Which one of the following formula represents a first order reaction?

Options

A. `K = x/t`
B. `K = 1/(2t)[1/((a-x)^2)-1/a^2]`
C. `K = 2.303 t log""a/((a_0-x))`
D. `K = 1/t xx x/(a(a-x))`

Answer

To determine which formula represents a first-order reaction, we need to analyze the characteristics of first-order kinetics.

### 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 first-order rate equation is:
\[ \frac{d[A]}{dt} = -k[A] \]
where \([A]\) is the concentration of the reactant, \(k\) is the rate constant, and \(t\) is time.

2. **Integrating the Rate Equation**:
To find the relationship between concentration and time, we integrate the rate equation:
\[ \int \frac{d[A]}{[A]} = -k \int dt \]
This gives us:
\[ \ln[A] = -kt + \ln[A_0] \]
where \([A_0]\) is the initial concentration of the reactant.

3. **Rearranging the Equation**:
Rearranging the integrated equation, we can express it in terms of the concentration at time \(t\):
\[ [A] = [A_0] e^{-kt} \]
This shows that the concentration of the reactant decreases exponentially over time.

4. **Alternative Form of the Equation**:
We can also express the first-order reaction in logarithmic form:
\[ kt = \ln[A_0] - \ln[A] \]
Using the properties of logarithms, this can be rewritten as:
\[ kt = \ln\left(\frac{[A_0]}{[A]}\right) \]

5. **Final Form of the First-Order Reaction**:
Another common form used for first-order reactions is:
\[ kt = 2.303 \log\left(\frac{[A_0]}{[A]}\right) \]
This is derived from converting the natural logarithm to base 10 logarithm.

6. **Identifying the Correct Formula**:
Based on the derived equations, we can identify that the formula representing a first-order reaction is:
\[ [A] = [A_0] e^{-kt} \]
or in logarithmic form:
\[ kt = 2.303 \log\left(\frac{[A_0]}{[A]}\right) \]

### Conclusion:
To determine which formula represents a first-order reaction, look for an equation that follows the exponential decay or logarithmic relationship as shown above.