Read the passage carefully and answer the questions.

The speed at which a chemical reaction takes place is called the rate of reaction. The rate of reaction depends on various factors like concentration of the reactants, temperature, etc. The relation between the rate of reaction and the concentration of reacting species is represented by the equation $r = k[A]^x[B]^y$, where $x$ and $y$ are the order of the reaction with respect to the reactants A and B, respectively. The overall order of the reaction is $x + y$. The rate of reaction can also be increased by the use of a catalyst which provides an alternate pathway of lower activation energy. It increases the rate of forward and backward reaction to an equal extent. It does not alter the Gibbs energy of the reaction.

The rate law of a reaction is given by $r = k[CH_3OCH_3]^{3/2}$, If the pressure is measured in bar and time in minutes, then the unit of rate constant will be

$bar^{-1/2}\, min^{-1}$

The correct answer is Option (3) → $bar^{-1/2}\, min^{-1}$

1. General Formula for Units of $k$

For a reaction of order $n$, the rate law is:

$\text{Rate} = k[\text{Reactant}]^n$

Rearranging for $k$:

$k = \frac{\text{Rate}}{[\text{Reactant}]^n}$

In terms of units (where concentration for gases is expressed in terms of pressure):

$\text{Units of } k = \frac{\text{bar} \cdot \text{min}^{-1}}{(\text{bar})^n} = \text{bar}^{(1-n)} \cdot \text{min}^{-1}$

2. Identifying the Order ($n$)

From the given rate law $r = k[CH_3OCH_3]^{3/2}$the order of the reaction is:

$n = \frac{3}{2}$

3. Calculating the Unit

Substitute $n = \frac{3}{2}$ into the general formula:

$\text{Unit of } k = \text{bar}^{(1 - 3/2)} \cdot \text{min}^{-1}$

$\text{Unit of } k = \text{bar}^{(-1/2)} \cdot \text{min}^{-1}$