The correct nature of plot for first order reaction is (are):

Options

A.
B.
C.
D.

Explanation

To determine the correct nature of plots for a first-order reaction, we need to analyze the relationships between concentration, time, and the rate of reaction.

### Step-by-Step Solution:

1. **Understanding First-Order Reactions**:
A first-order reaction is characterized by the rate being directly proportional to the concentration of one reactant. The rate law can be expressed as:
\[ \text{Rate} = -\frac{dC}{dt} = k[C] \]
where \( k \) is the rate constant and \( C \) is the concentration.

2. **Integrated Rate Law**:
The integrated rate law for a first-order reaction can be derived as follows:
\[ \ln[C] = \ln[C_0] - kt \]
Rearranging gives:
\[ t = \frac{1}{k} \ln\left(\frac{C_0}{C}\right) \]
This can be rewritten in a linear form:
\[ \ln[C] = -kt + \ln[C_0] \]
This indicates that if we plot \(\ln[C]\) versus time \(t\), we will get a straight line with a slope of \(-k\).

3. **Plotting Concentration vs. Time**:
If we plot concentration \(C\) versus time \(t\) for a first-order reaction, the plot will not be a straight line. Instead, it will be a curve that decreases exponentially.

4. **Plotting Log Concentration vs. Time**:
As derived earlier, plotting \(\ln[C]\) (or \(\log[C]\)) against time \(t\) yields a straight line. Thus, this plot is correct for a first-order reaction.

5. **Rate vs. Concentration Plot**:
The rate of reaction can also be expressed in terms of concentration:
\[ \text{Rate} = k[C] \]
Taking logarithms gives:
\[ \log(\text{Rate}) = \log(k) + \log[C] \]
This indicates that if we plot \(\log(\text{Rate})\) versus \(\log[C]\), we will also get a straight line with a slope of 1.

6. **Conclusion**:
From the analysis, the correct plots for a first-order reaction are:
- A straight line for \(\log[C]\) vs. \(t\) (Option A).
- A straight line for \(\log(\text{Rate})\) vs. \(\log[C]\) (Option D).

### Final Answer:
The correct nature of plots for a first-order reaction are:
- Option A: \(\log[C]\) vs. \(t\) (straight line with negative slope).
- Option D: \(\log(\text{Rate})\) vs. \(\log[C]\) (straight line with slope of 1).