If the conductivity of mercury at 0°C is $$1.07 \times 10^6$$ S m$$^{-1}$$ and the resistance of a cell containing mercury is 0.243$$\Omega$$, then the cell constant of the cell is $$x \times 10^4$$ m$$^{-1}$$. The value of x is _________. (Nearest integer)

We have been given the conductivity (also called specific conductance) of mercury at $$0^{\circ}{\rm C}$$ as $$\kappa = 1.07 \times 10^{6}\;{\rm S\,m^{-1}}$$ and the resistance of the conductivity cell filled with mercury as $$R = 0.243\;\Omega$$.

First, recall the fundamental relation that links conductivity, conductance and the cell constant:

$$\kappa \;=\; {\rm (cell\;constant)} \times {\rm (conductance)}.$$

Conductance $$G$$ is the reciprocal of resistance. Stating the definition,

$$G \;=\; \frac{1}{R}.$$

Substituting the given resistance, we obtain

$$G \;=\; \frac{1}{0.243\;\Omega}.$$

Carrying out the division step by step,

$$G \;=\; \frac{1.000}{0.243} \;=\; 4.115226337\;\text{S} \;( \text{siemens}).$$

Keeping an adequate number of significant digits for intermediate work, we may write

$$G \;\approx\; 4.115\;\text{S}.$$

Now insert this value of conductance into the conductivity relation:

$$\kappa \;=\; ({\rm cell\;constant}) \times G.$$

Hence, the cell constant becomes

$${\rm cell\;constant} \;=\; \frac{\kappa}{G} \;=\; \frac{1.07 \times 10^{6}\;{\rm S\,m^{-1}}}{4.115226337\;{\rm S}}.$$

Performing the division carefully,

$$\frac{1.07 \times 10^{6}}{4.115226337} \;\approx\; 2.600 \times 10^{5}\;{\rm m^{-1}}.$$

We are asked to express the result in the form $$x \times 10^{4}\;{\rm m^{-1}}.$$

Rewrite $$2.600 \times 10^{5}\;{\rm m^{-1}}$$ as

$$2.600 \times 10^{5} \;=\; 26.00 \times 10^{4}\;{\rm m^{-1}}.$$

Comparing with $$x \times 10^{4}\;{\rm m^{-1}},$$ we identify

$$x = 26.$$

So the nearest integer value required is $$26.$$

Hence, the correct answer is Option 26.