What is the unit of rate constant of `n^(th)` order of the reaction ?

Options

A. `(mol//L)^ns^(-1)`
B. `(mol//Ls^(-1))^(n-1)`
C. `((L)/(mol)s^(-1))^(n-1)`
D. `(L//mol)^(n-1)s^(-1)`

Answer

To determine the unit of the rate constant (k) for an nth order reaction, we can follow these steps:

### Step 1: Understand the Rate Law
The rate law for a reaction can be expressed as:
\[ \text{Rate} = k [A]^n \]
where:
- Rate is the change in concentration per unit time,
- \( k \) is the rate constant,
- \([A]\) is the concentration of the reactant, and
- \( n \) is the order of the reaction.

### Step 2: Define the Units
1. **Rate**: The rate of a reaction is typically expressed in terms of concentration change over time. The unit for concentration is moles per liter (mol/L), and the unit for time is seconds (s). Therefore, the unit for rate is:
\[ \text{Rate} = \frac{\text{mol}}{\text{L} \cdot \text{s}} \]

2. **Concentration**: The unit for concentration \([A]\) is also in moles per liter (mol/L).

### Step 3: Rearranging the Rate Law
From the rate law:
\[ k = \frac{\text{Rate}}{[A]^n} \]
Substituting the units into this equation gives:
\[ k = \frac{\frac{\text{mol}}{\text{L} \cdot \text{s}}}{\left(\frac{\text{mol}}{\text{L}}\right)^n} \]

### Step 4: Simplifying the Units
Now, we simplify the right-hand side:
\[ k = \frac{\frac{\text{mol}}{\text{L} \cdot \text{s}}}{\frac{\text{mol}^n}{\text{L}^n}} \]
This can be rewritten as:
\[ k = \frac{\text{mol}}{\text{L} \cdot \text{s}} \cdot \frac{\text{L}^n}{\text{mol}^n} \]
\[ k = \frac{\text{L}^{n-1}}{\text{mol}^{n-1} \cdot \text{s}} \]

### Step 5: Final Unit of Rate Constant
Thus, the unit of the rate constant \( k \) for an nth order reaction is:
\[ k = \frac{\text{L}^{n-1}}{\text{mol}^{n-1} \cdot \text{s}} \]

### Summary
The unit of the rate constant \( k \) for an nth order reaction is:
\[ \text{L}^{n-1} \cdot \text{mol}^{-(n-1)} \cdot \text{s}^{-1} \]

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