The table given below gives kinetic data for the following reaction at $298$ $K$ :
$O {C l}^{-} + I^{-} ⟶ O I^{-} + {C l}^{-}$
Expt.
No. $[O {C l}^{-}]$
$(i n m o l$ ${d m}^{- 3})$ $[I^{-}]$
$(i n m o l$ ${d m}^{- 3})$ $[O H^{-}]$
$(i n m o l$ ${d m}^{- 3})$ $d [{I O}^{-}] / d t$
$(i n 10^{- 4} m o l$ ${d m}^{- 3}$ $s^{- 1})$
$1.$ $0.0017$ $0.0017$ $1.0$ $1.75$
$2.$ $0.0034$ $0.0017$ $1.0$ $3.50$
$3.$ $0.0017$ $0.0034$ $1.0$ $3.50$
$4.$ $0.0017$ $0.0017$ $0.5$ $3.50$
What is the rate law and what is the value of rate constant?

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Solution

Let the rate law be $= k {[O {C l}^{-}]}^{x} {[I^{-}]}^{y} {[{O H}^{-}]}^{z}$
From expt. (1), $1.75 \times 10^{- 4} = k {[0.0017]}^{x} {[0.0017]}^{y} {[1.0]}^{z}$ ...(i)
From expt. (2), $3.50 \times 10^{- 4} = k {[0.0034]}^{x} {[0.0017]}^{y} {[1.0]}^{z}$ ...(ii)
Dividing equation (ii) by equation (i),
$\frac{3.50 \times 10^{- 4}}{1.75 \times 10^{- 4}} = \frac{{[0.0034]}^{x}}{{[0.0017]}^{x}}$
$2 = 2^{x}$
$x = 1$ , i.e., first order with respect to ${O C l}^{-}$
From expt. (1),
$1.75 \times 10^{- 4} = k {[0.0017]}^{x} {[0.0017]}^{y} {[1.0]}^{z}$ ...(i)
From expt. (3),
$3.50 \times 10^{- 4} = k {[0.0017]}^{x} {[0.0034]}^{y} {[1.0]}^{z}$ ...(iii)
Dividing equation (iii) by equation (i),
$\frac{3.50 \times 10^{- 4}}{1.75 \times 10^{- 4}} = \frac{{[0.0034]}^{y}}{{[0.0017]}^{y}}$
or $2 = 2^{y}$
or $y = 1$ , i.e., first order with respect to $I^{-}$
From expt. (1), $1.75 \times 10^{- 4} = k {[0.0017]}^{x} {[0.0017]}^{y} {[1.0]}^{z}$ ...(i)
From expt. (4),
$3.50 \times 10^{- 4} = k {[0.0017]}^{x} {[0.0017]}^{y} {[0.5]}^{z}$ ...(iv)
Dividing equation (i) by equation (iv),
$\frac{1.75 \times 10^{- 4}}{3.50 \times 10^{- 4}} = \frac{{[1.0]}^{z}}{{[0.5]}^{z}}$
or $\frac{1}{2} = 2^{z}$
or $2^{- 1} = 2^{z}$
$z = - 1$ , i.e., first order with respect to ${O H}^{-}$ is $- 1$ .
Rate law $= \frac{k [{O C l}^{-}] [I^{-}]}{[{O H}^{-}]}$
From expt. (1) $k = \frac{1.75 \times 10^{- 4} [{O H}^{-}]}{[{O C l}^{-}] [I^{-}]} = \frac{1.75 \times 10^{- 4} \times 1.0}{0.0017 \times 0.0017}$
$= 60.55 s^{- 1}$

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Expt. No.	$[O {C l}^{-}]$ $(i n m o l$ ${d m}^{- 3})$	$[I^{-}]$ $(i n m o l$ ${d m}^{- 3})$	$[O H^{-}]$ $(i n m o l$ ${d m}^{- 3})$	$d [{I O}^{-}] / d t$ $(i n 10^{- 4} m o l$ ${d m}^{- 3}$ $s^{- 1})$
$1.$	$0.0017$	$0.0017$	$1.0$	$1.75$
$2.$	$0.0034$	$0.0017$	$1.0$	$3.50$
$3.$	$0.0017$	$0.0034$	$1.0$	$3.50$
$4.$	$0.0017$	$0.0017$	$0.5$	$3.50$