Question

Two solutions of strong electrolytes A and B are taken separately in same conductivity cell, their resistances were found to be 100 ohm and 200 ohm respectively. If equal volumes of solution A and solution B are mixed, the resistance of the mixture, using the same conductivity cell is $\frac{y}{3}$ ohm. Find the value of y.

Answer 1

400

Answer 2

To solve this problem, we use the relationships between resistance, conductance, conductivity, and cell constant.

1. Definitions:

   • Conductance (G) is the reciprocal of resistance (R): $G = \frac{1}{R}$.
   • Conductivity ($\kappa$) is related to conductance (G) and cell constant ($G^*$) by: $\kappa = G \times G^*$.
   • Combining these, conductivity can also be expressed as: $\kappa = \frac{G^*}{R}$.

2. Individual Solutions:

   • For solution A: Resistance $R_A = 100 \text{ ohm}$. Conductivity $\kappa_A = \frac{G^*}{R_A} = \frac{G^*}{100}$.
   • For solution B: Resistance $R_B = 200 \text{ ohm}$. Conductivity $\kappa_B = \frac{G^*}{R_B} = \frac{G^*}{200}$.

   Since the same conductivity cell is used, the cell constant ($G^*$) is the same for both solutions.

3. Mixing Solutions: When equal volumes of two strong electrolyte solutions are mixed, and assuming no change in ionic mobility or volume additivity, the concentration of each component in the mixture becomes half of its original concentration. Consequently, the conductivity of the mixture ($\kappa_{mix}$) is the average of the individual conductivities: $\kappa_{mix} = \frac{\kappa_A + \kappa_B}{2}$

4. Calculate Conductivity of Mixture: Substitute the expressions for $\kappa_A$ and $\kappa_B$ into the equation for $\kappa_{mix}$: $\kappa_{mix} = \frac{1}{2} \left( \frac{G^*}{100} + \frac{G^*}{200} \right)$ Factor out $G^*$: $\kappa_{mix} = \frac{G^*}{2} \left( \frac{1}{100} + \frac{1}{200} \right)$ Find a common denominator for the fractions: $\kappa_{mix} = \frac{G^*}{2} \left( \frac{2}{200} + \frac{1}{200} \right)$ $\kappa_{mix} = \frac{G^*}{2} \left( \frac{3}{200} \right)$ $\kappa_{mix} = \frac{3G^*}{400}$

5. Calculate Resistance of Mixture: Let $R_{mix}$ be the resistance of the mixture. Using the relationship $\kappa_{mix} = \frac{G^*}{R_{mix}}$: $\frac{G^*}{R_{mix}} = \frac{3G^*}{400}$ Cancel out $G^*$ from both sides: $\frac{1}{R_{mix}} = \frac{3}{400}$ Solve for $R_{mix}$: $R_{mix} = \frac{400}{3} \text{ ohm}$

6. Find the Value of y: The problem states that the resistance of the mixture is $\frac{y}{3}$ ohm. So, $\frac{y}{3} = \frac{400}{3}$ Therefore, $y = 400$.