Question

The relation among conductance \((G)\), specific conductance \((\kappa)\) and cell constant \((l / A)\) is : (a) \(G=\kappa \frac{l}{A}\) (b) \(G=\kappa \frac{A}{l}\) (c) \(G \kappa=\frac{l}{A}\) (d) \(G=\kappa A l\) \(\therefore\)

Short answer

The correct answer is (b) G = \(\kappa \frac{A}{l}\).

Step-by-step solution

Understand the Concepts

Conductance (G) is the inverse of resistance (R), and it measures how easily electricity can pass through a material. Specific conductance (κ) is a measure of a material's ability to conduct electricity, normalized to its dimensions. The cell constant (l/A) is the inverse of the geometric factor of the cell, where 'l' is the distance between electrodes and 'A' is the area of the electrodes. The correct relation would provide G in terms of κ and the cell constant.

Recall the Relation

The correct relation is based on the formula for resistance R in terms of resistivity ρ, length l, and cross-sectional area A, which is R = ρ(l/A). Since conductance G is the inverse of resistance R, and specific conductance κ is the inverse of resistivity ρ, the relation should take the form of G = κ(A/l).

Identify the Correct Formula

Looking at the options, the formula that correctly represents the relation between G, κ, and l/A is the one that has G as the product of κ and the inverse of the cell constant. This leads us to option (b) which is G = κ(A/l), as it is the only formula that follows this structure.

Key concepts

Understanding Electrical Conductance (G)

Electrical conductance, denoted by the symbol (G), is a fundamental property in physical chemistry that quantifies the ease with which an electrical charge or current can flow through a conductor. It is the inverse of electrical resistance (R), meaning that materials with high conductance allow electricity to pass through them more easily than those with high resistance.
In mathematical terms, if R represents the resistance of a material, then conductance G is calculated as the reciprocal of R:
\[ G = \frac{1}{R} \]
This relationship also implies that a large value of R, which signifies poor electrical flow, will result in a smaller value for G, indicating low conductance. Conductance is measured in siemens (S), which is equivalent to the reciprocal ohm (\( \Omega^{-1} \)). Understanding conductance is crucial for analyzing electronic circuits and material properties in the field of electrochemistry.

Specific Conductance (\(\kappa\))

Specific conductance, represented by the Greek letter kappa (\(\kappa\)), takes the concept of conductance one step further by considering the dimensions of the material through which the current flows. This property measures a material's ability to conduct electricity relative to its size and is expressed as conductance per unit length (for a given cross-sectional area).
Mathematically, specific conductance is the conductance of a unit cube of material and is defined by the equation:
\[ \kappa = \frac{G}{l} \]where

• \( G \) is the conductance,
• \( l \) is the distance between the electrodes in the cell.

It is important to note that the specific conductance is dependent on both the material's intrinsic property and the cell's geometry. For practical applications, knowing the specific conductance allows for the comparison of materials' electrical properties regardless of their size and shape. It is commonly measured in siemens per meter (S/m).

Cell Constant (\(\frac{l}{A}\))

The cell constant is an important concept in electrochemistry, particularly in the context of measuring the conductance of electrolytic solutions. It is denoted by the ratio \(\frac{l}{A}\), where

• \(l\) is the distance between the electrodes, and
• \(A\) is the cross-sectional area of the electrodes through which current passes.

The cell constant effectively standardizes measurements by accounting for variations in electrode size and separation. This way, the specific conductance of different solutions can be compared without the influence of these physical factors. In practical terms, the cell constant is the inverse of the geometric factor—the physical shape and spacing of the electrodes, and it's a crucial part of translating between measured conductance and specific conductance:
\[ \kappa = G \cdot \frac{l}{A} \]Typically, cell constants are assessed in units of meters inverse (\(m^{-1}\)), and regular calibration of the cell constant is necessary to ensure accurate measurements in experiments involving conductance.

Resistance and Resistivity in Physical Chemistry

Resistance (R) and resistivity (\(\rho\)) are two closely related concepts in physical chemistry that describe how difficult it is for an electrical current to flow through a material. Resistance is the hindrance to the flow of charge and for a conductor of certain dimensions, it is given by the formula:
\[ R = \rho \frac{l}{A} \]Here,

• \(\rho\) is the resistivity of the material,
• \(l\) is the length of the conductor, and
• \(A\) is its cross-sectional area.

Resistivity, on the other hand, is a material-specific property, indicating the inherent ability of the material to resist the flow of current. It takes into account the nature of the substance and its temperature, but not the dimensions of the material sample. Resistivity is measured in ohm-meters (\(\Omega m\)). The relationship between resistance and resistivity is fundamental to understanding material behavior in the presence of an electric field and is instrumental for applications ranging from designing electrical components to analyzing chemical compositions.