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What are the units of equivalent conductivity of a solution?
A. \[mho\text{ }c{{m}^{-1}}\]
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. $mho\text{ }c{{m}^{2}}\text{ }g\text{ }equi{{v}^{-1}}$

Last updated date: 17th Jun 2024
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Hint: Think about what conductivity and what is the unit of conductivity before moving on to find the unit of equivalent conductivity. Remember that $mho$ is another name for the unit Siemens (S).

Complete solution:
Conductance is the ability of any solution to conduct electricity via the dissociated ions in the solution. This electricity is conducted between two parallel electrodes that are 1cm apart and contain the entirety of the solution present between them. We know that the unit of conductance is Siemens (S) which was formerly known as $mho$ or $oh{{m}^{-1}}$.
Conductivity on the other hand, also known as specific conductance, measures the conductance of a solution per unit meter. The unit of conductivity is $S/m$ or $S/cm$.
We have to find the equivalent conductivity of a solution, which can be called as the conductivity of a substance per gram equivalent of the same substance. This can be written as:
\[\text{Equivalent conductivity = }\dfrac{\text{conductivity}}{\text{gram equivalent}}\]
Now, we know that one gram equivalent of a substance is the number of gram equivalents per unit volume.
\[\text{Gram equivalents = }\dfrac{\text{No}\text{. of gram equivalents}}{\text{Volume}}\]
Here, we will consider the volume to be in ${{m}^{3}}$ as it is the same as the volume in liters which is regularly considered to calculate the gram equivalents. Thus, taking the units for the gram equivalent:
\[\text{Gram equivalents = }\dfrac{g\text{ }equiv}{{{m}^{3}}}\]
Substituting these units in the formula for equivalent conductivity we get\[\text{Equivalent conductivity = }\dfrac{\dfrac{S}{m}}{\dfrac{g\text{ }equiv}{{{m}^{3}}}}\]
Now solving and simplifying, we get:
\[\text{Equivalent conductivity = }\dfrac{S}{m}\times \dfrac{{{m}^{3}}}{g\text{ }equiv}\]
\[\text{Equivalent conductivity = }\dfrac{S{{m}^{2}}}{g\text{ }equiv}\]
This is the SI unit of equivalent conductivity. We know that the older unit for Siemens is $mho$ and we can switch from ${{m}^{2}}$ to $c{{m}^{2}}$ for ease in calculation in some problems. Thus, the unit for equivalent conductance will be:
\[\text{Equivalent conductance = }mho\text{ }c{{m}^{2}}\text{ }g\text{ }equi{{v}^{-1}}\]

Hence, the correct answer is ‘D. $mho\text{ }c{{m}^{2}}\text{ }g\text{ }equi{{v}^{-1}}$’

Note: Remember the gram equivalents considered here, the number of gram equivalents, and the equivalent weight are three different concepts. Gram equivalents relate to volume, the number of gram equivalents relate to the number of electrons exchanged during a reaction, and the equivalent weight refers to the molecular weight in relation with the number of gram equivalents. All these terms are interconnected but have different meanings.