Question

The ionic mobilities of \(\mathrm{Sr}^{2+}\) and \(\mathrm{Cl}^{-}\) at \(298 \mathrm{K}\) are \(6.16 \times 10^{-8}\) and \(7.91 \times 10^{-8} \mathrm{m}^{2} \mathrm{s}^{-1} \mathrm{V}^{-1}.\) Determine \(\Lambda_{\mathrm{m}}^{\infty}(298 \mathrm{K})\) for \(\mathrm{SrCl}_{2}.\)

Short answer

The molar conductivity, \( \Lambda_{\mathrm{m}}^{\infty} \), for \( \mathrm{SrCl}_2 \) at 298 K is approximately \( 2.713 \times 10^{-2} \ \mathrm{S \ m}^2 \ \mathrm{mol}^{-1} \).

Step-by-step solution

Understanding Ionic Mobility

The ionic mobility, denoted by \( u \), refers to the speed at which an ion moves through a solution under the influence of an electric field. It is measured in \( \mathrm{m}^2 \ \mathrm{s}^{-1} \ \mathrm{V}^{-1} \). The given mobilities for \( \mathrm{Sr}^{2+} \) and \( \mathrm{Cl}^{-} \) are \( 6.16 \times 10^{-8} \) and \( 7.91 \times 10^{-8} \) respectively.

Molar Conductivity at Infinite Dilution

The molar conductivity at infinite dilution, \( \Lambda_{\mathrm{m}}^{\infty} \), is the sum of the individual ionic conductivities at infinite dilution. It is computed using the equation:\[\Lambda_{\mathrm{m}}^{\infty} = u_{+} \lambda_{+}^{\infty} + u_{-} \lambda_{-}^{\infty} \]where \( u_{+} \) and \( u_{-} \) are the stoichiometric coefficients of the cation and anion respectively, and \( \lambda_{+}^{\infty} \) and \( \lambda_{-}^{\infty} \) are the ionic conductivities at infinite dilution, calculated as \( F \cdot u \).

Calculate Ionic Conductivities

Calculate the ionic conductivities using Faraday's constant \( F \) (\( 96485 \ \mathrm{C \ mol}^{-1} \)) and the given mobilities:\[ \lambda_{\mathrm{Sr}^{2+}}^{\infty} = z_{\mathrm{Sr}^{2+}} \cdot F \cdot u_{\mathrm{Sr}^{2+}} = 2 \times 96485 \times 6.16 \times 10^{-8} \]\[ \lambda_{\mathrm{Cl}^{-}}^{\infty} = z_{\mathrm{Cl}^{-}} \cdot F \cdot u_{\mathrm{Cl}^{-}} = 1 \times 96485 \times 7.91 \times 10^{-8} \]Calculate to get:- \( \lambda_{\mathrm{Sr}^{2+}}^{\infty} \approx 1.188 \times 10^{-2} \ \mathrm{S \ m}^2 \ \mathrm{mol}^{-1} \)- \( \lambda_{\mathrm{Cl}^{-}}^{\infty} \approx 7.631 \times 10^{-3} \ \mathrm{S \ m}^2 \ \mathrm{mol}^{-1} \)

Calculate Molar Conductivity

For \( \mathrm{SrCl}_2 \), \( u_{\mathrm{Sr}^{2+}}=1 \) and \( u_{\mathrm{Cl}^{-}}=2 \). Calculate \( \Lambda_{\mathrm{m}}^{\infty} \) using:\[ \Lambda_{\mathrm{m}}^{\infty} = 1 \cdot \lambda_{\mathrm{Sr}^{2+}}^{\infty} + 2 \cdot \lambda_{\mathrm{Cl}^{-}}^{\infty} \]Substituting the values:\[ \Lambda_{\mathrm{m}}^{\infty} = 1.188 \times 10^{-2} + 2 \times 7.631 \times 10^{-3} \]\[ \Lambda_{\mathrm{m}}^{\infty} \approx 2.713 \times 10^{-2} \ \mathrm{S \ m}^2 \ \mathrm{mol}^{-1} \]

Final Result

The molar conductivity at infinite dilution of \( \mathrm{SrCl}_2 \) at \( 298 \mathrm{K} \) is approximately \( 2.713 \times 10^{-2} \ \mathrm{S \ m}^2 \ \mathrm{mol}^{-1} \).

Key concepts

Molar Conductivity

Molar conductivity is an essential concept in electrochemistry, reflecting how well an electrolyte can conduct electricity when dissolved in a solvent, such as water. It is denoted by \( \Lambda_m \) and measures the conductance of a solution containing one mole of electrolyte, brought to infinite dilution. At infinite dilution, ions do not interact with each other, allowing for maximal conductivity. This property becomes helpful in comparing the efficiency of different electrolytes.

To calculate molar conductivity, especially at infinite dilution \( \Lambda_{m}^{\infty} \), ionic mobilities are a key consideration. Each ion type contributes to the total conductivity. Thus, the individual contributions of cations and anions (positive and negative ions) combine to form the overall molar conductivity at infinity. Recognizing this relationship helps in understanding the interaction of ions in a solution and their role in electricity conduction.

Ionic Conductivity

Ionic conductivity examines how well ions can carry an electric current through a solution. It is a crucial aspect of electrolytic solutions. Individual ions have intrinsic ionic conductivities, denoted \( \lambda \). At infinite dilution, the ionic conductivity refers to the ion's ability to conduct electricity when no ion-ion interaction occurs.

This value is determined by the product of the ion's charge \( z \), its mobility \( u \), and Faraday's constant (\( F \), approximately \( 96485 \ \mathrm{C \ mol}^{-1} \)). The formula for a singular ionic conductivity is:

• \( \lambda^{\infty} = z \cdot F \cdot u \)

At infinite dilution, each ion contributes independently to the overall conductivity, making individual ionic conductivities beneficial in predicting how substances in solutions behave under an electric field. Understanding ionic conductivity deepens one's insight into the movement and impact of ions within a solution.

Faraday's Constant

Faraday's constant is a fundamental number in electrochemistry, representing the electric charge of one mole of electrons. It is approximately \( 96485 \ \mathrm{C \ mol}^{-1} \). This constant is pivotal when dealing with calculations involving charge transfer in electrochemical reactions.

In the context of ionic mobility and conductivity, Faraday's constant links the mobility of ions to their ability to conduct electricity. It helps convert mobility (the speed of ions under an electric field) to conductivity. This conversion is crucial in equations that measure how effectively ions transfer charge in solutions.

Faraday's constant not only defines the charge carried by a mole of electrons but also serves as an integral measure when assessing how ions transport electrical energy through solutions. Thus, understanding Faraday's constant enriches comprehension of energy transfer processes in electro-chemical systems.