$(a)$ Write general reaction and derive the units of rate constant. $(b)$ Based on that write the rate constant for zero, first and $2^{nd}$ order reaction.
$(a)$ Write general reaction and derive the units of rate constant. $(b)$ Based on that write the rate constant for zero, first and $2^{nd}$ order reaction.
$(a)$ General Reaction : $a \mathrm{~A}+b \mathrm{~B} \rightarrow c \mathrm{C}+d \mathrm{D}$
The differential rate expression of general reaction is as under :
Rate $=-\frac{\mathrm{d}[\mathrm{R}]}{\mathrm{dt}}=k[\mathrm{~A}]^{x}[\mathrm{~B}]^{y} \quad \ldots(\mathrm{i})$
$\therefore k=\frac{\operatorname{Rate}}{[\mathrm{A}]^{x}}[\mathrm{~B}]^{y}$
Where, order of reaction $(x+y)=n$ and $n=0,1,2,3, \frac{1}{2}, \frac{3}{2} \ldots$ etc.
The $SI$ units of concentration is mol $\mathrm{L}^{-1}$ and time $\left(\mathrm{s}^{-1}\right)$ which is unit of concentration /time.
Order of reaction $=n$, so unit of $n$ is $\left(\mathrm{mol} \mathrm{L}^{-1}\right)^{n} .$
Put these value in equation (ii) and the unit of $k$
$\mathrm{k}=\frac{\text { concentration }}{\text { time }} \times \frac{1}{\text { concentration }}$
$=\frac{\text { mol } \mathrm{L}^{-1}}{\text { time }} \times \frac{1}{\left(\mathrm{~mol} \mathrm{~L}^{-1}\right)^{\mathrm{n}}}$
$\therefore$ Unit of $k=\frac{\left(\mathrm{mol} \mathrm{L}^{-1}\right)^{(1-\mathrm{n})}}{\mathrm{second}}$
Unit of $k=\left(\mathrm{mol} \mathrm{L}^{-1}\right)^{(1-n)} \mathrm{s}^{-1}$
Where, $n=$ order of the reaction.
|
Order of reaction $(n)$ |
unit of $k$ $\left(\operatorname{mol} \mathrm{L}^{-1}\right)^{(1-n)} \mathrm{s}^{-1}$ |
| zero $(0)$ | $\left(\operatorname{mol} \mathrm{L}^{-1}\right)^{1-0} \mathrm{~s}^{-1}=\operatorname{mol} \mathrm{L}^{-1} \mathrm{~s}^{-1}$ |
| one $(1)$ |
$\left(m o l \mathrm{~L}^{-1}\right)^{(1-1)} \mathrm{s}^{-1}=\left(\operatorname{mol} \mathrm{L}^{-1}\right)^{0} \mathrm{~s}^{-1}=\mathrm{s}^{-1}$ |
| two $(2)$ | $\begin{aligned}\left(\mathrm{mol} \mathrm{L}^{-1}\right)^{(1-2)} \mathrm{s}^{-1} &=\left(\mathrm{mol} \mathrm{L}^{-1}\right)^{-1} \mathrm{~s}^{-1} \\ &=\mathrm{mol}^{-1} \mathrm{~L} \mathrm{~s}^{-1} \end{aligned}$ |
| three $(3)$ | $\begin{aligned}\left(\mathrm{mol} \mathrm{L}^{-1}\right)^{(1-3)} \mathrm{s}^{-1} &=\left(\mathrm{mol} \mathrm{L}^{-1}\right)^{-2} \mathrm{~s}^{-1} \\ &=\mathrm{mol}^{-2} \mathrm{~L}^{2} \mathrm{~s}^{-1} \end{aligned}$ |
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