Question

Conductivity (Seimens \(\mathrm{S}\) ) is directly proportional to area of the vessel and the concentration of the solution it and is inversely proportional to the length of the vessel, then constant of proportionality is expressed in (a) \(\mathrm{S} \mathrm{m} \mathrm{mol}^{-1}\) (b) \(\mathrm{S}^{2} \mathrm{~m}^{2} \mathrm{~mol}^{2}\) (c) \(\mathrm{S} \mathrm{m}^{2} \mathrm{~mol}^{-1}\) (d) \(\mathrm{S}^{2} \mathrm{~m}^{2} \mathrm{~mol}\)

Short answer

(c) \( S \cdot m^2 \cdot mol^{-1} \)

Step-by-step solution

Understand conductivity proportionality

According to the problem statement, conductivity (\( C \)) is directly proportional to the area (\( A \)) and the concentration of the solution (\( c \)), and inversely proportional to the length (\( l \)). Mathematically, this can be expressed as \( C \propto \frac{A \cdot c}{l} \).

Introduce the constant of proportionality

To express the proportionality as an equation, introduce a constant of proportionality, \( k \). The relationship becomes \( C = k \cdot \frac{A \cdot c}{l} \).

Determine the units of each quantity

The units for each quantity are: conductivity (\( C \)) in Seimens (\( S \)), area (\( A \)) in square meters (\( m^2 \)), concentration (\( c \)) in moles per cubic meter (\( mol/m^3 \)), and length (\( l \)) in meters (\( m \)).

Analyze units of the equation

Substitute units into the equation \( S = k \cdot \frac{m^2 \cdot mol/m^3}{m} \). This simplifies to \( S = k \cdot \frac{mol}{m^2} \).

Solve for unit of constant of proportionality

Rearrange the equation to find \( k \). Thereby, \( k = S \cdot \frac{m^2}{mol} \). Therefore, the unit of \( k \) is \( S \cdot m^2 \cdot mol^{-1} \), which matches option (c).

Key concepts

Proportionality Constants

In the context of conductivity, a proportionality constant, often denoted as \( k \), is introduced to convert a proportional relationship into an equation. Proportional relationships express how one quantity changes in response to another. When dealing with conductivity, which is directly proportional to the area of the vessel and solution concentration and inversely proportional to the length, the relationship is initially represented as \( C \propto \frac{A \cdot c}{l} \). This means conductivity increases when either the area or concentration increases, but decreases if the length increases. To make calculations practical, we need to involve a constant \( k \) such that \( C = k \cdot \frac{A \cdot c}{l} \). This constant helps us quantify the relationship under specific conditions by providing a fixed multiplier, making the equation usable for predictions and measurements.

• Constant \( k \) adjusts the equation to match real-world data.
• Enables calculable predictions for changes in conductivity.

Units of Measurement

Understanding the units of measurement is crucial when working with equations involving physical quantities, such as conductivity. Each component in the equation \( C = k \cdot \frac{A \cdot c}{l} \) has specific units that must be compatible for calculations to make sense. In this scenario, conductivity \( (C) \) is measured in Siemens (\( S \)), which is the SI unit for electrical conductance. The area \( (A) \) is expressed in square meters (\( m^2 \)), the concentration \( (c) \) is given in moles per cubic meter (\( mol/m^3 \)), and the length \( (l) \) is measured in meters (\( m \)).
To solve for the constant \( k \), you must rearrange the equation to \( k = S \cdot \frac{m^2}{mol} \), showing its unit as Siemens meters squared per mole. This ensures that the units on both sides of the equation match, confirming the equation's validity. Understanding these units allows you to interpret and apply the conductivity formula correctly.

• Correct units are critical for meaningful equations.
• Consistent unit usage ensures coherence in scientific calculations.

Solution Concentration

Solution concentration is a key factor affecting conductivity and is part of the proportional relationship in the problem. It describes how much solute is dissolved in a given volume of solvent, typically measured in moles per cubic meter (\( mol/m^3 \)). High concentration means more charge carriers are available, potentially increasing conductivity. When integrated into the formula \( C = k \cdot \frac{A \cdot c}{l} \), concentration \( (c) \) directly affects the overall conductivity: as the concentration of the solution increases, conductivity tends to increase, assuming the area and length are constant.
The concentration is crucial because it dictates the number of charged particles present in the solution, thereby impacting the ease with which electrical current can flow. This makes it an essential parameter to control and measure in experiments involving conductivity, as shown in our understanding of the constant \( k \). Keeping track of solution concentration helps scientists tailor and optimize solutions for desired conductive properties.

• Higher concentration results in higher conductivity (up to saturation limits).
• Important for consistency in experiments measuring solution properties.