The rate of a gaseous reaction is given by the expression `k[A][B]`. If The volume of reaction vessel is suddenly reduced to one-fourth of the initial volume, the reaction rate relative to the original rate will be :

Options

A. `1//10`
B. `1//8`
C. `8`
D. 16

Answer

To solve the problem, we need to analyze how the rate of the reaction changes when the volume of the reaction vessel is reduced to one-fourth of its initial volume. The rate of the reaction is given by the expression:

\[ \text{Rate} = k[A][B] \]

where \( k \) is the rate constant, and \([A]\) and \([B]\) are the concentrations of reactants A and B, respectively.

### Step-by-Step Solution:

1. **Define Initial Concentrations**:
- Let the initial number of moles of A be \( a \) and the number of moles of B be \( b \).
- The initial volume of the reaction vessel is \( V \).
- Therefore, the initial concentrations are:
\[ [A] = \frac{a}{V} \quad \text{and} \quad [B] = \frac{b}{V} \]

2. **Calculate Initial Rate**:
- The initial rate of the reaction (\( r \)) can be expressed as:
\[ r = k[A][B] = k \left(\frac{a}{V}\right) \left(\frac{b}{V}\right) = \frac{kab}{V^2} \]

3. **Change in Volume**:
- The volume of the reaction vessel is reduced to one-fourth of its original volume:
\[ V' = \frac{V}{4} \]

4. **Calculate New Concentrations**:
- The new concentrations after the volume change will be:
\[ [A]' = \frac{a}{V'} = \frac{a}{\frac{V}{4}} = \frac{4a}{V} \]
\[ [B]' = \frac{b}{V'} = \frac{b}{\frac{V}{4}} = \frac{4b}{V} \]

5. **Calculate New Rate**:
- The new rate of the reaction (\( r' \)) can be expressed as:
\[ r' = k[A'][B'] = k \left(\frac{4a}{V}\right) \left(\frac{4b}{V}\right) = k \cdot \frac{16ab}{V^2} = 16 \cdot \frac{kab}{V^2} \]

6. **Relate New Rate to Original Rate**:
- Since the original rate \( r = \frac{kab}{V^2} \), we can express the new rate in terms of the original rate:
\[ r' = 16r \]

### Conclusion:
The new reaction rate \( r' \) is 16 times the original rate \( r \). Therefore, the reaction rate relative to the original rate will be:

\[ \text{Rate relative to the original rate} = 16 \]