Question

The ionic conductivity of \(\mathrm{Ba}^{2+}\) and \(\mathrm{Cl}^{-}\) at infinite dilution are 127 and \(76 \mathrm{ohm}^{-1} \mathrm{~cm}^{2} \mathrm{eq}^{-1}\) respectively. The equivalent conductivity of \(\mathrm{BaCl}_{2}\) at infinity dilution (in \(\mathrm{ohm}^{-1} \mathrm{~cm}^{2} \mathrm{eq}^{-1}\) ) would be : (a) 203 (b) 279 (c) \(101.5\) (d) \(139.5\)

Short answer

279 \(\mathrm{ohm}^{-1} \mathrm{~cm}^{2} \mathrm{eq}^{-1}\)

Step-by-step solution

Understanding Ionic Conductivity

Ionic conductivity at infinite dilution represents the maximum conductivity of an ion when it is sufficiently apart from other ions so that there is no inter-ionic interaction affecting its mobility. The given ionic conductivities are for the ions \(\mathrm{Ba}^{2+}\) and \(\mathrm{Cl}^{-}\) at infinite dilution.

Calculate Equivalent Conductivity for \(\mathrm{BaCl}_{2}\)

The equivalent conductivity of a salt at infinite dilution can be found by summing the ionic conductivities of its constituent ions, considering their stoichiometric coefficients in the formula unit of the salt. In the case of \(\mathrm{BaCl}_{2}\), this means the equivalent conductivity \(\Lambda_{\infty}\) is calculated by adding the conductivity of one \(\mathrm{Ba}^{2+}\) ion to the conductivity of two \(\mathrm{Cl}^{-}\) ions.

Perform the Calculation

Using the provided conductivities, \(\Lambda_{\infty}(\mathrm{Ba}^{2+}) = 127 \(\mathrm{ohm}^{-1} \mathrm{~cm}^{2} \mathrm{eq}^{-1}\)\) and \(\Lambda_{\infty}(\mathrm{Cl}^{-}) = 76 \(\mathrm{ohm}^{-1} \mathrm{~cm}^{2} \mathrm{eq}^{-1}\)\), the calculation for \(\mathrm{BaCl}_{2}\) at infinite dilution is:\[\Lambda_{\infty}(\mathrm{BaCl}_{2}) = 127 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~eq}^{-1} + 2 \times 76 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~eq}^{-1} = 279 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~eq}^{-1}\].

Key concepts

Ionic Conductivity

Ionic conductivity is a measure of an ion's ability to conduct electricity in a solution. At infinite dilution, where each ion is isolated due to an extremely low concentration, ionic conductivity reaches a maximum value because there are no inter-ionic interactions to hinder the movement of the ions.

For an ion in solution, the ionic conductivity is influenced by factors such as the charge of the ion, the size of the ion, and the temperature of the solution. The unit of ionic conductivity commonly used is \(\text{ohm}^{-1} \text{cm}^{2} \text{eq}^{-1}\), which represents the conductance of one equivalent of an ion in a cube of solution with a length of one centimeter.

When we refer to the ionic conductivity of \(\text{Ba}^{2+}\) and \(\text{Cl}^{-}\), we discuss how well these ions can carry electrical charge when they are not influenced by other ions in the solution. This concept is critical in determining the equivalent conductivity of salts, such as \(\text{BaCl}_{2}\), at infinite dilution.

Stoichiometric Coefficients

Stoichiometric coefficients are numbers used to balance the ratio of atoms in a chemical reaction, reflecting the proportion of each substance involved. They are essential in converting between different measures of chemical quantities – like molecules, grams, and moles – and play a crucial role in the calculation of equivalent conductivity at infinite dilution.

In the context of \(\text{BaCl}_{2}\), the stoichiometric coefficient for \(\text{Ba}^{2+}\) is 1, and for \(\text{Cl}^{-}\) is 2, indicating that one barium ion pairs with two chloride ions to form the compound. When computing the equivalent conductivity of \(\text{BaCl}_{2}\), these coefficients must be taken into account since they determine how many units of each ion's conductivity should be included in the total.

For example, the total equivalent conductivity of \(\text{BaCl}_{2}\) at infinite dilution is the sum of the conductivity of one barium ion and twice the conductivity of a chloride ion, due to their stoichiometric ratios in the compound.

Inter-Ionic Interaction

Inter-ionic interaction refers to the effects ions have on each other's movement in a solution. When ions are concentrated, they interact more frequently, which can slow down their mobility and reduce conductivity. However, at infinite dilution, these interactions are minimized as each ion is effectively isolated.

The extent of inter-ionic interaction depends on factors such as the size of the ions, their charges, and the properties of the solvent. Strong interactions can significantly affect the ions’ ability to move freely, thereby influencing the ionic conductivity of the solution.

In electrochemistry, we take advantage of infinite dilution to measure the intrinsic conductivity of ions without interference from these interactions. When calculating the equivalent conductivity of a salt at infinite dilution, the assumption is made that the ions are sufficiently separated, thus inter-ionic interactions are negligible. This enables us to calculate a value that solely reflects the combined conductivities of the individual ions as they would behave in an ideal scenario.