When the rate determined by the change in concentration of two different reactants, then the kinetic equation may be expressed as

Options

A. `k_(2) = (2.303)/((a-b)t)log.((a-x)b)/((b-x)a)`
B. `k_(2) = (2.303)/((a-b)t)log.((a-x))/((b+x))`
C. `k_(2) = (2.303)/((a-b)t)log.((a-x))/((b-x))`
D. `k_(2) = (1)/(t) xx (x)/((a-x))`

Explanation

To solve the question regarding the kinetic equation when the rate is determined by the change in concentration of two different reactants, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Reaction**:
Consider the reaction:
\[ A + B \rightarrow \text{Products} \]
Here, \( A \) and \( B \) are the two different reactants.

2. **Define Initial Concentrations**:
At time \( t = 0 \):
- Concentration of \( A \) = \( [A]_0 = A \)
- Concentration of \( B \) = \( [B]_0 = B \)
- Concentration of products = 0

3. **Define Concentrations at Time \( t \)**:
At time \( t \):
- Concentration of \( A \) = \( [A] = A - x \)
- Concentration of \( B \) = \( [B] = B - x \)
- Concentration of products = \( x \)

4. **Write the Rate Equation**:
The rate of the reaction can be expressed as:
\[ \text{Rate} = -\frac{d[A]}{dt} = -\frac{d[B]}{dt} = k [A][B] \]
Substituting the expressions for \( [A] \) and \( [B] \):
\[ \text{Rate} = k (A - x)(B - x) \]

5. **Integrate the Rate Equation**:
We can express the rate as:
\[ \frac{dx}{dt} = k (A - x)(B - x) \]
Rearranging gives:
\[ \frac{dx}{(A - x)(B - x)} = k dt \]

6. **Set Up the Integral**:
Integrate both sides:
\[ \int_0^x \frac{dx}{(A - x)(B - x)} = \int_0^t k dt \]

7. **Perform the Integration**:
The left side can be integrated using partial fractions:
\[ \frac{1}{(A - x)(B - x)} = \frac{1}{B - A} \left( \frac{1}{A - x} - \frac{1}{B - x} \right) \]
Thus, the integral becomes:
\[ \frac{1}{B - A} \left( \ln|A - x| - \ln|B - x| \right) \bigg|_0^x = kt \]

8. **Evaluate the Limits**:
Evaluating the limits gives:
\[ \frac{1}{B - A} \left( \ln|A - x| - \ln|B - x| - (\ln|A| - \ln|B|) \right) = kt \]

9. **Rearranging the Equation**:
This can be rearranged to find:
\[ kt = \frac{1}{B - A} \left( \ln\left(\frac{(A - x)}{(B - x)}\right) + \ln\left(\frac{B}{A}\right) \right) \]

10. **Final Expression**:
The final expression can be simplified to:
\[ k = \frac{2.303}{B - A} \cdot \frac{t}{\log\left(\frac{(A - x)}{(B - x)}\right)} \]