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.

Q) The rate law of a reaction is given by r = k[CH₃OCH₃]^3/2 , If the pressure is measured in bar and time in minutes, then the unit of rate constant will be

The correct option is:
bar^-1/2 min^-1

Step-by-step Explanation:

1. Understand the given Rate Law:
The rate law for the reaction is given as:
$$r=k{\left[{\mathrm{CH}}_3{\mathrm{OCH}}_3\right]}^{3/2}$$
where:
• $r$ is the rate of reaction.
• $k$ is the rate constant.
• $\left[{\mathrm{CH}}_3{\mathrm{OCH}}_3\right]$ represents the pressure of the reactant (dimethyl ether).

2. Determine the Units of the Quantities:
• The pressure of the reactant is measured in bar.
• Time is measured in minutes (min).
• The rate of reaction ($r$) is defined as the change in pressure per unit time:
$$\text{Unit of}r=\frac{\text{bar}}{\text{min}}={\text{bar min}}^{-1}$$

3. Calculate the Unit of the Rate Constant ($k$):
Rearranging the rate law equation to solve for the rate constant $k$:
$$k=\frac{r}{{\left[{\mathrm{CH}}_3{\mathrm{OCH}}_3\right]}^{3/2}}$$
Now, substitute the corresponding units into the equation:
$$\text{Unit of}k=\frac{{\text{bar min}}^{-1}}{{\text{(bar)}}^{3/2}}$$
Simplify the powers of the bar unit:
$$\text{Unit of}k={\text{bar}}^{1-\frac{3}{2}}{\text{min}}^{-1}$$
Since $1-\frac{3}{2}=-\frac{1}{2}$, we get:
$$\text{Unit of}k={\text{bar}}^{-1/2}{\text{min}}^{-1}$$