Which expression is wrong for first order reaction

Options

A. `k = (2.303)/(t) "log" ((A_(0))/(At))`
B. `k = (t)/(2.303) "log" ((A_(0))/(At))`
C. `-k = (t)/(2.303) "log" ((At)/(A_(0)))`
D. Rate = k[A]

Explanation

To determine which expression is wrong for a first-order reaction, we need to analyze the expressions related to the rate constant \( K \) and the integrated rate law for first-order reactions.

### Step-by-Step Solution:

1. **Understanding First-Order Reactions:**
- A first-order reaction is characterized by a rate that is directly proportional to the concentration of one reactant. The general form of the rate equation is:
\[ \text{Rate} = k[A] \]
- Where \( k \) is the rate constant and \( [A] \) is the concentration of the reactant.

2. **Integrated Rate Law for First-Order Reactions:**
- The integrated rate law for a first-order reaction can be expressed as:
\[ \ln[A] = \ln[A_0] - kt \]
- This can also be rearranged to:
\[ k = \frac{2.303}{t} \log\left(\frac{[A_0]}{[A]}\right) \]
- Here, \( [A_0] \) is the initial concentration and \( [A] \) is the concentration at time \( t \).

3. **Analyzing the Given Expressions:**
- We need to check each expression provided in the options to identify which one is incorrect.

- **Expression 1:**
\[ k = \frac{2.303}{t} \log\left(\frac{[A_0]}{[A]}\right) \]
- This expression is correct.

- **Expression 2:**
\[ k = \frac{2.303}{t} \log\left(\frac{[A]}{[A_0]}\right) \]
- This expression is incorrect because the logarithm should have the initial concentration in the numerator and the concentration at time \( t \) in the denominator.

- **Expression 3:**
\[ -k = \frac{2.303}{t} \log\left(\frac{[A]}{[A_0]}\right) \]
- This expression can be rearranged to show that it is equivalent to the correct form, thus it is correct.

- **Expression 4:**
\[ R = k[A_0]^1 \]
- This expression is also correct as it represents the rate of a first-order reaction.

4. **Conclusion:**
- The wrong expression for a first-order reaction is **Expression 2**.

### Final Answer:
The incorrect expression for a first-order reaction is **Expression 2**: \( k = \frac{2.303}{t} \log\left(\frac{[A]}{[A_0]}\right) \).