Which of the following solutions have highest resistance? — Electrochemistry Chemistry Question

💡 Solution & Explanation

Step 1 - Understand the Relationship Between Resistance and Conductivity For any electrical conductor, the electrical resistance ($R$) is defined by Ohm's Law and relates to the physical dimensions of the conductor: $$R = \rho \frac{l}{A} = \frac{1}{\kappa} \left(\frac{l}{A}\right)$$ Where: * $R$ is the electrical resistance in ohms ($\Omega$). * $\rho$ is the specific resistance (resistivity) in $\Omega\text{ m}$. * $\kappa$ is the specific conductance (conductivity) in $\Omega^{-1}\text{ m}^{-1}$ (or $\text{S m}^{-1}$). * $l$ is the distance between the electrodes. * $A$ is the cross-sectional area of the electrodes. * $\frac{l}{A}$ is the cell constant, which remains constant for a given conductivity cell. From this expression, we observe that for a fixed cell constant, the resistance of the solution is inversely proportional to its conductivity ($\kappa$): $$R \propto \frac{1}{\kappa}$$ Step 2 - Relate Conductivity ($\kappa$) to Electrolytic Concentration ($C$) The specific conductivity ($\kappa$) of an electrolytic solution is defined as the conductance of a unit volume (such as $1\text{ cm}^3$ or $1\text{ m}^3$) of the solution enclosed between two parallel electrodes of unit area separated by a unit distance. Therefore, the value of $\kappa$ depends directly on the **concentration of charge-carrying free ions per unit volume** (ion density): $$\kappa \propto \text{Number of ions per unit volume}$$ For a strong electrolyte like sodium chloride ($\ce{NaCl}$), dissociation into sodium ($\ce{Na^+}$) and chloride ($\ce{Cl^-}$) ions is virtually complete in dilute solutions: $$\ce{NaCl(aq) -> Na^+(aq) + Cl^-(aq)}$$ As the concentration ($C$) of the electrolyte solution decreases (i.e., as the solution is diluted): 1. The total volume of the solution containing a given mass of solute increases. 2. The number of ions per unit volume of the solution decreases. 3. Consequently, the specific conductivity ($\kappa$) decreases. Thus, specific conductivity is directly proportional to concentration: $$\kappa \propto C$$ Step 3 - Combine the Proportionalities to Relate Resistance ($R$) to Concentration ($C$) By combining the relationships from Step 1 and Step 2: $$R \propto \frac{1}{\kappa} \quad \text{and} \quad \kappa \propto C \implies R \propto \frac{1}{C}$$ The electrical resistance ($R$) of an electrolytic solution is inversely proportional to its molar or normal concentration ($C$). Therefore: * The **most concentrated** solution will have the highest ion density, the highest conductivity, and the **lowest resistance**. * The **most dilute** (lowest concentration) solution will have the lowest ion density, the lowest conductivity, and the **highest resistance**. Step 4 - Compare the Concentrations of the Given Solutions We are given four solutions of sodium chloride ($\ce{NaCl}$) with different normalities ($N$): * (A) $1\text{ N}$ * (B) $0.05\text{ N}$ * (C) $2\text{ N}$ * (D) $0.1\text{ N}$ Let us arrange these concentrations in decreasing order: $$2\text{ N} > 1\text{ N} > 0.1\text{ N} > 0.05\text{ N}$$ Comparing these values, the solution with the lowest concentration is $0.05\text{ N}$. Since $0.05\text{ N}$ is the most dilute solution among the choices, it contains the lowest number of ions per unit volume, possesses the lowest specific conductivity ($\kappa$), and consequently has the **highest electrical resistance ($R$)**. Step 5 - Evaluate the Options * **Option (A) is incorrect:** A $1\text{ N}$ solution has a relatively high concentration, which translates to a high ionic density, high conductivity, and low electrical resistance compared to the more dilute solutions. * **Option (B) is correct:** As mathematically shown, the $0.05\text{ N}$ solution is the most dilute among all options. This minimum concentration results in the lowest specific conductivity and therefore the highest resistance. * **Option (C) is incorrect:** A $2\text{ N}$ solution has the highest concentration among all options, resulting in the highest specific conductivity and the lowest electrical resistance. * **Option (D) is incorrect:** A $0.1\text{ N}$ solution is more concentrated than the $0.05\text{ N}$ solution, meaning its specific conductivity is higher and its resistance is lower than that of the $0.05\text{ N}$ solution. $$\text{Correct Option: } \boxed{\text{B}}$$