Question

If \(\lambda_{\infty}\) and \(\lambda_{v}\) are the equivalent conductances at infinite dilution and at \(V\) dilution, the degree of dissociation, \(\alpha\) is given by (a) \(\alpha=\frac{\lambda_{\infty}}{\lambda_{v}}\) (b) \(\alpha=\frac{\lambda_{c 0}}{\lambda_{v}^{2}}\) (c) \(\alpha=\frac{\lambda_{v}}{\lambda_{\mathrm{cos}}}\) (d) None of these

Short answer

The correct option is (d) None of these.

Step-by-step solution

Refer to the basic formula

The basic formula for degree of dissociation (\(\alpha\)) is given as \(\alpha=\frac{\lambda_{v}}{\lambda_{\infty}}\) and not the inverse. This formula is derived from the fundamentals of physical chemistry.

Compare with given options

Compare this formula with the options given in the exercise. It can be seen that the first option \(\alpha=\frac{\lambda_{\infty}}{\lambda_{v}}\) is the inverse of the basic formula, hence it is incorrect.

Check remaining options

The other options such as \(\alpha=\frac{\lambda_{c 0}}{\lambda_{v}^{2}}\) and \(\alpha=\frac{\lambda_{v}}{\lambda_{\mathrm{cos}}}\) are not matching the basic formula. Hence, these options are also incorrect.

Confirming the correct answer

As none of the given options match the basic formula for degree of dissociation (\(\alpha\)), we can conclude that 'None of these' is the correct answer.

Key concepts

Equivalent Conductance

In physical chemistry, equivalent conductance is a measure of the conductivity of an electrolyte solution normalized for the concentration of the electrolyte. It is denoted by the symbol \(\Lambda_v\) and is measured in Siemens per meter (S/m). The equivalent conductance of a solution at a specific concentration is defined as the conductance of one equivalent of the electrolyte contained in a volume of the solution that, when placed between two parallel electrodes one centimeter apart, provides a conductance of one Siemens.

To understand how it changes with concentration, it's important to note that as the concentration of an electrolyte decreases, the ions present in the solution are more widely spaced and their interactions reduce, leading to an increase in equivalent conductance. At infinite dilution, when the concentration approaches zero, the ions are farthest apart, leading to the highest possible equivalent conductance for that electrolyte, denoted as \(\Lambda_{\infty}\).

Infinite Dilution

Infinite dilution refers to a hypothetical condition where a solution's solute concentration is reduced to almost zero, effectively eliminating interactions between the solute particles. In the case of electrolytes, this means that the ionic species are far enough apart that their conductive properties are not impeded by each other's presence.

From the perspective of physical chemistry, it's significant because at infinite dilution, the ionic mobility reaches a maximum, and the equivalent conductance (\(\Lambda_{\infty}\)) of the electrolyte is at its peak. This condition is used as a reference point for studying and comparing the behavior of different electrolytes because it represents the innate conductive potential of ions without interionic interactions.

Physical Chemistry

Physical chemistry is the branch of chemistry concerned with the underlying principles and laws that govern the physical properties and transformations of chemical substances. Not only does it focus on the macroscopic phenomena observed in chemical systems but also on the submicroscopic (atomic and molecular) level to explain them. Concepts like equivalent conductance and degree of dissociation are critical to this field because they demonstrate how fundamental principles, such as electrolyte dissociation and ion mobility, explain observable properties like conductivity and reactivity.

Understanding these concepts requires a solid grasp of the relationships between concentration, conductance, and the degree of dissociation of ionic species in solution. Physical chemistry provides the statistical and thermodynamic frameworks needed to analyze these relationships quantitatively.

Dilution Formulas

In solving problems related to solution concentration and conductance, dilution formulas play a prominent role. They are mathematical expressions that allow us to calculate changes in properties when the concentration of a solution is altered, typically by adding a solvent. One of the keys to understanding these properties is the application of the dilution formula to calculate the degree of dissociation (\(\alpha\)) using \(\Lambda_v\) and \(\Lambda_{\infty}\).

According to the formula, the degree of dissociation, which indicates the fraction of dissolved molecules that have dissociated into ions, can be expressed as \(\alpha = \frac{\Lambda_v}{\Lambda_{\infty}}\). This formula highlights the importance of quantifying how many electrolyte molecules are present as free ions at a particular concentration.

Applying Dilution Principles

Understanding dilution allows students not only to find the concentration of an ion in a solution after dilution but also to predict how the conductive properties of an electrolyte will change with concentration adjustments.