# Lecture 27: Reaction Rates and Orders

Order of the reaction is an experimental quantity

• Not possible to know ﻿$n$﻿ and ﻿$m$﻿ from a balanced equation

Rate ﻿$= k[A]^n$﻿

• Zero Order - Rate of reaction ﻿$\rightarrow$﻿ Always the same
• First Order - Rate proportional to the reactant concentration
• Doubling ﻿$[A]$﻿ ﻿$\rightarrow$﻿ Double the rate of the reaction
• Second Order - Rate directly proportional to square toot of the reactant concentration
• A rate quadruples when doubling ﻿$[A]$﻿

#### Integrated Rate Law

• Relationship between ﻿$k$﻿ and the time

﻿$[A] = -kt + [A]_{initial}$﻿ ﻿$\rightarrow$﻿ Straight line with slope ﻿$-k$﻿

#### First Order Reactions

Rate ﻿$=$﻿ ﻿$k[A]^4 = k [A]$﻿

﻿$\ln [A] = -kt + \ln [A]_{initial}$﻿

﻿$kt = \ln [A]_{initial}-ln[A]$﻿

﻿$kt = \ln (\frac{[A]_{initial}}{[A]})$﻿

﻿$kt_{1/2} = \frac {\ln [A]_{initial} }{[A]_{initial} / 2}$﻿ ﻿$\rightarrow$﻿ Half-Life

﻿$kt_{1/2} = \ln (2)$﻿

﻿$kt_{1/2} = 0.693$﻿

﻿$t_{1/2} = \frac { 0.693}{k}$﻿

The half life of a first order reactions is constant when:

Rate ﻿$=$﻿ ﻿$M/s, \ k = s^{=1}$﻿

### Zero Order

Rate ﻿$= K[A]^\circ = K$﻿

• Constant rate reactions

﻿$[A] = -kt + [A]_{initial}$﻿

Rate does not change if concentration changes.

When Rate ﻿$= M/s, \ k = M/s$﻿

﻿$t_{1/2}= \frac{[A_{initial}]}{2k}$﻿

### First Order

Rate ﻿$=k[A]^4 = k[A]$﻿

﻿$\ln [A] = -Kt + \ln [A]_{initial}$﻿

Derivation: ﻿$\ln [A]_{initial} - \ln [A]$﻿

﻿$Kt = \ln ([A]_{initial}- \ln [A])$﻿

﻿$Kt = \ln (\frac{[A]_{initial}}{[A]_{initial / 2}})$﻿

﻿$Kt_{1/2} = \ln(2)$﻿

﻿$Kt_{1/2} = 0.693$﻿

﻿$t_{1/2} = \frac{0.693}{k}$﻿

The half life of first order reaction is constant.

When Rate ﻿$= m/s, \ K= s ^{-1}$﻿

﻿$*$﻿The half life, ﻿$t_{1/2}$﻿ of a reaction is the length f time it takes for the concentration of the reaction to fall to ﻿$\frac{1}{2}$﻿ its initial value. The half life for First order reaction, the half life is constant and independent and independent of concentration.

### Second Order

Rate ﻿$= K[A]^2$﻿

﻿$\frac{1}{[A]} = Kt + \frac {1}{[A]_{initial}}$﻿

﻿$t_{1/2} = \frac{1}{K[A_0]}$﻿

When Rate ﻿$﻿= m/s, \ K=M^{-1} \cdot S^{-1}$﻿