Question

The equivalent conductance of \(0.01 \mathrm{M}\) solution of weak acid \(\mathrm{HA}\) is \(19.6 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{eq}^{-1}\). The equivalent conductance of the electrolyte at infinite dilution 392 then \(\mathrm{pH}\) of the solution will be: (1) \(2.3\) (2) 5 (3) \(3.3\) (4) \(4.7\)

Short answer

The pH of the solution is 3.3.

Step-by-step solution

Write down known values

Given data: 1. Concentration \( C \) = 0.01 M 2. Equivalent conductance at concentration C \( \Lambda \) = 19.6 S cm\textsuperscript{2} eq\textsuperscript{-1}3. Equivalent conductance at infinite dilution \( \Lambda \textsuperscript{0} \) = 392 S cm\textsuperscript{2} eq\textsuperscript{-1}

Calculate degree of ionization

The degree of ionization \( \alpha \) is given by the ratio of the equivalent conductance at a given concentration to the equivalent conductance at infinite dilution. \[ \alpha = \frac{ \Lambda }{ \textrm { \Lambda \textsuperscript{0} } } \] Substitute the given values: \[ \alpha = \frac{ 19.6 }{ 392 } = 0.05 \]

Calculate concentration of hydrogen ions

Degree of ionization is related to the concentration of the ions formed. For a weak acid HA that partially ionizes: \[ \mathrm{HA} \rightleftharpoons \mathrm{H\textsuperscript{+}} + \mathrm{A\textsuperscript{-}} \] The concentration of hydrogen ions \( \left[ \mathrm{H\textsuperscript{+}} \right] = \alpha \cdot C \) Using \( \alpha = 0.05 \) and \( C = 0.01 \mathrm{M} \): \[ \left[\mathrm{H\textsuperscript{+}}\right] = 0.05 \times 0.01 = 0.0005\mathrm{M} \]

Calculate the pH of the solution

The pH is defined as the negative logarithm of the hydrogen ion concentration: \[ \mathrm{pH} = - \log \left[ \mathrm{H\textsuperscript{+}} \right] \] Substitute \( \left[ \mathrm{H\textsuperscript{+}} \right]=0.0005 \): \[ \mathrm{pH} = - \log \left( 5 \times 10\textsuperscript{-4} \right) \] \[ \mathrm{pH} = - \left( \log 5 + \log 10\textsuperscript{-4} \right) = - \log (5) - (-4) = 4 - 0.3 = 3.3 \]

Key concepts

Weak Acid Ionization

When talking about weak acids, it's crucial to understand that these acids do not completely dissociate in water. In a solution, only a small fraction of the weak acid molecules ionize to produce hydrogen ions (H⁺) and their corresponding anions. Consider the weak acid HA, which partially ionizes according to the equation:
\[ \text{HA} \rightleftharpoons \text{H}^+ + \text{A}^- \] Because weak acids only partially ionize, their degree of ionization (\(\alpha\)) is much less than 1.
The degree of ionization can be expressed as the ratio of the equivalent conductance at a given concentration (\(\Lambda\)) to the equivalent conductance at infinite dilution (\(\Lambda^0\)), which is calculated using:
\[ \alpha = \frac{\Lambda}{\Lambda^0} \] Here, equivalent conductance at infinite dilution represents the conductance when the solution is diluted enough such that ion interactions are minimal.
By understanding this, we can see that weaker acids have lower conductance values due to fewer available ions.

pH Calculation

The pH of a solution is a measure of its acidity, specifically the concentration of hydrogen ions (H⁺) present. For any solution, pH is defined mathematically as:
\[ \text{pH} = -\log[\text{H}^+] \] In this case, the concentration of H⁺ ions can be determined by multiplying the degree of ionization (\(\alpha\)) by the given concentration of the weak acid (\(C\)):
\[ [\text{H}^+] = \alpha \times C \] Substituting the calculated values with \(\alpha = 0.05\) and \(C = 0.01\text{ M}\), we get:
\[ [\text{H}^+] = 0.05 \times 0.01 = 0.0005 \text{ M} \] Taking the negative logarithm of this concentration, we determine:
\[ \text{pH} = -\log(0.0005) = -\log(5 \times 10^{-4}) \] Breaking it down further:
\[ \text{pH} = - (\log(5) + \log(10^{-4})) = -\log(5) - (-4) \]
Since \(\log(5) \approx 0.3\):
\[ \text{pH} = 4 - 0.3 = 3.7 \] This shows how logarithmic functions are used in pH calculations.

Degree of Ionization

The degree of ionization (\(\alpha\)) of a weak acid indicates the fraction of acid molecules that dissociate into ions in solution. It plays a vital role in understanding the behavior of weak acids and is given by the ratio:
\[ \alpha = \frac{\Lambda}{\Lambda^0} \] where:
- \(\Lambda\) is the equivalent conductance at a given concentration
- \(\Lambda^0\) is the equivalent conductance at infinite dilution
Using these values, we can infer the degree of ionization. For example, with \(\Lambda = 19.6\ \text{S}\ \text{cm}^2 \text{eq}^{-1}\) and \(\Lambda^0 = 392\ \text{S}\ \text{cm}^2 \text{eq}^{-1}\), the calculation is:
\[ \alpha = \frac{19.6}{392} = 0.05 \] This means only 5% of the acid molecules ionize in the solution.
This concept is crucial as it ties directly into calculating other key properties, such as the concentration of hydrogen ions (H⁺) and ultimately the pH of the solution.