Question
The decomposition of dimethyl ether is a fractional order reaction. The rate of reaction is given by $${\text{rate}} = {\text{k}}{\left( {{p_{_{C{H_3}OC{H_3}}}}} \right)^{\frac{3}{2}}}.$$ If the pressure is measured in bar and time in minutes, then what are the units of rate and rate constant?
A.
\[\text{bar}\,{{\min }^{-1}},\,\text{ba}{{\text{r}}^{2}}\,{{\min }^{-1}}\]
B.
\[\text{bar}\,{{\min }^{-1}},\,\text{ba}{{\text{r}}^{-\frac{1}{2}}}\,{{\min }^{-1}}\]
C.
\[\text{ba}{{\text{r}}^{-\frac{1}{2}}}\,{{\min }^{-1}},\,\text{ba}{{\text{r}}^{2}}\,{{\min }^{-1}}\]
D.
\[\text{bar}\,{{\min }^{-1}},\,\text{ba}{{\text{r}}^{\frac{1}{2}}}\,{{\min }^{-1}}\]
Answer :
\[\text{bar}\,{{\min }^{-1}},\,\text{ba}{{\text{r}}^{-\frac{1}{2}}}\,{{\min }^{-1}}\]
Solution :
\[\begin{align}
& \text{In terms of pressure,} \\
& \text{Rate}=k{{\left( {{p}_{C{{H}_{3}}OC{{H}_{3}}}} \right)}^{\frac{3}{2}}} \\
& \text{Unit of rate = bar mi}{{\text{n}}^{-1}} \\
& \text{Unit of rate constant} \\
& =\frac{\text{rate}}{{{\left( {{p}_{C{{H}_{3}}OC{{H}_{3}}}} \right)}^{\frac{3}{2}}}} \\
& =\frac{\text{bar}\,\,{{\min }^{-1}}}{\text{ba}{{\text{r}}^{\frac{3}{2}}}} \\
& =\text{ba}{{\text{r}}^{-\frac{1}{2}}}{{\min }^{-1}} \\
\end{align}\]