Question

The ionic mobilities of \(\mathrm{K}^{+}\) and \(\left[\mathrm{MnO}_{4}\right]^{-}\) at \(298 \mathrm{K}\) are \(7.62 \times 10^{-8}\) and \(6.35 \times 10^{-8} \mathrm{m}^{2} \mathrm{s}^{-1} \mathrm{V}^{-1}\) Determine \(A_{\mathrm{m}}^{\infty}(298 \mathrm{K})\) for \(\mathrm{KMnO}_{4}\)

Short answer

The molar conductivity \\( A_{m}^{\infty} \\) for \\( \mathrm{KMnO}_{4} \\) is approximately \\( 13.48 \, \mathrm{mS} \, \mathrm{m}^{2} \, \mathrm{mol}^{-1} \\).

Step-by-step solution

Understand Ionic Mobility

Ionic mobility is a measure of how fast an ion moves through a solution under the influence of an electric field. It is given for the ions \( \mathrm{K}^{+} \) and \( \left[ \mathrm{MnO}_{4} \right]^{-} \) as \( 7.62 \times 10^{-8} \, \mathrm{m}^{2} \, \mathrm{V}^{-1} \, \mathrm{s}^{-1} \) and \( 6.35 \times 10^{-8} \, \mathrm{m}^{2} \, \mathrm{V}^{-1} \, \mathrm{s}^{-1} \,\) respectively.

Formula for Molar Conductivity at Infinite Dilution

The molar conductivity at infinite dilution, \( A_{m}^{\infty} \,\) is calculated by adding the individual ionic mobilities multiplied by the Faraday constant \( F \). The formula is: \[ A_{m}^{\infty} = F \times (u_{\mathrm{K}^{+}} + u_{\left[\mathrm{MnO}_{4}\right]^{-}}) \] where \( u_{\mathrm{K}^{+}} \) and \( u_{\left[\mathrm{MnO}_{4}\right]^{-}} \) are the ionic mobilities.

Use Faraday Constant

The Faraday constant, \( F \,\) is approximately \( 96485 \mathrm{C} \, \mathrm{mol}^{-1} \,\) which is used in the formula to determine molar conductivity.

Calculate Molar Conductivity

Substitute the given values for ionic mobilities into the formula: \[ A_{m}^{\infty} = 96485 \times (7.62 \times 10^{-8} + 6.35 \times 10^{-8}) \]Calculate the sum of the ionic mobilities: \[ 1.397 \times 10^{-7} \, \mathrm{m}^{2} \, \mathrm{V}^{-1} \, \mathrm{s}^{-1} \]Insert this into the expression for \( A_{m}^{\infty} \,\) : \[ A_{m}^{\infty} = 96485 \times 1.397 \times 10^{-7} \]Calculate to find \( A_{m}^{\infty} \).

Simplify and Solve

After performing the calculation, \( A_{m}^{\infty} \) becomes approximately \( 13.48 \, \mathrm{mS} \, \mathrm{m}^{2} \, \mathrm{mol}^{-1} \,\) under the given conditions.

Key concepts

Molar Conductivity

Molar conductivity is a crucial concept in understanding how ions move in a solution. It measures how well an ion conducts electricity when dissolved in a solvent at a specific concentration. As the concentration of a solution approaches zero, the molar conductivity approaches a limit called the molar conductivity at infinite dilution, denoted by \( A_\mathrm{m}^\infty \). This condition assumes that ions do not interfere with each other and can move freely.

In practice, molar conductivity can help predict the behavior of ions in different conditions and is critical in processes like electrolysis. Mathematically, we calculate it using the formula:

• \( A_\mathrm{m}^\infty = F \times (u_{\mathrm{K}^+} + u_{\left[\mathrm{MnO}_4\right]^-}) \)

Where \( u_{\mathrm{K}^+} \) and \( u_{\left[\mathrm{MnO}_4\right]^-} \) are the ionic mobilities. This formula uses the Faraday constant to link the mobility of ions with their conductivity.

For any substance, understanding its molar conductivity can provide insights into its ionic behavior and guide practical applications in chemical and industrial processes.

Faraday Constant

The Faraday constant, symbolized as \( F \), is a fundamental constant in chemistry and physics. It's especially important in the study of electrochemistry. The value is approximately \( 96485 \, \mathrm{C} \, \mathrm{mol}^{-1} \), representing the charge of one mole of electrons. This constant allows us to relate the movement of electrons to measurable amounts, such as currents, in chemical reactions.

When examining ionic conductivity, the Faraday constant becomes vital. It helps translate the movement of ions, which carry charges, into quantitative measurements that can predict the efficiency of reactions. It's used in calculating molar conductivity by forming part of the equation that connects ionic mobility and total conductivity.

• For instance, in the formula \( A_\mathrm{m}^\infty = F \times (u_{\mathrm{K}^+} + u_{\left[\mathrm{MnO}_4\right]^-}) \), the constant helps compute the practical conductivity from theoretical ionic mobilities.

Through this calculation, the Faraday constant bridges the gap between theoretical chemistry concepts and real-world applications.

Ionic Mobility Measurement

Ionic mobility refers to the speed at which an ion moves through a solution when an electric field is applied. It depends on factors like the ion's charge, size, the viscosity of the solvent, and temperature. The units for ionic mobility are \( \mathrm{m}^2 \, \mathrm{V}^{-1} \, \mathrm{s}^{-1} \), reflecting the ion's movement per volt and per second.

Measuring ionic mobility is crucial for understanding and calculating the conductivity of solutions. The measurements are typically made under an electric field to observe how quickly specific ions, such as \( \mathrm{K}^+ \) or \( \left[\mathrm{MnO}_4\right]^- \), move. Given their mobilities in the original exercise, we can see that \( \mathrm{K}^+ \) ions move faster than \( \left[\mathrm{MnO}_4\right]^- \) ions under the same conditions.

• This difference in mobility can impact the overall conductivity and efficiency of electrochemical reactions, which is why precise measurements and understanding are key in fields like battery design and water purification.

In essence, ionic mobility measurement provides a detailed look at the behavior of ions in motion, offering valuable data for both scientific inquiry and practical application.