Question

A conductance cell when filled with \[{{0}}{{.5 M KC}}l\] solution (specific conductance $ = 6.67 \times {10^{ - 3}}{\Omega ^{ - 1}}c{m^{ - 1}}$ ) registers a resistance of $243\Omega $ . Its cell constant is:
A. \[1.62{{ }}cm\]
B. \[1.62{{ }}c{m^{ - 1}}\]
C. $1.62d{m^{ - 1}}$
D. \[1.62{{ }}{m^{ - 1}}\]

Answer 1

Hint:First from the formula of resistance $R = \rho l/a$ we will find the relation between cell constant and conductivity (or specific conductance). Then using that relation we can find the cell constant value.
Formula used:
$R = \rho l/a$ where, $R$ is resistance, $\rho $ is resistivity, $l$ is length and $a$ is area of the cell.
Also, $l/a$ is the cell constant
$K = G \times l/a$ where, $K$ is the conductivity, $G$ is conductance.

Complete step by step answer:
First let us find the relation between cell constant and conductivity (or specific conductance):
We know that $R = \rho l/a$
And $l/a$ $ = $ cell constant
Therefore we can also write it as $l/a$ $ = $ $R/\rho $
We also are aware that $1/\rho = G$ , i.e., reciprocal of resistivity is conductivity.
Therefore we have the final relation $l/a = R \times G$
Now let’s see the given values, we have
 $G$ $ = 6.67 \times {10^{ - 3}}{\Omega ^{ - 1}}c{m^{ - 1}}$
$R$ $ = $ $243\Omega $
Now we put the values in the final relation that we derived to get the value of cell constant
$l/a = 6.67 \times {10^{ - 3}}{\Omega ^{ - 1}}c{m^{ - 1}} \times 243\Omega $
Upon doing the calculations we have the answer as $l/a = 1.62c{m^{ - 1}}$ .
Therefore the correct option is Option B.

Additional Information:
Cell constant is the ratio of the distance between the cell electrodes to the area (cross-sectional) that the electrodes occupy. It is usually measured for a cell containing a solution whose conductivity is already known to us. Because of this reason we use $KCl$ solution since its conductivity can be known to accurately measure at various concentrations and temperatures.

Note:
For a particular cell the value of $l/a$ is constant. This value is generally written on the cell and can be experimentally determined too. Cell constant is actually a multiplier constant which when multiplied with the measured current gives the electrical conductivity of the solution.