The resistance of a solution `A` is `50 ohm` and that of solution `B` is `100ohm,` both solutions are taken in the same conductivity cell. If equal volumes of solution`A` and `B` are mixed, what is the resistance of the mixture using the same cell ? `(` Assume there is no change or increase in the `prop` of `A` and `B` on mixing `)`.

Explanation

To solve the problem of finding the resistance of the mixture of solutions A and B, we can follow these steps:

### Step 1: Understand the given values
- Resistance of solution A, \( R_A = 50 \, \Omega \)
- Resistance of solution B, \( R_B = 100 \, \Omega \)

### Step 2: Calculate the specific conductance (κ) for each solution
The specific conductance (κ) is related to resistance (R) by the formula:
\[ \kappa = \frac{G^*}{R} \]
Where \( G^* \) is the cell constant, which remains constant for the same conductivity cell.

For solution A:
\[ \kappa_A = \frac{G^*}{R_A} = \frac{G^*}{50} \]

For solution B:
\[ \kappa_B = \frac{G^*}{R_B} = \frac{G^*}{100} \]

### Step 3: Mix equal volumes of solutions A and B
When equal volumes of solutions A and B are mixed, the specific conductance of the mixture (κ_mix) can be calculated as the average of the specific conductances of A and B:
\[ \kappa_{mix} = \frac{\kappa_A + \kappa_B}{2} \]

Substituting the values of κ_A and κ_B:
\[ \kappa_{mix} = \frac{\frac{G^*}{50} + \frac{G^*}{100}}{2} \]

### Step 4: Simplify the expression for κ_mix
To simplify:
\[ \kappa_{mix} = \frac{G^* \left(\frac{1}{50} + \frac{1}{100}\right)}{2} \]
Finding a common denominator for the fractions:
\[ \frac{1}{50} + \frac{1}{100} = \frac{2}{100} + \frac{1}{100} = \frac{3}{100} \]
Thus,
\[ \kappa_{mix} = \frac{G^* \cdot \frac{3}{100}}{2} = \frac{3G^*}{200} \]

### Step 5: Relate κ_mix to the resistance of the mixture
Using the relationship between specific conductance and resistance:
\[ \kappa_{mix} = \frac{G^*}{R_{mix}} \]
We can set the two expressions for κ equal to each other:
\[ \frac{3G^*}{200} = \frac{G^*}{R_{mix}} \]

### Step 6: Solve for R_mix
Cancelling \( G^* \) from both sides (assuming \( G^* \neq 0 \)):
\[ \frac{3}{200} = \frac{1}{R_{mix}} \]
Taking the reciprocal gives:
\[ R_{mix} = \frac{200}{3} \approx 66.67 \, \Omega \]

### Final Answer
The resistance of the mixture of solutions A and B is approximately \( 66.67 \, \Omega \).

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