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12.3 Rate laws

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By the end of this section, you will be able to:
  • 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:

rate = k [ A ] m [ B ] n [ C ] p …

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:

rate = k [ A ] m [ B ] n

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:

rate = k [ H 2 O 2 ]

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

rate = k [ C 4 H 6 ] 2

describes a reaction that is second order in C 4 H 6 and second order overall. The rate law:

rate = k [ H + ] [ OH − ]

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

Writing rate laws from reaction orders

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

NO 2 ( g ) + CO( g ) ⟶ NO( g ) + CO 2 ( g )

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:

rate = k [ NO 2 ] m [ CO ] n

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:

rate = k [ NO 2 ] 2 [ CO ] 0 = k [ NO 2 ] 2

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

The rate law for the reaction:

H 2 ( g ) + 2 NO( g ) ⟶ N 2 O( g ) + H 2 O( g )

has been determined to be rate = k [NO] 2 [H 2 ]. What are the orders with respect to each reactant, and what is the overall order of the reaction?

Answer:

order in NO = 2; order in H 2 = 1; overall order = 3

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:

CH 3 OH + CH 3 CH 2 OCOCH 3 ⟶ CH 3 OCOCH 3 + CH 3 CH 2 OH

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

rate = k [ CH 3 OH ]

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

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Source:  OpenStax, Chemistry. OpenStax CNX. May 20, 2015 Download for free at http://legacy.cnx.org/content/col11760/1.9
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