Understanding Electrical Conductivity and Resistivity Relations

Let's analyze each given relation to determine which ones are incorrect. We need to recall the standard definitions and formulas related to electrical resistance, specific resistance (resistivity), specific conductance (conductivity), and equivalent conductivity.

Analysis of Statement I: Specific Conductance Formula

Statement I provides the relation for specific conductance \(k\) as \(k = \frac{1}{R}\left( {l/A} \right)\). Let's verify this.

• Resistance \(R\) of a conductor is related to its specific resistance (resistivity) \(\rho\), length \(l\), and area of cross-section \(A\) by the formula: \(R = \rho \frac{l}{A}\).
• Specific resistance \(\rho\) is defined as \(\rho = R \frac{A}{l}\).
• Specific conductance \(k\) (or \(\kappa\)) is the reciprocal of specific resistance \(\rho\). So, \(k = \frac{1}{\rho}\).

Substituting the expression for \(\rho\) into the formula for \(k\):

$$k = \frac{1}{\rho} = \frac{1}{R \frac{A}{l}}$$

$$k = \frac{1}{R} \times \frac{l}{A}$$

This derived relation \(k = \frac{1}{R}\left( {l/A} \right)\) matches the one given in Statement I. Therefore, Statement I is correct.

Analysis of Statement II: Equivalent Conductivity Formula

Statement II provides the relation for equivalent conductivity \(\lambda\) as \(\lambda = \frac{{100\;K}}{C}\). Let's check the standard formula.

• Equivalent conductivity (\(\Lambda_{eq}\) or \(\lambda_{eq}\), often denoted by \(\lambda\)) relates specific conductance (\(K\) or \(k\)) to the concentration of the electrolyte solution, typically expressed in terms of normality (\(N\)).
• The standard relation is: \(\Lambda_{eq} = \frac{1000 \times K}{N}\), where \(K\) is in S cm^-1, \(N\) is in equivalents L^-1 (Normality), and \(\Lambda_{eq}\) is in S cm² equiv^-1.

Comparing the given relation \(\lambda = \frac{{100\;K}}{C}\) with the standard formula \(\lambda_{eq} = \frac{1000 \times K}{N}\), we see that the constant is 100 in the given relation, whereas it should be 1000 (assuming C represents normality or equivalent concentration and K is specific conductance in typical units). Therefore, Statement II is wrong.

Analysis of Statement III: Specific Resistance Formula

Statement III provides the relation for specific resistance \(\rho\) as \(\rho = \frac{{Rl}}{A}\). Let's verify this.

• As established earlier, the resistance \(R\) of a conductor is given by \(R = \rho \frac{l}{A}\).
• To find specific resistance \(\rho\), we need to rearrange this formula.

Multiplying both sides by \(\frac{A}{l}\):

$$R \times \frac{A}{l} = \rho \frac{l}{A} \times \frac{A}{l}$$

$$R \frac{A}{l} = \rho$$

So, the correct relation for specific resistance is \(\rho = R \frac{A}{l}\). The relation given in Statement III is \(\rho = \frac{{Rl}}{A}\), which is incorrect. Therefore, Statement III is wrong.

Summary of Analysis:

• Statement I: \(k = \frac{1}{R}\left( {l/A} \right)\) - Correct
• Statement II: \(\lambda = \frac{{100\;K}}{C}\) - Wrong (should be 1000)
• Statement III: \(\rho = \frac{{Rl}}{A}\) - Wrong (should be \(R \frac{A}{l}\))

The relations that are wrong are Statement II and Statement III.