Rate law | How to write Rate law

For a chemical reaction to go on, one of the most important controlling factors is reactants concentration. As the reactant concentration generally decreases with the proceeding reaction, it is safe to assume that the reactant concentration is in some ways related to the rate of reaction. An expression which shows how rate depends on the concentration of reacting species is termed rate law.

Rate of a reaction ∝ [reactants]

Rate law is used to calculate the rate of reaction for any particular set of reactant concentrations. It can figure out two things about a particular chemical reaction.

  1. Speed of reaction
  2. Mechanism of reaction

Rate law

The rate law is a mathematical expression that relates the rate of a reaction with the concentration of reactants. It involves the concentration of reactants, not product because we generally consider the reverse reaction to be unimportant in reversible reactions. 

Rate law expression defines the order of the reaction as well. Although, the order of a reaction of a reaction is known for sure, once the experiment is performed. 

Consider a general reaction.

aA + bB  →  cC + dD

The rate law for this reaction is,

Rate ∝ [A]a [B]b

Rate = k [A]a [B]b

such that a + b = order of the reaction.

In the above reaction, A and B are reactants and the square brackets represent their concentrations. ‘k’ is the rate constant, which is specific for a particular reaction. It is independent of the concentration of reactants or products but depends on temperature.

The units of rate constant is ( mol dm-3 sec-1). 

There are two types of rate law:

  1. Differential rate law
  2. Integerated rate law

Differential Rate Law

‘Differential rate law is a mathematical term’

A differential rate law expresses the rate of a reaction in terms of changes in the concentration of reactants over a specific interval of time. The differential rate law is used to find out about what is happening on molecular level during a chemical reaction.

Rate = d[A] / dt

In the above formula, rate = d[A] / dt = k.[A]n. Thsi means that a differential rate law is an average rate law with the function of change in concentration being studied with time as a reference.

Integerated Rate Law

‘Integerated rate law relates concentration to reaction time’

Integerated rate laws express the rate fo a reaction as a function of initial and measured actual concentration of reactanst after a time interval ‘t’. Thjis type of rate laws are used to determine rate constant and order of a reaction by experimental data.

ln [A] = -kt + ln [A]o

where,

[A]= Initial concentration fo reactant A.

[A] = Concentration of reactant A after a time ‘t’.

Rate of a reaction

The rate of reaction is the speed at which the reaction proceeds. It is defined as the change in concentration of reactants or products per unit time. For example, reactant A goes to product B in this reaction.

A  →  B

BNow the rate of a reaction is the decrease in concentration of reactant A over time ‘t’. It can also be evaluated as the increase in concentration of product B over time ‘t’.

Rate of reaction = – dA/dt

Rate of reaction = dB/dt

The negative sign with a concentration of reactants indicates that the reactants’ concentration decreases with the passage of time. Where the positive sign with the products’ concentration indicates an increases in concentration over time.

Order of a reaction

Order of reaction is the sum of exponents in concentration terms. If the rate of reaction is independent of the concentration of reactants, the reaction is zero order. If the rate of reaction depends on the first power of concentration terms, it shows the order of reaction to be 1st order. Similarly, if the rate of reaction depends on the second power of concentration terms, it is a 2nd order reaction.

In some cases an exponent can even be negative. It would mean that increasing the concentration of that reactant will decrease the reaction rate.

How to write Rate law?

The Steps to writing the rate law are as following:

  1. Find the rate determining step (RDS) from the mechanism of the reaction. Although the reaction mechanism is found by rate law, most tentative mechanism may be used.
  2. The reactants of the rate determining step must be involved in the rate law.
  3. Write the rate constant along with concentration of reactanst into the rate formula.
  4. Coefficients of reactants (from a balanced chemical equation) become the power of concentration terms. These coefficients later on, sum up to be order of the reaction.
  5. In case of reversible reactions, the reverse reactions must be rendered insignificant. This step is done to ensure the primary focus of a chemist on the concentration of reactants of one side.
  6. Th rate law equation must contain a negative sign if concentration of reactants is being used in a differential rate law. It is positive for products instead.
  7. Actual concentration of reactants must be known for both differential and integrated rate laws.

The rate law for Elementary reactions

The reactions which take place in one step are called elementary reactions. The rate laws of such reactions involve obvious reactants only. Some illustrations for generalized reactions are shown for different types of conditions in a chemical reaction.

1. The rate law for the elementary unimolecular reaction.

→  P

Rate = k [A]

2. The rate law for an elementary bimolecular reaction with similar reactants.

A + A  →  P

Rate = k [A]2

3. The rate law for an elementary bimolecular reaction with totally different reactants.

A + B  →  P

Rate = k [A] [B]

4. The rate law for an elementary trimolecular reaction with all reactants similar.

A + A + A  →  P

Rate = k [A]3

5. The rate law for an elementary trimolecular reaction with two similar reactanst and one different reactant.

A + A + B  →  P

Rate = k [A]2 [B]

6. The rate law for an elementary trimolecular reaction with all three reactants different from each other.

A + B + C  →  P

Rate = k [A] [B] [C]

The rate law for multi-step reactions

The reactions which take place in more than one step are called multi-step reactions. Rate law does not involve every single (elementary) step of multi step reactants, but it necessarily involves the rate determining step (RDS).

The slowest step which determines the rate of reaction is called the rate determining step. This step determines the rate of a multi step reaction and its order as well. 

Consider the decomposition of nitrogen dioxide (NO2) into nitric oxide (NO) and oxygen (O2).

2NO2 (g)  →  2NO (g) + O2 (g)

The net balanced chemical equation is the sum of all the elementary steps involved in a multi step chemical reaction. The mechanism of this reaction is as;

2NO2  →  NO3 + NO — (slow reaction)

NO3  →  NO + O2 — (fast reaction)

NO3 is an intermediate unstable species (experimentally proved) which makes the next step relatively fast. The rate law equation therefore becomes.

Rate = k [NO2]2

Examples of Rate law equations

Now, considering some examples of  reactions whose reaction rates are given in the forms of rate laws.

1. Hydrogen and hydroxide ion combine to form water

H+ + OH  →  H2O   —–   Rate = k [H+] [OH]

The order of this reaction is 2. 

2. Decomposition of acetaldehyde into methane and carbon monoxide

CH3CHO  →  CH4 + CO   —–   Rate =k [CH3CHO]2

In this reaction, the exponent in rate equation is different from that of coefficients. This is because, this reaction is complex and happens in more than one step.

3. Reaction between nitrogen dioxide and fluorine to form nitryl fluoride.

2NO2 + F2  →  2NO2F   —–   Rate = k [NO2]2 [F2]

4. The reaction between nitric oxide and hydrogen gas to make nitrogen gas and water

2NO + H2  →  2N2 + 2H2O   —–   Rate = k [NO]2 [H2]

5. Reaction between ozone and chlorine

O3 + Cl  →  O2 + ClO   —–   Rate = k [O3] [Cl]

Key Takeaway(s)
How to write the rate law - Rate law - Rate constant

Concepts Berg

What is the Rate Law?

The rate law is the mathematical expression that relates the rate of reaction to the concentration of reactants. Mathematically, rate law can be written as;

Rate of reaction = k x reactants

Where k is the rate constant for a particular reaction.

How do you write a rate law for an overall reaction?

If a reaction involves a series of steps then the rate law for the overall reaction can be written as the reactants f the rate determining step (RDS). 

What is the statement of rate law?

The rate law is a mathematical expression that gives a relationship between the rate of reaction with the concentration of reactants involved in that reaction.  The rate law for a general reaction

A → P

Rate of reaction = k [A]

How do you write a rate law for an elementary reaction?

The rate law for elementary reactions can be written from a balanced chemical equation. For example, the rate law for following elementary reaction;

A + B + 2C → P

Rate of reaction = k [A] [B] [C]2

How to determine rate law from a table?

Rate law can be determined by reaction table by figuring out the order of the reaction by the difference in two concentrations of same reactant. Now, this order value is plugged into the rate equation to determine rate constant. This rate constant is then again used to make the final rate law equation with specific rate constant and order values.

How to write a rate law for a mechanism?

If a reaction involves a mechanism or multistep reaction, we have to find the rate determining step (RDS) firstly. The slowest step in a reaction is the rate determining step. Then rate law can be written according to reactants concentration involved in the rate determining step. For example,

First step —  H2O2 + I–  →  H2O + IO

Second step  —   IO+ H2O→  H2O + O2 + I

The net reaction is therefore,

2H2O→  2 H2O + O2

Since the first step is slow, it is definitely the rate determining step (RDS). Hence, the rate law is;

Rate = k [H2O2]

What are the units of rate constant (k)?

The units of rate constant depend on the order of the reaction. The general formula to find the units of the rate constant is;

k (unit) = (concentration of reactants)1-n (time)-1

where, units are as, 

  • Concentration of reactants = mol dm-3
  • n = order of reaction 
  • Time = sec

For zero order reaction, put [n = 0] in the above formula,  

k = (mol dm-3)1-0 (sec)-1

k (for zero order) = mol dm-3 sec-1

and so on.

How to determine the order of reaction?

Order of reaction is the sum of exponents of concentration terms in the rate equation. There are five methods to determine the order of a reaction. 

  1. Graphical method
  2. Half life method
  3. Initial rate method
  4. Ostwald dilution method
  5. Differential rate method

What is the differential rate law?

The differential rate law is a mathematical expression that gives a rate of change of concentration of reactants with reference to time. 

What is first order rate law?

A first order reaction is a chemical reaction in which the rate of reaction depends on the first power of concentration terms. The rate law for first order reaction can be written as, 

A → P

Rate = k [A]

What are the units of the rate of reaction in chemistry?

The rate of reaction is usually expressed as the change in the concentration of reactants or products per unit time. Hence, its units are;

= mol.dm-3 sec-1

Reference books

  • Physical Chemistry| Third edition, by Robert G. Mortimer (Rhodes College, Memphis, Tennessee)
  • General Chemistry Principles and Structure | Third edition, by James E. Brandy (St. John’s University, Jamaica, New York) and Gerard E. Humiston (Widener University, West Chester, Pennsylvania) – [SI version, Prepared by Henry Heikkinen (University of Maryland, USA)]
  • Chemistry | fifth edition, by Steven S. Zumdhal and Susan A. Zumdhal (University of Illinois, Urbana Champaign, IL, USA)

Reference links

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