Question

The resistance of a solution A is 50 ohms and that of solution B is 100 ohms, both solutions being taken in the same conductivity cell. If equal volumes of solution A and B are mixed, the resistance (in ohms) of the mixture using the same cell will be $13x+2$. Then $x$ is:
Assume that there is no increase in the degree of dissociation of A and B on mixing.

(write the value to the nearest integer)

Answer 1

Let $K_1$ and $K_2$ represent the conductivities of two solutions A and B respectively.
Let $x$ be the cell constant.
For solution A,
The conductivity is the product of the conductance and the cell constant.
$K_1=\frac{x}{50}$......(1)
For solution B
$K_2=\frac{x}{100}$......(2)
Equal volumes of A and B are mixed. Both the solutions are doubly diluted.
Their individual contribution towards the conductivity of the mixture will be $\frac{K_1}{2}$ and $\frac{K_2}{2}$ respectively and the conductivity of the mixture will be $\frac{K_1+K_2}{2}$
For the mixture,
$\frac{K_1+K_2}{2}=\frac{x}{R}$......(3).
Here, $R$ is the resistance of the mixture.
Solving equations (1), (2) and (3), we get
$R=66.67\approx 67$ ohm.

Hence, $x$=5