Chemical Kinetics: Rate Laws | General Chemistry 2

Rate laws in chemical kinetics are studied in this chapter: reaction rates, first-order and second-order reactions (specific rate of reaction, relation between concentration and time, half-time), radioactive decay

Reaction Rates

Reaction rate of a reactant:

1α [reactant]t

α = stochiometric coefficient
[reactant] = concentration of the reactant (in mol.L-1)


Reaction rate of a product:

1α [product]t

α = stochiometric coefficient
[product] = concentration of the product (in mol.L-1)


A + B → C
reaction rate = - 1a [A]t = - 1b [B]t = 1c [C]t

NB: reaction rate is in mol.L-1.s-1 (or -1.s-1)



Reaction rate can also be expressed as a product of the concentration of reactants:

a A + B → products

reaction rate = k [A]a [B]b

k = rate constant
a = order of reaction with respect to A
b = order of reaction with respect to B
a + b = overall order of reaction


First-Order Reactions

Reaction rate of first-order reactions: k [A]
Units of rate constant k: s-1

Dependence of [A] on time: - [A]t = k [A]

After integrating:

ln[A] = ln[A]0 – kt

⇒ ln [A][A]0 = -kt

⇒ [A][A]0 = e-kt


Test plot of first-order reactions: ln[A] versus t


Half-time t1/2:  time it takes for one-half of the reactant to react

At t = t1/2[A] = [A]02 

ln [A]02 [A]0 = - k t1/2 

⇒ t1/2ln 2k

⇒ t1/2 is independent of [A](half-time of a first-order reaction is independent of the initial concentration of the reactant)


Second-Order Reactions

Reaction rate of second-order reactions: k [A]2
Units of rate constant k: -1.s-1 (mol.L-1.s-1)

Dependence of [A] on time: - [A]t = k [A]2
After integrating:

1[A] = 1[A]0 + kt


Test plot of second-order reactions: 1[A] versus t


At t = t1/2: [A] = [A]02

2[A]0 = 1[A]0 + kt1/2

⇒ t1/2 = 1k [A]0

⇒ t1/2 is dependent of [A]0


Radioactive Decay

Radioactive decay: process by which an unstable atomic nucleus loses energy by emitting small particles (α-particles, β-particles, γ-rays)

Radioactive decay is a first-order process:

ln [A][A]0 = -kt and t1/2ln 2k

⇒ k = ln 2t1/2 

⇒ ln [A][A]0 = - ln 2t1/2 x t

Relationship between half-life and number of particles in first-order nuclear decay:
Number of radioactive nuclei N is proportional to the concentration of the radioactive species

 [A][A]0 = NN0 

⇒ ln NN0 = ln [A][A]0 = - ln 2t1/2 x t