Question

What is the correct official unit measurement scale used to represent Molar Conductivity (\( \Lambda_m \)) profiles within electrolytic systems?

• A. \( \text{S cm}^{-2} \text{ mol}^{-1} \)
• B. \( \text{S}^{-1} \text{ cm}^2 \text{ mol} \)
• C. \( \Omega \text{ cm}^2 \text{ mol}^{-1} \)
• D. \( \text{S cm}^2 \text{ mol}^{-1} \)

Hint:

When using concentration values in moles per liter (\(\text{mol L}^{-1}\)), apply a scaling factor of 1000 to keep your units consistent: \( \Lambda_m = \frac{\kappa \times 1000}{\text{Molarity}} \), which yields the final value directly in \( \text{S cm}^2 \text{ mol}^{-1} \).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Conductivity (\(\kappa\)) is a measure of a solution's ability to conduct an electric current.
Molar conductivity (\(\Lambda_m\)) is the conducting power of all the ions produced by dissolving one mole of an electrolyte in a solution.
It allows scientists to compare the conductivities of different electrolytes while accounting for the differences in their concentrations.
Step 2: Key Formula or Approach:
The molar conductivity is mathematically related to specific conductivity (\(\kappa\)) and concentration (\(C\)) by the following expression:
\[ \Lambda_m = \frac{\kappa}{C} \]
To find the units, we must perform a dimensional analysis using the base units of the variables involved.
Step 3: Detailed Explanation:
1. Analyze \(\kappa\) (Specific Conductivity): Specific conductivity is the conductance of a solution of 1 cm length and 1 cm\(^2\) area of cross-section. Its units are Siemens per centimeter (\(S \cdot cm^{-1}\)).
(Note: \( S = \Omega^{-1} = \text{mho} \))

2. Analyze \(C\) (Concentration): In the context of molar conductivity calculations, concentration is typically expressed in terms of the number of moles per unit volume. To keep lengths consistent with cm, we use moles per cubic centimeter (\(mol \cdot cm^{-3}\)).

3. Combine Units:
\[ \text{Units of } \Lambda_m = \frac{S \cdot cm^{-1}}{mol \cdot cm^{-3}} \]
Using the rules for exponents (\(\frac{a^m}{a^n} = a^{m-n}\)):
\[ \text{Units of } \Lambda_m = S \cdot cm^{-1} \cdot cm^3 \cdot mol^{-1} \]
\[ \text{Units of } \Lambda_m = S \cdot cm^2 \cdot mol^{-1} \]

4. Alternative Forms: While \(S \cdot cm^2 \cdot mol^{-1}\) is the most common laboratory unit, in SI units, it would be \(S \cdot m^2 \cdot mol^{-1}\). Since the question asks for the standard "official unit measurement scale" usually found in textbook profiles, the cm-based version is expected.
Step 4: Final Answer:
The derived units match option (D).