Which of the following curves represents a first order reaction.

Options

A.
B.
C.
D. All of these

Explanation

To determine which of the given curves represents a first-order reaction, we need to analyze the integrated rate law for first-order reactions.

### Step-by-Step Solution:

1. **Understanding the Rate Law for First-Order Reactions**:
The rate of a first-order reaction can be expressed as:
\[ -\frac{d[R]}{dt} = k[R] \]
where \([R]\) is the concentration of the reactant, \(t\) is time, and \(k\) is the rate constant.

2. **Integrated Rate Law**:
For a first-order reaction, the integrated rate law is given by:
\[ \ln\left(\frac{[R_0]}{[R]}\right) = kt \]
where \([R_0]\) is the initial concentration of the reactant and \([R]\) is the concentration at time \(t\).

3. **Identifying the Graphs**:
- **First Curve**: This curve shows the concentration of the reactant \([R]\) versus time \(t\). For a first-order reaction, this curve would typically show an exponential decay.
- **Second Curve**: This curve shows \(\ln[R]\) versus time \(t\). For a first-order reaction, this plot should yield a straight line with a negative slope.
- **Third Curve**: This curve shows \(\ln\left(\frac{[R_0]}{[R]}\right)\) versus time \(t\). According to the integrated rate law, this should also yield a straight line with a positive slope.

4. **Conclusion**:
The second and third curves represent first-order reactions, as they yield straight lines when plotted. However, the most common representation of a first-order reaction is the plot of \(\ln[R]\) versus time, which is the second curve.

### Final Answer:
The second curve (log concentration of reactant at time \(t\) versus time \(t\)) represents a first-order reaction.

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