**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

\(\frac{-d{{C}_{R}}}{dt}=\frac{-d\left[ A \right]}{dt}=\frac{-d\left[ a-x \right]}{dt}=k{{\left( a-x \right)}^{0}}=\frac{dx}{dt}\)

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 mole/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.

\(M\xrightarrow{1{{t}_{1/2}}}\frac{M}{2}\xrightarrow{2{{t}_{1/2}}}\frac{M}{4}\xrightarrow{3{{t}_{1/2}}}\frac{M}{8}\)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

\({{K}_{\eta }}=\frac{1}{\left( n-1 \right)t}\left[ \frac{1}{{{\left( a-x \right)}^{n-1}}}-\frac{1}{{{a}^{n-1}}} \right]\)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

\(\frac{{{\left( t\frac{1}{2} \right)}_{1}}}{{{\left( t\frac{1}{2} \right)}_{2}}}={{\left( \frac{{{a}_{2}}}{{{a}_{1}}} \right)}^{n-1}}\)

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 |