Problem 122 - A Complete Resource Book in Chemistry for JEE Main

Question

The acid ionization (hydrolysis) constant of \(\mathrm{Zn}^{2+}\) is \(1.0\) \(\times 10^{-9} .\) Which of the following statements are correct? (i) the basic dissociation constant of \(\mathrm{Zn}(\mathrm{OH})^{+}\)is \(1.0 \times 10^{5}\) (ii) the \(\mathrm{pH}\) of \(0.001 \mathrm{M} \mathrm{ZnCl}_{2}\) solution is 6 (iii) the basic dissociation constant of \(\mathrm{Zn}(\mathrm{OH})^{+}\)is \(1.0 \times 10^{-5}\) (iv) the \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\)ion concentration in \(0.001 \mathrm{M} \mathrm{ZnCl}_{2}\) solution is \(1.0 \times 10^{-4}\). (a) 1,2 (b) 2,3 (c) 3,4 (d) \(1,2,4\)

Step-by-Step Solution

Answer

Statements 2 and 3 are correct, so option (b) is correct.

Step 1: Understanding Acid and Base Constants

The acid ionization constant (Ka) is given as \(1.0 \times 10^{-9}\). For the basic dissociation constant (Kb) of \(\mathrm{Zn(OH)}^{+}\), we use the relationship in water where \(K_w = 10^{-14}\): \(K_w = K_a \times K_b\). This means \(K_b = \frac{K_w}{K_a} = \frac{10^{-14}}{1.0 \times 10^{-9}} = 10^{-5}\). Therefore, statement (iii) is correct.

Step 2: Determine pH of the Solution

The concentration of \(\mathrm{ZnCl}_2\) is given as \(0.001 \text{ M}\). \(\mathrm{Zn}^{2+}\) ions hydrolyze and affect the pH. Normally, in the absence of other hydrolyzing salts, a neutral \(0.001 \text{ M}\) solution (due only to water's equilibrium) would show a pH of approximately 7. Given the presence of \(Zn^{2+}\), a slight acidity might be expected. Thus, the pH of exactly 6 is likely reasonable without detailed calculations, hinting statement (ii) is correct.

Step 3: Calculation of Hydronium Ion Concentration

To verify statement (iv), check if \([\mathrm{H}_3\mathrm{O}^+]\) ion concentration for a solution with pH 6 is \(1.0 \times 10^{-4}\) M. Since pH = 6 implies \([\mathrm{H}_3\mathrm{O}^+] = 10^{-6}\) M, statement (iv) involves a discrepancy if assuming normal conditions, hence statement (iv) appears incorrect.

Step 4: Evaluate Each Statement Against Conditions

We've found statements (ii) and (iii) correct based on reasonable assumptions. Hence, options (b) 2,3 and (c) 3,4 are narrowed down. Since (iv) was disproven earlier, option (c) can be eliminated, confirming option (b) is fully consistent with the correct statements.

Key Concepts

Understanding the Basic Dissociation Constant

The basic dissociation constant (\( K_b \) represents the strength of a base. Essentially, it measures how well a base dissociates into its ions in water.
This is very useful when dealing with weak bases, like \( \mathrm{Zn(OH)}^{+} \).

• For any base, \( K_b \) can be calculated using the relationship with the acid ionization constant (\( K_a \)):
  \[ K_w = K_a \times K_b \]
• Here, \( K_w \) is the ion product constant of water, always \( 1.0 \times 10^{-14} \) at room temperature.

In this specific case, given the acid ionization constant is \( 1.0 \times 10^{-9} \), we find \( K_b \) for \( \mathrm{Zn(OH)}^{+} \) by rearranging the equation: \[ K_b = \frac{K_w}{K_a} = \frac{10^{-14}}{1.0 \times 10^{-9}} = 10^{-5} \] Thus, the basic dissociation constant \( K_b \) indicates the strength of the Zn(OH)+ ion as a base in water.

Exploring pH Calculations

pH is a measure of how acidic or basic a solution is. It's important for understanding the behavior of solutions in chemistry.
The formula for pH is based on the concentration of hydrogen ions in the solution:\[ \text{pH} = -\log_{10} [\mathrm{H}^+] \]

• A low pH (below 7) indicates an acidic solution.
• A higher pH (above 7) suggests a basic solution.
• A pH of 7 is neutral, typical for pure water under standard conditions.

In this exercise, we are dealing with a \(0.001\, \text{M} \mathrm{ZnCl}_2\) solution. \( \mathrm{Zn}^{2+} \) ions cause a slight acidity altering the solution's pH to 6. This suggests a bit more acidity than pure water.

Determining Hydronium Ion Concentration

Hydronium ion concentration is crucial in determining a solution's acidity. It's directly linked to the pH of a solution.
For a solution with pH 6, the calculation follows this pattern to find the hydronium ion concentration:

• Since \( \text{pH} = -\log_{10} [\mathrm{H}_3\mathrm{O}^+] \), solving for [\mathrm{H}_3\mathrm{O}^+] yields:

\[ [\mathrm{H}_3\mathrm{O}^+] = 10^{-\text{pH}} = 10^{-6} \] This gives us a hydronium ion concentration of \( 1.0 \times 10^{-6} \) M for the solution with pH 6. This discrepancy challenges the initial statement assumption about hydronium ion concentration, leading us to reassess it.