Question

Resistance of a conductivity cell filled with a solution of an electrolyte of concentration \[{\text{0}}{\text{.1 M}}\] is \[{\text{ 100}}\Omega \] The conductivity of this solution is \[{\text{1}}{\text{.29 S/m}}\]. Resistance of the same cell when filled with \[{\text{0}}{\text{.02 M}}\] is \[{\text{520}}\Omega \]. The molar conductivity of \[{\text{0}}{\text{.02 M}}\] solution of the electrolyte will be.
1. \[{\text{1}}{\text{24}} \times {\text{1}}{{\text{0}}^{ - 4}}{\text{ S}}{{\text{m}}^2}{\text{mo}}{{\text{l}}^{ - 1}}\]
2. \[{\text{1240 }} \times {\text{1}}{{\text{0}}^{ - 4}}{\text{ S}}{{\text{m}}^2}{\text{mo}}{{\text{l}}^{ - 1}}\]
3. \[{\text{1}}{\text{.24}} \times {\text{1}}{{\text{0}}^{ - 4}}{\text{ S}}{{\text{m}}^2}{\text{mo}}{{\text{l}}^{ - 1}}\]
4. \[{\text{12}}{\text{.4}} \times {\text{1}}{{\text{0}}^{ - 4}}{\text{ S}}{{\text{m}}^2}{\text{mo}}{{\text{l}}^{ - 1}}\]

Answer 1

Hint: Cell constant is the product of conductivity and resistance.
Use the following formula for the cell constant of the conductivity cell
\[{\text{Cell constant = Conductivity }} \times {\text{ resistance}}\].
Use the following formula to calculate molar conductivity.
\[{\text{Molar conductivity}} = \dfrac{{{\text{Conductivity}}}}{{{\text{1000 }} \times {\text{ molarity}}}}\]

Complete answer:
Resistance of a conductivity cell filled with a solution of an electrolyte of concentration \[{\text{0}}{\text{.1 M}}\] is \[{\text{ 100}}\Omega \] The conductivity of this solution is \[{\text{1}}{\text{.29 S/m}}\]. Calculate the cell constant.
\[{\text{Cell constant = Conductivity }} \times {\text{ resistance}} \\
{\text{Cell constant = }}1.29 \times {\text{ 100}} \\
{\text{Cell constant = }}129{\text{ }}{{\text{m}}^{ - 1}} \\\]
The value of the cell constant of the conductivity cell is \[129{\text{ }}{{\text{m}}^{ - 1}}\]
Resistance of the same cell when filled with \[{\text{0}}{\text{.02 M}}\] is \[{\text{520}}\Omega \].
Calculate the conductivity of the second solution
\[{\text{Conductivity}} = \dfrac{{{\text{Cell constant}}}}{{{\text{Resistance}}}} \\
{\text{Conductivity}} = \dfrac{{129}}{{520}}{\text{ S }}{{\text{m}}^{ - 1}} \\\]
Hence, the conductivity of the second solution is \[\dfrac{{129}}{{520}}{\text{ S }}{{\text{m}}^{ - 1}}\].
Calculate the molar conductivity of second solution
\[{\text{Molar conductivity}} = \dfrac{{{\text{Conductivity}}}}{{{\text{1000 }} \times {\text{ molarity}}}} \\
{\text{Molar conductivity}} = \dfrac{{\dfrac{{129}}{{520}}{\text{ S }}{{\text{m}}^{ - 1}}}}{{{\text{1000 }} \times {\text{ 0}}{\text{.02 M}}}} \\
{\text{Molar conductivity}} = 124 \times {10^{ - 4}}{\text{ S }}{{\text{m}}^2}{\text{ mo}}{{\text{l}}^{ - 1}} \\\]
Hence the molar conductivity of second solution is \[124 \times {10^{ - 4}}{\text{ S }}{{\text{m}}^2}{\text{ mo}}{{\text{l}}^{ - 1}}\].

Hence, the correct option is the option (1).

Additional Information In the equation for molar conductivity, the molarity is multiplied with 1000 to convert the unit of molarity from \[{\text{mol d}}{{\text{m}}^{ - 3}}{\text{ to mol }}{{\text{m}}^{ - 3}}\]. This is done because in the numerator, the unit of conductivity is Siemens per metre.

Note: The cell constant of the conductivity cell is independent of the concentration of the electrolyte. It depends on the dimensions of the electrode. For a given electrode, the cell constant is a constant. The cell constant is the ratio of distance between the electrodes to surface area of the electrodes.