The unit of rate constant and rate of a reaction are same for

Options

A. First order
B. Zero order
C. Second order
D. Third order

Answer

To determine when the unit of the rate constant (k) and the rate of a reaction are the same, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Rate of Reaction**:
The rate of a reaction is defined as the change in concentration of a reactant or product per unit time. Mathematically, it can be expressed as:
\[ \text{Rate} = \frac{\Delta [A]}{\Delta t} \]
where \([A]\) is the concentration of the reactant.

2. **Unit of Rate**:
The concentration is typically measured in moles per liter (mol/L), and time is measured in seconds (s). Therefore, the unit of rate is:
\[ \text{Unit of Rate} = \frac{\text{mol}}{\text{L} \cdot \text{s}} = \text{mol L}^{-1} \text{s}^{-1} \]

3. **Rate Law Expression**:
According to the rate law, the rate of a reaction can also be expressed as:
\[ \text{Rate} = k [A]^n \]
where \(k\) is the rate constant and \(n\) is the order of the reaction.

4. **Finding the Unit of Rate Constant (k)**:
Rearranging the rate law to solve for \(k\):
\[ k = \frac{\text{Rate}}{[A]^n} \]
Substituting the units:
\[ k = \frac{\text{mol L}^{-1} \text{s}^{-1}}{(\text{mol L}^{-1})^n} \]
This simplifies to:
\[ k = \text{mol L}^{-1} \text{s}^{-1} \cdot \text{(mol L}^{-1})^{-n} = \text{mol L}^{-1} \text{s}^{-1} \cdot \text{mol}^{-n} \text{L}^n \]
\[ k = \text{mol}^{1-n} \text{L}^{n-1} \text{s}^{-1} \]

5. **Setting Units Equal**:
For the units of \(k\) and the rate to be the same, we need:
\[ \text{mol}^{1-n} \text{L}^{n-1} \text{s}^{-1} = \text{mol L}^{-1} \text{s}^{-1} \]
This leads to two equations:
- For moles: \(1 - n = -1\) → \(n = 2\)
- For liters: \(n - 1 = -1\) → \(n = 0\)

6. **Conclusion**:
The only case where the units of the rate constant and the rate of reaction are the same is when the reaction is of **zero order** (n = 0). Thus, the unit of the rate constant \(k\) for a zero-order reaction is:
\[ k = \text{mol L}^{-1} \text{s}^{-1} \]

### Final Answer:
The unit of the rate constant and the rate of a reaction are the same for **zero-order reactions**.