# What are the units of equivalent conductivity of a solution?

(A) $mho\text{ }c{{m}^{-1}}$$mho\text{ }c{{m}^{-1}}$

(B) $ohm\text{ }c{{m}^{-1}}\text{ }g\text{ }equi{{v}^{-1}}$

(C) $mho\text{ }c{{m}^{-2}}\text{ }g\text{ }equi{{v}^{-1}}$

(D) $oh{{m}^{-1}}\text{ c}{{\text{m}}^{2}}\text{ g equi}{{\text{v}}^{-1}}$

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**Hint:**The conductance of a solution of different electrolytes varies with their concentration. To compare the conductance of different electrolytes. It is convenient to define a quantity which is called equivalent conductance.

**Complete answer:**

- The equivalent conductance can be defined as the net conductance of every ion that is produced from one gram equivalent of a given substance.

- If we consider two large parallel electrodes set 1 cm apart and the whole of the solution containing 1g equivalent of an electrolyte is placed between the electrodes. If V is the volume of the solution containing 1 g equivalent of an electrolyte. The equivalent conductivity is given as:

\[\lambda =kV\]

Where k is the specific conductance,

If C is concentration of the solution in $\left( g\text{ }equi\text{ }c{{m}^{-3}} \right)$, then we can write the relation of Volume to concentration, that is $volume=\dfrac{1}{concentration}$.

Hence, Then the equation is,

\[\lambda =k\times \dfrac{1}{C}\]

- $\lambda $ is never determined directly, but is calculated from its specific conductance and concentration.

- Specific conductance k is the reciprocal of specific resistance $\rho $ (it is the resistance offered by a material 1cm in length and having an area of cross section $1c{{m}^{2}}$). Specific resistance has unit ohm cm, specific conductance has unit of$oh{{m}^{-1}}\text{ }c{{m}^{-1}}$

-Hence we can see that the unit of equivalent conductance is found to be:

\[\lambda =k\times \dfrac{1}{C}\]

\[\begin{align}

& \dfrac{oh{{m}^{-1}}c{{m}^{-1}}}{g\text{ equiv c}{{\text{m}}^{3}}} \\

& =oh{{m}^{-1}}\text{ }g\text{ equi}{{\text{v}}^{-1}}\text{ c}{{\text{m}}^{2}}\text{ } \\

\end{align}\]

Or we can write it as $oh{{m}^{-1}}\text{ c}{{\text{m}}^{2}}\text{ g equi}{{\text{v}}^{-1}}$

- Hence we can conclude that the option (d) is the correct answer that is the equivalent conductivity has the unit $oh{{m}^{-1}}\text{ c}{{\text{m}}^{2}}\text{ g equi}{{\text{v}}^{-1}}$.

**Additional information:**

- Experimental measurement of a solution is reciprocal of the resistance, therefore, the experimental determination of the conductance of a solution involves the measurement of its resistance.

- We have seen that conductivity k is the reciprocal of resistivity, that is $\rho $that is:

\[\begin{align}

& k=\dfrac{1}{\rho } \\

& and\text{ }\rho \text{=R}\dfrac{a}{l} \\

& k=\dfrac{1}{R}\left( \dfrac{1}{a} \right) \\

& k=G\left( \dfrac{l}{a} \right) \\

\end{align}\]

Where G is the conductance of the cell, l is the distance of separation of two electrodes, and $\dfrac{l}{a}$ cell constant.

**Note:**

- We should not get confused in terms of specific and equivalent conductance. Specific conductance is denoted by symbol k and equivalent conductance is denoted by symbol $\lambda $

- We can see that$\lambda $ is never determined directly, but always calculated from its specific conductivity and concentration.