The decomposition of dimethyl ether leads to the formation of `CH_(4),H_(2),` and `CO` and the reaction rate is given by
Rate `=k[CH_(3)OCH_(3)]^(3//2)`
The rate of reaction is followed by increase in the pressure in a closed vessel , so the rate can also be expressed in terms of the partial pressure of dimethyl either, `i.e., `
Rate `=k[p_(CH_(3)OCH_(3))]^(3//2)`
If the pressure is measured in bar and time in minutes, then what are the units of rate and rate constant ?

Explanation

To determine the units of the rate and the rate constant for the decomposition of dimethyl ether, we can follow these steps:

### Step 1: Understand the Rate Expression
The rate of the reaction is given by:
\[ \text{Rate} = k [\text{CH}_3\text{OCH}_3]^{3/2} \]
This indicates that the rate depends on the concentration of dimethyl ether raised to the power of \( \frac{3}{2} \).

### Step 2: Define the Units of Rate
The rate of a reaction can be defined as the change in concentration over time. In this case, since we are dealing with gases, we can express concentration in terms of pressure. Therefore, the units of rate can be expressed as:
\[ \text{Rate} = \frac{\Delta P}{\Delta t} \]
Where:
- \( \Delta P \) is the change in pressure (in bar)
- \( \Delta t \) is the change in time (in minutes)

Thus, the units of rate are:
\[ \text{Units of Rate} = \text{bar} \cdot \text{min}^{-1} \]

### Step 3: Determine the Units of the Rate Constant (k)
From the rate expression:
\[ \text{Rate} = k [\text{CH}_3\text{OCH}_3]^{3/2} \]
We can rearrange this to find the units of \( k \):
\[ k = \frac{\text{Rate}}{[\text{CH}_3\text{OCH}_3]^{3/2}} \]

### Step 4: Determine the Units of Concentration
The concentration of dimethyl ether can also be expressed in terms of pressure (in bar). Therefore, the units of concentration are:
\[ [\text{CH}_3\text{OCH}_3] = \text{bar} \]

### Step 5: Substitute the Units into the Rate Constant Expression
Now substituting the units into the expression for \( k \):
\[ k = \frac{\text{bar} \cdot \text{min}^{-1}}{(\text{bar})^{3/2}} \]
This simplifies to:
\[ k = \frac{\text{bar} \cdot \text{min}^{-1}}{\text{bar}^{3/2}} = \text{bar}^{1 - \frac{3}{2}} \cdot \text{min}^{-1} = \text{bar}^{-1/2} \cdot \text{min}^{-1} \]

### Final Results
- **Units of Rate**: \( \text{bar} \cdot \text{min}^{-1} \)
- **Units of Rate Constant (k)**: \( \text{bar}^{-1/2} \cdot \text{min}^{-1} \)