Question

At \[{25^ \circ }C\] specific conductance of a normal solution of \[KCl\] is \[0.002765\;mhos\;c{m^{ - 1}}\]. The resistance of the cell is \[400\;ohms\]. The cell constant is (\[c{m^{ - 1}}\]):
(a) 0.815
(b) 1.016
(c) 1.106
(d) 2.106

Answer 1

Hint: The cell constant is the measure of the theoretical distance between electrodes and the area of the cross-section. Whereas specific conductance is defined as the conductance of a substance to conduct electricity. And it is the reciprocal value of specific resistance.

Complete Step by Step Solution:
The value of the cell constant (\[K\]) can be calculated in the following way.

We know cell constant is \[K = \frac{l}{A}\], where l is the length in \[{\rm{cm}}\]and A is the area in \[{\rm{c}}{{\rm{m}}^2}\].
Therefore, the value of cell constant will be in \[{\rm{c}}{{\rm{m}}^{ - 1}}\] i.e, \[K = \frac{{cm}}{{c{m^2}}} = c{m^{ - 1}}\].
Also, \[R \propto \frac{l}{A}\] or \[R = \rho \frac{l}{A}\] where\[\rho \] is specific resistant.

The reciprocal value of specific resistance is known as specific conductance (\[k\]) or conductivity i.e., \[k = \frac{l}{\rho }\].
Hence mathematically, \[\frac{l}{R} = \frac{l}{\rho }.\frac{A}{l}\]
On rearranging,
\[\frac{l}{R} = k.\frac{A}{l}\]
\[\frac{l}{A} = k.R\]
\[K = k.R\]
or
(Cell constant \[ = \] Specific conductance\[ \times \] Resistant) … Eq. 1
Here, \[K = \] cell constant
\[k = \] Specific conductance
\[R = \] Resistant

Given in question,
The specific conductance (\[k\]) for \[KCl\] is \[0.002765\;{\Omega ^{ - 1}}c{m^{ - 1}}\] and resistant of the cell is \[400\Omega \]. By putting these values in equation 1, the value of the cell constant can easily be calculated.
\[K = 0.002765\;{\Omega ^{ - 1}}c{m^{ - 1}} \times 400\Omega \]
Hence, \[K = 1.106c{m^{ - 1}}\].
Hence, the above explanation denotes that options C is correct.

Note: The value of cell constant is depending upon the cross-section area of electrodes, the distance between the electrodes and on the nature of the electric fields. The value of the cell constant of a conductivity cell stays constant for a cell.