$A \rarr P$

$r=k[A]^2……(1)$

$r=-\frac{d[A]}{dt}……(2)$

$-\frac{d[A]}{dt}=k[A]^2……(3)$

$-\frac{d[A]}{[A]^2}=kdt$

$\int_{[A]_0}^{[A]_t}$ ${\frac{d[A]}{[A]^2}}$

$= - \int_{t=0}^t k d t$

$-[\frac{1}{[A]_t}-\frac{1}{[A]_0}]=-kt$

$\frac{1}{[A]_t}-\frac{1}{[A]_0}=kt$

$\frac{1}{[A]_t}=\frac{1}{[A]_0}+kt……(4)$

$t_{1/2} \rArr [A]_0 \rarr \frac{1}{2}[A]_0$

$\frac{1}{[A]_t}=\frac{1}{[A]_0}+kt……(4)$

$\frac{1}{\frac{1}{2}[A]_0} - \frac{1}{[A]_0}$

$=kt_{1/2}$

$kt_{1/2}=\frac{2}{[A]_0} - \frac{1}{[A]_0}$

$t_{1/2}=\frac{1}{k[A]_0}……(5)$

$t_{1/2}=\frac{1}{k[A]_0}$

Half-life is proportional to the reciprocal of concentration

Longer the concentration, lesser is the half-life

Half-life increases as the reaction proceeds

Second order reaction

$t_{1/2}\rarr$ Preliminary check

$A \rarr P$

$aA \rarr P$

$r=-\frac{1}{a} \frac{d[A]}{dt}=k[A]^2$

Derive the rate law

For second order reaction when reactants are different

$A \rarr P$

$A+B \rarr P$

$r=k[A][B]……(6)$

if $[A]=[B]$

$r=k[A]^2……(7)$

If $[A]≠[B]$

A+B$\rarr P$

$r=k[A][B]$

Derive the integrated rate law for a reaction like mentioned above!

aA + bB $\rarr P……(8)$

We need to establish whether the rate equation is-

$r=k[A]^{\alpha} [B]^{\beta}……(9)$

Problem- Rate of reaction now depends on the concentration of both the reactants

Solution- Difficult to disentangle the effect of one reactant from the other

We arrange our experimental conditions in such a way that data analysis will be simplified.

Two ways

Isolation method

Initial Rate method

aA + bB$\rarr P$

$ClO^-(aq)+ Br^-(aq) \rarr BrO^-(aq)+Cl^-(aq)……(9)$

A Plausible rate equation

$r=k[ClO^-]^\alpha [Br^-]^\beta……(10)$

Let $[ClO^-] =0.1 molL^{-1}$

$[Br^-]=2.0×10^{-3} molL^{-1}$

$[ClO^-]»[Br^-]$

$ClO^-$ is in excess

$\frac{[ClO^-]}{[Br^-]}$

$=\frac{0.1 molL^{-1}}{2.0×10^{-3}molL^{-1}}$ = 50

$[ClO^-]$ is in fifty-fold excess

For hypochloride

For bromide

$ClO^-$ is being consumed to a very little extent

$[ClO^-]$ can be treated to be constant

$Br^-$ ion is almost fully consumed!

$[ClO^-]$ Has essentially remained constant throughout the reaction

$[ClO^-]≈[ClO^-]_0$

$r=k[ClO^-]^\alpha [Br^-]^\beta……(10)$

$r=k[ClO^-]^\alpha_0 [Br^-]^\beta……(11)$

Where $k[ClO^-]^\alpha_0$ is effectively a constant

$r=k^{’}[Br^-]^\beta……(12)$

$k^{’}=k[ClO^-]^{\alpha}_0……(13)$

Where $r=k^{’}[Br^-]^\beta$ is Depending only on $[Br^-]$

Kinetics contribution of $[Br^-]$ has been isolated

$r=k^{’}[Br^-]^\beta……(13)$

Experimentally $\rarr \beta=1$

$r=k{’}[Br^-]^{1}$

$r=k^{’}[Br^-]……(14)$

Keeping $[ClO]^-$ in large excess

$[Br^-]»[ClO^-]$

$r=k[ClO^-]^\alpha [Br^-]^\beta_0$

$r=\underbrace{k[Br^-]^\beta_0}_{k’’} [ClO^-]^\alpha$

$r=k"[ClO^-]^\alpha……(15)$

$\alpha$=1

$r=k[ClO^-][Br^-]……(16)$