For which of the following reactions, the units of rate constant and rate of reaction are same?

Options

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

Explanation

To determine for which of the following reactions the units of the rate constant (k) and the rate of reaction (R) are the same, we need to analyze the relationship between the rate of reaction and the rate constant based on the order of the reaction.

### Step-by-Step Solution:

1. **Understanding Rate of Reaction**:
The rate of a chemical reaction is defined as the change in concentration of a reactant or product per unit time. The general form of the rate can be expressed as:
\[ R = k[A]^n \]
where:
- \( R \) = rate of reaction
- \( k \) = rate constant
- \( [A] \) = concentration of reactant A
- \( n \) = order of the reaction with respect to A

2. **Units of Rate of Reaction**:
The unit of the rate of reaction (R) is typically expressed as:
\[ \text{Molarity per second} = \text{mol L}^{-1} \text{s}^{-1} \]

3. **Units of Rate Constant (k)**:
The units of the rate constant (k) depend on the order of the reaction (n). The units of k can be derived from the rate equation:
\[ R = k[A]^n \implies k = \frac{R}{[A]^n} \]
Therefore, the units of k can be expressed as:
\[ k = \frac{\text{mol L}^{-1} \text{s}^{-1}}{\text{mol L}^{-1}} = \text{mol L}^{-1} \text{s}^{-1} \cdot \text{(mol L}^{-1})^{-n} \]
This simplifies to:
\[ k = \text{mol L}^{-1} \text{s}^{-1} \cdot \text{mol}^{-n} \cdot \text{L}^{n} = \text{mol L}^{-1-n} \text{s}^{-1} \]

4. **Finding When Units are the Same**:
For the units of k and R to be the same, we need:
\[ \text{mol L}^{-1} \text{s}^{-1} = \text{mol L}^{-1-n} \text{s}^{-1} \]
This implies:
\[ -1 = -1 - n \implies n = 0 \]
Therefore, the reaction must be a **zero-order reaction** for the units of the rate constant and the rate of reaction to be the same.

5. **Conclusion**:
In a zero-order reaction, both the rate of reaction and the rate constant have the same units of \( \text{mol L}^{-1} \text{s}^{-1} \).

### Final Answer:
The correct answer is that the units of the rate constant and the rate of reaction are the same for **zero-order reactions**.

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