If `R_(1)` and `R_(2)` are the resistance of two solution of equal volume, then the conductance of the mixture in the same conductivity cell is:

Options

A. `(1)/(2)((1)/(R_1) +(1)/(R_2))`
B. `(1)/(2)((1)/(R_(1)+R_(2)))`
C. `(1)/(sqrtR_(1) R_(2)`
D. none of these

Explanation

To solve the problem regarding the conductance of a mixture of two solutions with equal volumes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Conductance and Resistance**:
- Conductance (G) is the reciprocal of resistance (R). Therefore, \( G = \frac{1}{R} \).
- The specific conductance (κ) is defined as \( \kappa = \frac{1}{R} \cdot G^* \), where \( G^* \) is the cell constant.

2. **Calculate Specific Conductance for Each Solution**:
- For the first solution, the specific conductance \( \kappa_1 \) can be expressed as:
\[ \kappa_1 = \frac{G^*}{R_1} \]
- For the second solution, the specific conductance \( \kappa_2 \) is:
\[ \kappa_2 = \frac{G^*}{R_2} \]

3. **Mixing the Solutions**:
- When equal volumes of the two solutions are mixed, the new specific conductance \( \kappa_{mix} \) for the mixture will be the average of the specific conductances of the two solutions, each halved due to dilution:
\[ \kappa_{mix} = \frac{\kappa_1}{2} + \frac{\kappa_2}{2} \]

4. **Substituting the Values**:
- Substitute the expressions for \( \kappa_1 \) and \( \kappa_2 \):
\[ \kappa_{mix} = \frac{G^*}{2R_1} + \frac{G^*}{2R_2} \]
- Factor out \( G^* \):
\[ \kappa_{mix} = G^* \left( \frac{1}{2R_1} + \frac{1}{2R_2} \right) \]

5. **Finding the Conductance of the Mixture**:
- The conductance \( G_{mix} \) can be expressed as:
\[ G_{mix} = \kappa_{mix} \cdot \text{Cell Constant} \]
- Since the cell constant remains the same, we can write:
\[ G_{mix} = \frac{G^*}{2} \left( \frac{1}{R_1} + \frac{1}{R_2} \right) \]
- This can be simplified to:
\[ G_{mix} = \frac{G^*}{2} G_{net} \]
- Where \( G_{net} = \frac{1}{R_1} + \frac{1}{R_2} \).

6. **Final Expression**:
- Thus, the final expression for the conductance of the mixture is:
\[ G_{mix} = \frac{G^*}{2} \left( \frac{1}{R_1} + \frac{1}{R_2} \right) \]