Problem 117 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The molar conductance of a \(0.01 \mathrm{M}\) solution of acetic acid was found to be \(16.30 \Omega^{-1} \mathrm{~cm}^{-1} \mathrm{~mol}^{-1}\) at \(25^{\circ} \mathrm{C}\). The ionic conductances of hydrogen and acetate ions at infinite dilution are \(349.8\) and \(40.9 \Omega^{-1}\) \(\mathrm{cm}^{-1} \mathrm{~mol}^{-1}\), respectively, at the same temperature. What percentage of acetic acid is dissociated at this concentration? (a) \(0.04172 \%\) (b) \(4.172 \%\) (c) \(41.72 \%\) (d) \(0.4172 \%\)

Step-by-Step Solution

Answer

4.172%

Step 1 - Calculate Limiting Molar Conductivity

Combine the ionic conductances of the hydrogen and acetate ions to find the limiting molar conductivity (\r\(\Lambda_m^\circ\)) of acetic acid. \r\[\Lambda_m^\circ = \Lambda_{+}^\circ + \Lambda_{-}^\circ\] \rFor hydrogen ion (\r\(\Lambda_{+}^\circ\)), it is \r\(349.8\ \r\Omega^{-1} \r\mathrm{cm}^{-1} \r\mathrm{mol}^{-1}\), and for acetate ion (\r\(\Lambda_{-}^\circ\)), it is \r\(40.9 \ \r\Omega^{-1} \r\mathrm{cm}^{-1} \r\mathrm{mol}^{-1}\).

Step 2 - Calculate Degree of Dissociation

Using the molar conductance of the acetic acid solution (\r\(\Lambda_m\)) and the limiting molar conductivity (\r\(\Lambda_m^\circ\)), calculate the degree of dissociation (\r\(\alpha\)) of acetic acid. \r\[\alpha = \frac{\Lambda_m}{\Lambda_m^\circ}\] \rThe molar conductance \r\(\Lambda_m\) is given as \r\(16.30\ \r\Omega^{-1} \r\mathrm{cm}^{-1} \r\mathrm{mol}^{-1}\).

Step 3 - Calculate Percentage Dissociation

To find the percentage of dissociation, multiply the degree of dissociation (\r\(\alpha\)) by 100. \r\[\text{Percentage of dissociation} = \alpha \times 100\%\]

Key Concepts

Degree of Dissociation

The concept of the degree of dissociation is central in understanding the behavior of electrolytes in a solution. It denotes the fraction of solute molecules that split into constituent ions. For weak electrolytes like acetic acid, not all molecules dissociate, leading to a value less than 1 for the degree of dissociation, symbolized by the Greek letter \( \alpha \). Mathematically, it's calculated as the ratio of the molar conductance of the electrolyte at a given concentration (\( \Lambda_m \)) to the limiting molar conductivity (\( \Lambda_m^\circ \)). The formula used is:
\[ \alpha = \frac{\Lambda_m}{\Lambda_m^\circ} \]
In simpler terms, by understanding the degree of dissociation, we get insights into the extent to which a substance dissociates into ions, which is a measure of its strength as an electrolyte. This value is crucial for various applications in chemistry, including determining reaction mechanisms, calculating equilibrium constants, and in the field of electrochemistry.

Limiting Molar Conductivity

Limiting molar conductivity, represented as \( \Lambda_m^\circ \), is an essential parameter in physical chemistry that reveals the conductive capacity of an ion when the concentration approaches zero and inter-ionic interactions are negligible. This theoretical value is significant because it allows chemists to compare the ionic conductivities of different electrolytes under standard conditions.
It's determined by extrapolating the plot of molar conductivity versus the square root of concentration to zero concentration. However, for ions, \( \Lambda_m^\circ \) is the sum of the conductivities of the individual ions at infinite dilution. The equation used to find the limiting molar conductivity of acetic acid is:
\[ \Lambda_m^\circ = \Lambda_{+}^\circ + \Lambda_{-}^\circ \]
The value of \( \Lambda_m^\circ \) provides a baseline for comparison and allows for the calculation of the degree of dissociation of an electrolyte when combined with the measured molar conductance.

Ionic Conductance

Ionic conductance refers to the ability of an ion to carry an electric charge through a solution. Each ion in solution has its unique conductance value, often measured at infinite dilution to ensure no interaction between ions and a stable, consistent measurement.
In the context of our exercise, we have the ionic conductances of hydrogen (\( \Lambda_{+}^\circ \)) and acetate (\( \Lambda_{-}^\circ \)) ions, given as \(349.8 \Omega^{-1} \mathrm{cm}^{-1} \mathrm{mol}^{-1}\) and \(40.9 \Omega^{-1} \mathrm{cm}^{-1} \mathrm{mol}^{-1}\), respectively. These values highlight the conductivity if each of these ions were present alone at infinite dilution. However, in an acetic acid solution, both ions contribute to overall conductivity in accordance with their mole fraction and inherent conductance.