{s}^{-1} & k(0.10\;\text{mol\;L}^{-1})^2(0.10\;\text{mol\;L}^{-1})^1 \\[0.5em] k & 3.0\;\text{mol}^{-2}\text{L}^2\text{s}^{-1} \end{array}[/latex]

##### Interactive Video Quiz

# Reaction Order and Rate Constant Units

In some of our examples, the reaction orders in the rate law happen to be the same as the coefficients in the chemical equation for the reaction. This is merely a coincidence and very often not the case.

Rate laws may exhibit fractional orders for some reactants, and negative reaction orders are sometimes observed when an increase in the concentration of one reactant causes a decrease in reaction rate. A few examples illustrating these points are provided:

It is important to note that *rate laws are determined by experiment only and are not reliably predicted by reaction stoichiometry.*

Reaction orders also play a role in determining the units for the rate constant *k*. In Example 2, a second-order reaction, we found the units for *k* to be [latex]\text{L\;mol}^{-4}\text{s}^{-1}[/latex], whereas in Example 3, a third order reaction, we found the units for *k* to be mol^{−2} L^{2}/s. More generally speaking, the units for the rate constant for a reaction of order [latex](m\;+\;n)[/latex] are [latex]\text{mol}^{1\;-\;(m\;+\;n)}\text{L}^{(m\;+\;n)\;-\;1}\text{s}^{-1}[/latex].Table 7 summarizes the rate constant units for common reaction orders.

Reaction Order | Units of k |
---|---|

[latex](m\;+\;n)[/latex] | [latex]\text{mol}^{1\;-\;(m\;+\;n)}\text{L}^{(m\;+\;n)\;-\;1}\text{s}^{-1}[/latex] |

zero | mol/L/s |

first | s^{−1} |

second | L/mol/s |

third | mol^{−2} L^{2} s^{−1} |

Table 7. Rate Constants for Common Reaction Orders |

Note that the units in the table can also be expressed in terms of molarity (*M*) instead of mol/L. Also, units of time other than the second (such as minutes, hours, days) may be used, depending on the situation.

# Key Concepts and Summary

Rate laws provide a mathematical description of how changes in the amount of a substance affect the rate of a chemical reaction. Rate laws are determined experimentally and cannot be predicted by reaction stoichiometry. The order of reaction describes how much a change in the amount of each substance affects the overall rate, and the overall order of a reaction is the sum of the orders for each substance present in the reaction. Reaction orders are typically first order, second order, or zero order, but fractional and even negative orders are possible.

##### Interactive Video Quiz

### Chemistry End of Chapter Exercises

- How do the rate of a reaction and its rate constant differ?
- Doubling the concentration of a reactant increases the rate of a reaction four times. With this knowledge, answer the following questions:
2 A.

2 B.

- Tripling the concentration of a reactant increases the rate of a reaction nine times. With this knowledge, answer the following questions:
(a) What is the order of the reaction with respect to that reactant?

(b) Increasing the concentration of a reactant by a factor of four increases the rate of a reaction four times. What is the order of the reaction with respect to that reactant?

- How much and in what direction will each of the following affect the rate of the reaction: [latex]\text{CO}(g)\;+\;\text{NO}_2(g)\;{\longrightarrow}\;\text{CO}_2(g)\;+\;\text{NO}(g)[/latex] if the rate law for the reaction is [latex]\text{rate} = k[\text{NO}_2]^2[/latex]?
(a) Decreasing the pressure of NO

_{2}from 0.50 atm to 0.250 atm.(b) Increasing the concentration of CO from 0.01

*M*to 0.03*M*. - How will each of the following affect the rate of the reaction: [latex]\text{CO}(g)\;+\;\text{NO}_2(g)\;{\longrightarrow}\;\text{CO}_2(g)\;+\;\text{NO}(g)[/latex] if the rate law for the reaction is [latex]\text{rate} = k[\text{NO}_2][\text{CO}][/latex]?
(a) Increasing the pressure of NO

_{2}from 0.1 atm to 0.3 atm(b) Increasing the concentration of CO from 0.02

*M*to 0.06*M*. - Regular flights of supersonic aircraft in the stratosphere are of concern because such aircraft produce nitric oxide, NO, as a byproduct in the exhaust of their engines. Nitric oxide reacts with ozone, and it has been suggested that this could contribute to depletion of the ozone layer. The reaction [latex]\text{NO}\;+\;\text{O}_3\;{\longrightarrow}\;\text{NO}_2\;+\;\text{O}_2[/latex] is first order with respect to both NO and O
_{3}with a rate constant of 2.20 × 10^{7}L/mol/s. What is the instantaneous rate of disappearance of NO when [NO] = 3.3 × 10^{−6}*M*and [O_{3}] = 5.9 × 10^{−7}*M*? - Radioactive phosphorus is used in the study of biochemical reaction mechanisms because phosphorus atoms are components of many biochemical molecules. The location of the phosphorus (and the location of the molecule it is bound in) can be detected from the electrons (beta particles) it produces:

[latex]_{15}^{32}\text{P}\;{\longrightarrow}\;_{16}^{32}\text{S}\;+\;\text{e}^{-}[/latex]

[latex]\text{Rate} = 4.85\;\times\;10^{-2}\text{day}^{-1}[^{32}\text{P}][/latex]

What is the instantaneous rate of production of electrons in a sample with a phosphorus concentration of 0.0033*M*? - The rate constant for the radioactive decay of
^{14}C is 1.21 × 10^{−4}year^{−1}. The products of the decay are nitrogen atoms and electrons (beta particles):[latex]_{14}^{6}\text{C}\;{\longrightarrow}\;_{14}^{6}\text{N}\;+\;\text{e}^{-}[/latex]

[latex]\text{rate} = k[_{14}^{6}\text{C}][/latex]What is the instantaneous rate of production of N atoms in a sample with a carbon-14 content of 6.5 × 10

^{−9}*M*? - The decomposition of acetaldehyde is a second order reaction with a rate constant of 4.71 × 10
^{−8}L/mol/s. What is the instantaneous rate of decomposition of acetaldehyde in a solution with a concentration of 5.55 × 10^{−4}*M*? - Alcohol is removed from the bloodstream by a series of metabolic reactions. The first reaction produces acetaldehyde; then other products are formed. The following data have been determined for the rate at which alcohol is removed from the blood of an average male, although individual rates can vary by 25–30%. Women metabolize alcohol a little more slowly than men:

[C _{2}H_{5}OH] (*M*)4.4 × 10 ^{−2}3.3 × 10 ^{−2}2.2 × 10 ^{−2}Rate (mol/L/h) 2.0 × 10 ^{−2}2.0 × 10 ^{−2}2.0 × 10 ^{−2}**Table 8.**Determine the rate equation, the rate constant, and the overall order for this reaction.

- Under certain conditions the decomposition of ammonia on a metal surface gives the following data:

[NH _{3}] (*M*)1.0 × 10 ^{−3}2.0 × 10 ^{−3}3.0 × 10 ^{−3}Rate (mol/L/h ^{1})1.5 × 10 ^{−6}1.5 × 10 ^{−6}1.5 × 10 ^{−6}**Table 9.**Determine the rate equation, the rate constant, and the overall order for this reaction.

- Nitrosyl chloride, NOCl, decomposes to NO and Cl
_{2}.

[latex]2\text{NOCl}(g)\;{\longrightarrow}\;2\text{NO}(g)\;+\;\text{Cl}_2(g)[/latex]Determine the rate equation, the rate constant, and the overall order for this reaction from the following data:

[NOCl] ( *M*)0.10 0.20 0.30 Rate (mol/L/h) 8.0 × 10 ^{−10}3.2 × 10 ^{−9}7.2 × 10 ^{−9}**Table 10.** - From the following data, determine the rate equation, the rate constant, and the order with respect to
*A*for the reaction [latex]A\;{\longrightarrow}\;2C[/latex].

[ *A*] (*M*)1.33 × 10 ^{−2}2.66 × 10 ^{−2}3.99 × 10 ^{−2}Rate (mol/L/h) 3.80 × 10 ^{−7}1.52 × 10 ^{−6}3.42 × 10 ^{−6}**Table 11.** - Nitrogen(II) oxide reacts with chlorine according to the equation:

[latex]2\text{NO}(g)\;+\;\text{Cl}_2(g)\;{\longrightarrow}\;2\text{NOCl}(g)[/latex]The following initial rates of reaction have been observed for certain reactant concentrations:

[NO] (mol/L ^{1})[Cl _{2}] (mol/L)Rate (mol/L/h) 0.50 0.50 1.14 1.00 0.50 4.56 1.00 1.00 9.12 **Table 12.**What is the rate equation that describes the rate’s dependence on the concentrations of NO and Cl

_{2}? What is the rate constant? What are the orders with respect to each reactant? - Hydrogen reacts with nitrogen monoxide to form dinitrogen monoxide (laughing gas) according to the equation: [latex]\text{H}_2(g)\;+\;2\text{NO}(g)\;{\longrightarrow}\;\text{N}_2\text{O}(g)\;+\;\text{H}_2\text{O}(g)[/latex]
Determine the rate equation, the rate constant, and the orders with respect to each reactant from the following data:

[NO] ( *M*)0.30 0.60 0.60 [H _{2}] (*M*)0.35 0.35 0.70 Rate (mol/L/s) 2.835 × 10 ^{−3}1.134 × 10 ^{−2}2.268 × 10 ^{−2}**Table 13.** - For the reaction [latex]A\;{\longrightarrow}\;B\;+\;C[/latex], the following data were obtained at 30 °C:

[ *A*] (*M*)0.230 0.356 0.557 Rate (mol/L/s) 4.17 × 10 ^{−4}9.99 × 10 ^{−4}2.44 × 10 ^{−3}**Table 14.**(a) What is the order of the reaction with respect to [

*A*], and what is the rate equation?(b) What is the rate constant?

- For the reaction [latex]Q\;{\longrightarrow}\;W\;+\;X[/latex], the following data were obtained at 30 °C:

[ *Q*]_{initial}(*M*)0.170 0.212 0.357 Rate (mol/L/s) 6.68 × 10 ^{−3}1.04 × 10 ^{−2}2.94 × 10 ^{−2}**Table 15.**(a) What is the order of the reaction with respect to [

*Q*], and what is the rate equation?(b) What is the rate constant?

- The rate constant for the first-order decomposition at 45 °C of dinitrogen pentoxide, N
_{2}O_{5}, dissolved in chloroform, CHCl_{3}, is 6.2 × 10^{−4}min^{−1}.

[latex]2\text{N}_2\text{O}_5\;{\longrightarrow}\;4\text{NO}_2\;+\;\text{O}_2[/latex]What is the rate of the reaction when [N

_{2}O_{5}] = 0.40*M*? - The annual production of HNO
_{3}in 2013 was 60 million metric tons Most of that was prepared by the following sequence of reactions, each run in a separate reaction vessel.(a) [latex]4\text{NH}_3(g)\;+\;5\text{O}_2(g)\;{\longrightarrow}\;4\text{NO}(g)\;+\;6\text{H}_2\text{O}(g)[/latex]

(b) [latex]2\text{NO}(g)\;+\;\text{O}_2(g)\;{\longrightarrow}\;2\text{NO}_2(g)[/latex]

(c) [latex]3\text{NO}_2(g)\;+\;\text{H}_2\text{O}(l)\;{\longrightarrow}\;2\text{HNO}_3(aq)\;+\;\text{NO}(g)[/latex]

The first reaction is run by burning ammonia in air over a platinum catalyst. This reaction is fast. The reaction in equation (c) is also fast. The second reaction limits the rate at which nitric acid can be prepared from ammonia. If equation (b) is second order in NO and first order in O

_{2}, what is the rate of formation of NO_{2}when the oxygen concentration is 0.50*M*and the nitric oxide concentration is 0.75*M*? The rate constant for the reaction is 5.8 × 10^{−6}L^{2}/mol^{2}/s. - The following data have been determined for the reaction:

[latex]\text{I}^{-}\;+\;\text{OCl}^{-}\;{\longrightarrow}\;\text{IO}^{-}\;+\;\text{Cl}^{-}[/latex]Equation 1 2 3 [latex][\text{I}^{-}]_{\text{initial}}[/latex] ( *M*)0.10 0.20 0.30 [latex][\text{OCl}^{-}]_{\text{initial}}[/latex] ( *M*)0.050 0.050 0.010 Rate (mol/L/s) 3.05 × 10 ^{−4}6.20 × 10 ^{−4}1.83 × 10 ^{−4}**Table 16.**Determine the rate equation and the rate constant for this reaction.

## Glossary

- method of initial rates
- use of a more explicit algebraic method to determine the orders in a rate law

- overall reaction order
- sum of the reaction orders for each substance represented in the rate law

- rate constant (
*k*) - proportionality constant in the relationship between reaction rate and concentrations of reactants

- rate law
- (also, rate equation) mathematical equation showing the dependence of reaction rate on the rate constant and the concentration of one or more reactants

- reaction order
- value of an exponent in a rate law, expressed as an ordinal number (for example, zero order for 0, first order for 1, second order for 2, and so on)

### Solutions

**Answers to Chemistry End of Chapter Exercises**

2. (a) 2; (b) 1

4. (a) The process reduces the rate by a factor of 4. (b) Since CO does not appear in the rate law, the rate is not affected.

6. 4.3 × 10^{−5} mol/L/s

8. 7.9 × 10^{−13} mol/L/year

10. rate = *k*; *k* = 2.0 × 10^{−2} mol/L/h (about 0.9 g/L/h for the average male); The reaction is zero order.

12. rate = *k*[NOC]^{2}; *k* = 8.0 × 10^{−8} L/mol/s; second order

14. rate = *k*[NO]^{2}[Cl]_{2}; *k* = 9.12 L^{2} mol^{−2} h^{−1}; second order in NO; first order in Cl_{2}

16. (a) The rate equation is second order in A and is written as rate = *k*[*A*]^{2}. (b) *k* = 7.88 × 10^{−13} L mol^{−1} s^{−1}

18. (a) 2.5 × 10^{−4} mol/L/min

20. rate = *k*[I^{−}][OCl^{−1}]; *k* = 6.1 × 10^{−2} L mol ^{−1} s^{−1}