# 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:

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

Reaction rate of a product:

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

A + B → C
reaction rate = -  = -  =

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: - $\frac{∆\left[\mathrm{A}\right]}{∆\mathrm{t}}$ = k [A]

After integrating:

ln[A] = ln[A]0 – kt

⇒ ln $\left(\frac{\left[\mathrm{A}\right]}{{\left[\mathrm{A}\right]}_{0}}\right)$ = -kt

⇒ $\frac{\left[\mathrm{A}\right]}{{\left[\mathrm{A}\right]}_{0}}$ = 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] = $\frac{{\left[\mathrm{A}\right]}_{0}}{2}$

ln  = - k t1/2

⇒ t1/2

⇒ 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: - $\frac{∆\left[\mathrm{A}\right]}{∆\mathrm{t}}$ = k [A]2
After integrating:

$\frac{1}{\left[\mathrm{A}\right]}$ = $\frac{1}{{\left[\mathrm{A}\right]}_{0}}$ + kt

Test plot of second-order reactions: $\frac{1}{\left[\mathrm{A}\right]}$ versus t

At t = t1/2: [A] = $\frac{{\left[\mathrm{A}\right]}_{0}}{2}$

$\frac{2}{{\left[\mathrm{A}\right]}_{0}}$ = $\frac{1}{{\left[\mathrm{A}\right]}_{0}}$ + kt1/2

⇒ t1/2 =

⇒ t1/2 is dependent of [A]0

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 $\left(\frac{\left[\mathrm{A}\right]}{{\left[\mathrm{A}\right]}_{0}}\right)$ = -kt and t1/2

⇒ k =

⇒ ln $\left(\frac{\left[\mathrm{A}\right]}{{\left[\mathrm{A}\right]}_{0}}\right)$ = -  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

$\frac{\left[\mathrm{A}\right]}{{\left[\mathrm{A}\right]}_{0}}$ = $\frac{\mathrm{N}}{{\mathrm{N}}_{0}}$

⇒ ln $\left(\frac{\mathrm{N}}{{\mathrm{N}}_{0}}\right)$ = ln $\left(\frac{\left[\mathrm{A}\right]}{{\left[\mathrm{A}\right]}_{0}}\right)$ = -  x t