Question

Resistance of a decimolar solution between two electrodes \(0.02\) meter apart and \(0.0004 \mathrm{~m}^{2}\) in \(\cdots\) area was found to be 50 ohm. Specific conductance \((\kappa)\) is : (a) \(0.1 \mathrm{Sm}^{-1}\) (b) \(1 \mathrm{~S} \mathrm{~m}^{-1}\) (c) \(10 \mathrm{~S} \mathrm{~m}^{-1}\) (d) \(4 \times 10^{-4} \mathrm{~S} \mathrm{~m}^{-1}\)

Short answer

1 \mathrm{~S} \mathrm{~m}^{-1}

Step-by-step solution

Understand the Concept of Specific Conductance

Specific conductance, denoted as \(\kappa\), is defined as the conductance of a solution per unit length and per unit cross-sectional area perpendicular to the current flow. It is the reciprocal of specific resistance (\(\rho\)). The formula to find specific conductance is given by \(\kappa = 1/\rho\) or \(\kappa = L/(RA)\) where \(L\) is the distance between the electrodes, \(R\) is the resistance, and \(A\) is the cross-sectional area.

Calculate the Specific Conductance

Use the given values to calculate specific conductance.\[\kappa = \frac{L}{RA}\]\[\kappa = \frac{0.02}{50 \times 0.0004}\]Substitute the values into the formula and calculate the specific conductance.

Compute the Final Value

After substituting, perform the division to find \(\kappa\).\[\kappa = \frac{0.02}{50 \times 0.0004}\]\[\kappa = \frac{0.02}{0.02}\]\[\kappa = 1 \mathrm{~S} \mathrm{~m}^{-1}\]

Key concepts

Understanding Electrolytic Conductance

Electrolytic conductance is a measure of an electrolyte's ability to conduct electricity. When an electric current passes through an electrolytic solution, ions move towards the electrodes, allowing the current to flow through the liquid. The larger the number of ions present and the more mobile they are, the higher the electrolytic conductance of the solution.

Factors affecting electrolytic conductance include the nature of the ions (their size and charge), the temperature (conductance usually increases with temperature), and the concentration of the electrolyte (conductance generally increases with concentration up to a certain point). In most cases, a higher temperature or concentration leads to increased conductance due to increased ion activity or number.

It's also worth noting that the unit of electrolytic conductance is the siemens (S), previously known as mho, which is the reciprocal of resistance (ohm).

Resistance and Resistivity

While 'conductance' refers to a material's ability to conduct electrical current, 'resistance' is the measure of the opposition to current flow in an electrical circuit. The resistance of a conductor is determined by its size, shape, and the material it is made of, characterized by a property called 'resistivity'.

The formula for resistance (R) is given by: \[ R = \rho \frac{L}{A} \] where

• \( \rho \) (rho) is the resistivity of the material, measured in ohm-meters (\(\Omega m\)),
• L is the length of the conductor, and
• A is the cross-sectional area.

Resistivity is a fundamental property of materials and varies with temperature. Conductors have low resistivity, while insulators have high resistivity. This property helps in the selection of materials for different electrical components based on their required resistance levels.

Solution Conductance Calculation

The specific conductance of a solution, commonly denoted as \(\kappa\), is a practical way to understand solution conductance on a standardized scale. Calculating specific conductance is critical for fields such as electrochemistry and in applications like desalination or water purification.

To calculate this value, the formula used is: \[ \kappa = \frac{1}{\rho} = \frac{L}{RA} \] where:

• \(\rho\) is the specific resistance,
• L is the distance between the electrodes,
• R is the resistance of the solution, and
• A is the cross-sectional area.

In the exercise provided, when you work out the calculation with the resistance of 50 ohms and the given electrode distance and area, the answer attained is \(1 S m^{-1}\), which is choice (b). This calculation gives an insight into the ionic mobility and concentration within the solution, both of which are pivotal for understanding the conductive properties of the solution under study.

The clear interpretation of this calculation allows us to quantify the conductive ability of the solution, helping in a range of practical applications and theoretical studies.