The Rate Law

Objectives: • To understand what a rate law is• To determine the overall reaction order from a rate law

CLE 3224.3.3

Initial Reaction Rate

One way of studying the effect of concentration on reaction rate is to determine the way in which the rate at the beginning of the reaction(the initial rate) depends on the starting concentrations.

Consider the following reaction:NH4

+ (aq) + NO2-(aq) N2(g) + 2H2O(l)

We might study the rate of this reaction by measuring the concentration of NH4

+ or NO2-

as a function of time or by measuring the volume of N2 collected. Because the stoichiometric coefficients on NH4

+ , NO2- , and

N2 are the same, all of the rates will be equal.

The data indicates that changing either [NH4+] or [NO2

−] changes the reaction rate. Notice that if we double [NH4

+] while holding [NO2−] constant, the rate doubles( look at experiments 1 and 2). If [NH4

+] is increased by a factor of 4 with [NO2

−] left unchanged(look at experiments 1 and 3), the rate changes by a factor of 4.

NH4+ (aq) + NO2

-(aq) N2(g) + 2H2O(l)

When [NO2−] is varied while [NH4

+] is held constant, the rate is affected in the same manner. The reaction rate is directly proportional to the concentration of both [NO2

−] and[NH4+]. We express the way the rate

depends on the concentration of these reactants as Rate = k [NO2

−] [NH4+]

An equation such as this which shows how the rate depends on the concentrations of the reactants is called a rate law.

General Form of the Rate Law

For a general reaction,aA + bB cC + dD

The rate law generally has the formRate = k[A]m[B]n

The constant k in the rate law is called the rate constant. The exponents m and n are typically small whole numbers.

Calculating k

If we know the rate law for a reaction and its rate for a set of concentrations, we can calculate the value of k, the rate constant.

If we know the rate law for a reaction and its rate for a set of concentrations, we can calculate the rate law.Rate = k [NO2

−] [NH4+]

We can use any of the data from the six experiments to calculate k.

NH4+ (aq) + NO2

-(aq) N2(g) + 2H2O(l)

Study Check

Use the data from any of the six experiments to calculate k for this reaction:NH4

+ (aq) + NO2-(aq) N2(g) + 2H2O(l)

Rate law =k [NO2−] [NH4

+]

Other Facts About k, the Specific Rate Constant

• The value of k does not change for different concentrations of reactants or products

• The value of k is for the reaction at a specific temperature ; if we increase the temperature of the reaction, the value of k increases.

• The value of k becomes larger if a catalyst is present

Study Check

1. What is a rate law?2. What is the name of k in any rate law?

Reaction Orders: The Exponents in the Rate Law

The rate laws for most reactions have the general formRate = k[reactant 1]m[reactant 2]n

The exponents m and n in a rate law are called reaction orders.

Determination of Overall Reaction Order

Consider the following rate law:Rate = k [NO2

−] [NH4+]

Because the exponent of [NH4+] is 1, the rate is first

order in NH4+. The rate is also first order in

NO2− . (The exponent 1 is not shown in rate laws)

The overall reaction order is the sum of the orders with respect to each reactant in the rate law. This rate law has an overall reaction order of 1+ 1=2, and the reaction is second order overall.

The exponents in a rate law indicate how the rate is affected by the concentration of each reactant. Because the rate at which NH4

+ reacts with NO2

− depends on [NH4+ ] raised to the first

power, the rate doubles when [NH4+ ] doubles,

triples when [NH4+ ] triples, and so forth.

If a rate law is second order with respect to a reactant( i.e.[A]2), then doubling the concentration of [A] causes the reaction rate to quadruple(22=4 ) and tripling the concentration causes the reaction rate to increase by 9 times(32= 9).

Zero Order

If the concentration of a reactant has no effect on the reaction rate, the reaction is zero order in that reactant. Anything raised to the zero power is one, so changing the concentration of a reactant that has a zero order will not affect the rate.

Additional Rate Laws

2N2O5(g) -- 4NO2(g) +O2(g) rate = k[N2O5]

H2(g) + I2(g) 2HI(g) rate = k[H2][I2 ]

Note that although the exponents in a rate law are sometimes the same as the coefficients in the balanced equation, this is not necessarily the case. The values of these exponents must be determined experimentally.

Study Check

For the following reaction, 2NO(g) + 2H2(g) N2(g) + 2H2O(g)

the rate law is Rate = k[NO]2[H2]

Determine:1. The order of each reactant in the rate law2. The overall reaction order