Integrated Rate Equations

1. Zero order reactions:

⇒ Order of the reaction is zero.

⇒ The reaction in which the rate of reaction is independent of concentration of the reactants.

⇒ Rate of reaction remains constant during the course of reaction.

⇒ No concentration term in the rate law.

A → Products

Initially t = 0                    a           0

t = t₁                               a - x        x

dx/ dt = k

dx = k dt

x = kt + c

When t = 0 x = 0 ⇒ C= 0

∴ x = kt

t = t_½

x = a/2

a/2 = kt_½

t_½ = (a/2) k

a. Graph for x = kt

b. Graph for t_½ = a/2K

c. [A₀] - [A] = kt

[A₀] = initial concentration, t = 0

[A]_t = concentration, t = t

[A₀] - [A]_t = Kt

+ [A]_t = -Kt + [A₀]

Half-life of reaction: The time required for the completion of 50% of the reaction is called half-life of the reaction.

Units of rate constant: n A → Products

Rate law is R = K [Conc]ⁿ

K = Rate/ (Concentration)ⁿ

= mole¹¯ⁿ literⁿ¯¹ sec¯¹

Where n is order of reaction

2. First order reaction: The reaction in which the rate of R x n depends only on one concentration is doubled. Rate of reaction will also be doubled.

Equations: A → Products

t = 0          a mol/lit          0

t = t             a - x              x

Rate = K [A]¹

Rate = - dC_R/dt = d [A]/dt = + K [A]

- d (a - x)/ dt = - K (a - x)

dx/ dt = K (a - x)

∫dx/ (a - x) = ∫ Kdt

- log (a - x) = Kt + c

When t = 0, x = 0

- log a = c

- log (a - x) = Kt - log a

Kt = 2030 log (a/a - x)

a = is initial concentration

x = is dissociated concentration.

log (a - x) = - Kt/2.303 + log a

And also

m (a/a-x) = Kt

a/(a - x) = e^Kt ⇒ (a - x)/a = e¯^Kt

x = a (1 - e¯^Kt)

Half-life of first order Reaction:

log (a/a-x) = Kt/2.303

t = 2.303/K log (a/a-x)

t = t_½ ⇒ x = a/2

K = (2.303/ t_½) log2 = 0.693/ t_½

t_½ = 0.693/K [It is independent of initial concentration]

For the first order reaction.

Amount of reaction left after time t = Initial Amount/ 2ⁿ

η = t/t_½ = Number of half - lifes

3. n^th Order Reaction:

A → Products

Rate law is (dn/dt) = K [A]ⁿ = K (a - x)ⁿ

⇒ Time T is required to complete a particular fraction of reaction.

T α (a)¹¯ⁿ

⇒ If concentration is changed m times new rate will be mⁿ

Half-life of n^th order reaction:

⇒ t_½ α 1/aⁿ¯¹

a is initial concentration

η is order of Reaction

⇒ Half-life of n^th order reaction

Reaction Order	Differential Rate Law	Integrated Rate Law	Characteristics Kinetic Plot	Slope of Kinetic Plot	Units of Rate Constant
Zero	- d[A]/dt = K	[A] = [A]₀ - Kt	[A] vs t	- K	Mole L¯¹ sec¯¹
First	- d[A]/dt = K[A]	[A] = [A]₀ e-Kt	ln [A] vs t	- K	sec¯¹
Second	- d[A]/dt = K[A]²	[A] = [A]₀/1 + Kt[A]₀	1/[A] vs t	K	L Mole sec¯¹