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By the end of this section, you will be able to:
  • Explain the form and function of an integrated rate law
  • Perform integrated rate law calculations for zero-, first-, and second-order reactions
  • Define half-life and carry out related calculations
  • Identify the order of a reaction from concentration/time data

The rate laws we have seen thus far relate the rate and the concentrations of reactants. We can also determine a second form of each rate law that relates the concentrations of reactants and time. These are called integrated rate laws . We can use an integrated rate law to determine the amount of reactant or product present after a period of time or to estimate the time required for a reaction to proceed to a certain extent. For example, an integrated rate law is used to determine the length of time a radioactive material must be stored for its radioactivity to decay to a safe level.

Using calculus, the differential rate law for a chemical reaction can be integrated with respect to time to give an equation that relates the amount of reactant or product present in a reaction mixture to the elapsed time of the reaction. This process can either be very straightforward or very complex, depending on the complexity of the differential rate law. For purposes of discussion, we will focus on the resulting integrated rate laws for first-, second-, and zero-order reactions.

First-order reactions

An equation relating the rate constant k to the initial concentration [ A ] 0 and the concentration [ A ] t present after any given time t can be derived for a first-order reaction and shown to be:

ln ( [ A ] t [ A ] 0 ) = k t


ln ( [ A ] 0 [ A ] t ) = k t


[ A ] = [ A ] 0 e k t

The integrated rate law for a first-order reaction

The rate constant for the first-order decomposition of cyclobutane, C 4 H 8 at 500 °C is 9.2 × 10 −3 s −1 :

C 4 H 8 2C 2 H 4

How long will it take for 80.0% of a sample of C 4 H 8 to decompose?


We use the integrated form of the rate law to answer questions regarding time:

ln ( [ A ] 0 [ A ] ) = k t

There are four variables in the rate law, so if we know three of them, we can determine the fourth. In this case we know [ A ] 0 , [ A ], and k , and need to find t .

The initial concentration of C 4 H 8 , [ A ] 0 , is not provided, but the provision that 80.0% of the sample has decomposed is enough information to solve this problem. Let x be the initial concentration, in which case the concentration after 80.0% decomposition is 20.0% of x or 0.200 x. Rearranging the rate law to isolate t and substituting the provided quantities yields:

t = ln [ x ] [ 0.200 x ] × 1 k = ln 0.100 mol L −1 0.020 mol L −1 × 1 9.2 × 10 −3 s −1 = 1.609 × 1 9.2 × 10 −3 s −1 = 1.7 × 10 2 s

Check your learning

Iodine-131 is a radioactive isotope that is used to diagnose and treat some forms of thyroid cancer. Iodine-131 decays to xenon-131 according to the equation:

I-131 Xe-131 + electron

The decay is first-order with a rate constant of 0.138 d −1 . All radioactive decay is first order. How many days will it take for 90% of the iodine−131 in a 0.500 M solution of this substance to decay to Xe-131?


16.7 days

Got questions? Get instant answers now!

We can use integrated rate laws with experimental data that consist of time and concentration information to determine the order and rate constant of a reaction. The integrated rate law can be rearranged to a standard linear equation format:

Questions & Answers

what are oxidation numbers
Idowu Reply
pls what is electrolysis
Idowu Reply
Electrolysis is the process by which ionic substances are decomposed (broken down) into simpler substances when an electric current is passed through them. ... Electricity is the flow of electrons or ions. For electrolysis to work, the compound must contain ions.
what is the basicity of an atom
Eze Reply
basicity is the number of replaceable Hydrogen atoms in a Molecule. in H2SO4, the basicity is 2. in Hcl, the basicity is 1
how to solve oxidation number
Mr Reply
mention some examples of ester
Chinenye Reply
do you mean ether?
what do converging lines on a mass Spectra represent
Rozzi Reply
would I do to help me know this topic ?
what the physic?
Bassidi Reply
who is albert heistein?
similarities between elements in the same group and period
legend Reply
what is the ratio of hydrogen to oxulygen in carbohydrates
Nadeen Reply
what is poh and ph
Amarachi Reply
please what is the chemical configuration of sodium
2, 6, 2, 1
1s2, 2s2, 2px2, 2py2, 2pz2, 3s1
what is criteria purity
Austin Reply
cathode is a negative ion why is it that u said is negative
Michael Reply
cathode is a negative electrode while cation is a positive ion. cation move towards cathode plate.
CH3COOH +NaOH ,complete the equation
david Reply
compare and contrast the electrical conductivity of HCl and CH3cooH
Sa Reply
The must be in dissolved in water (aqueous). Electrical conductivity is measured in Siemens (s). HCl (aq) has higher conductivity, as it fully ionises (small portion of CH3COOH (aq) ionises) when dissolved in water. Thus, more free ions to carry charge.
HCl being an strong acid will fully ionize in water thus producing more mobile ions for electrical conduction than the carboxylic acid
differiante between a weak and a strong acid
how can I tell when an acid is weak or Strong
an aqueous solution of copper sulphate was electrolysed between graphite electrodes. state what was observed at the cathode
Bakanya Reply
write the equation for the reaction that took place at the anode
what is enthalpy of combustion
Enthalpy change of combustion: It is the enthalpy change when 1 mole of substance is combusted with excess oxygen under standard conditions. Elements are in their standard states. Conditions: pressure = 1 atm Temperature =25°C
Observation at Cathode: Cu metal deposit (pink/red solid).
Equation at Anode: (SO4)^2- + 4H^+ + 2e^- __> SO2 + 2H2O
Equation : CuSO4 -> Cu^2+ + SO4^2- equation at katode: 2Cu^2+ + 4e -> 2Cu equation at anode: 2H2O -> 4H+ + O2 +4e at the anode which reacts is water because SO4 ^ 2- cannot be electrolyzed in the anode
Practice Key Terms 2

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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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