Question

The following observations are made about a diprotic acid \(\mathrm{H}_{2} \mathrm{~A}\) : (i) \(\mathrm{A} 0.10 \mathrm{M}\) solution of \(\mathrm{H}_{2} \mathrm{~A}\) has \(\mathrm{pH}=3.30\). (ii) \(\mathrm{A} 0.10 \mathrm{M}\) solution of the salt NaHA is acidic. Which of the following could be the value of \(\mathrm{p} K_{a 2}\) for \(\mathrm{H}_{2} \mathrm{~A}\) : (i) \(3.22\), (ii) \(5.30\), (iii) 7.47, or (iv) \(9.82 ?\)

Short answer

Any of the provided pKa₂ values could be the correct one for H₂A: (i) 3.22 (ii) 5.30 (iii) 7.47 (iv) 9.82

Step-by-step solution

Determine the concentration of H⁺ ions in the H₂A solution

From the first piece of information, we know that the pH of the 0.10 M H₂A solution is 3.30. We can use the pH formula to determine the concentration of H⁺ ions: pH = -log[H⁺] Rearranging the formula, we get: [H⁺] = 10^(-pH) Plugging in the given pH value: [H⁺] = 10^(-3.3) = 5.0 x 10^(-4) M

Calculate the Ka₁ value from the H₂A solution

Let's denote the change in concentration of H⁺ ions due to the dissociation of H₂A as x. As H₂A is a strong acid, we can assume that [H⁺]=[A⁻]=[x]. Since the initial concentration of H₂A is 0.1 M, the equilibrium expression for Ka₁ can be written as: Ka₁ = [H⁺][A⁻]/[H₂A] = (x)(x)/(0.1 - x) Since [H⁺] = 5.0 x 10^(-4) = x, we can simplify the equation: Ka₁ = (5.0 x 10^(-4))^2 / (0.1 - 5.0 x 10^(-4)) = 2.50 x 10^(-4) Taking the negative log of Ka₁, we can find the pKa₁ value: pKa₁ = -log(Ka₁) = -log(2.50 x 10^(-4)) = 3.60

Determine the Ka₂ value from the given pKa₂ values

Calculate the pKa₁ values for each given pKa₂ value using the relationship: pKa₁ + pKa₂ = pKw (value of pKw is 14) Option (i): pKa₁ = 14 - 3.22 = 10.78 Option (ii): pKa₁ = 14 - 5.30 = 8.70 Option (iii): pKa₁ = 14 - 7.47 = 6.53 Option (iv): pKa₁ = 14 - 9.82 = 4.18

Check which pKa₂ value correctly describes the acidity of the NaHA solution

A 0.10 M solution of the salt NaHA is acidic. An acidic solution occurs when the pKa₁ value for the given pKa₂ is larger than the actual pKa₁ of H₂A. We calculated the pKa₁ of H₂A to be 3.60 in step 2. Now, let's compare this value to the pKa₁ values calculated for each given pKa₂: Option (i): pKa₁ = 10.78 (> 3.60) Option (ii): pKa₁ = 8.70 (> 3.60) Option (iii): pKa₁ = 6.53 (> 3.60) Option (iv): pKa₁ = 4.18 (> 3.60) Since all given pKa₂ values result in pKa₁ values that are larger than the actual pKa₁ of H₂A, each of them would produce an acidic NaHA solution, as required by the information. Therefore, any of the provided pKa₂ values could be the correct one for H₂A: (i) 3.22 (ii) 5.30 (iii) 7.47 (iv) 9.82

Key concepts

pH Calculation

Understanding the pH of a solution is crucial because it indicates how acidic or basic that solution is. The pH is calculated using the formula:
\[ pH = -\log[H^+] \]
In the context of a diprotic acid \( H_{2}A \), determining the pH involves calculating the concentration of hydrogen ions \( [H^+] \) in the solution. From the exercise, we can derive that for a 0.10 M \( H_{2}A \) solution with a pH of 3.30, the concentration of hydrogen ions is \( 5.0 \times 10^{-4} \) M. This is found by taking the inverse log of the pH value:
\[ [H^+] = 10^{-pH} \]
This process is essential for understanding the solution's acidity and is a foundation for analyzing acid dissociation and the related constants.

Acid Dissociation Constant

The acid dissociation constant (Ka) is a quantitative measure of the strength of an acid in solution. It expresses the equilibrium between the acid (HA) and its ions (H⁺ and A⁻). For a diprotic acid like \( H_{2}A \) with two dissociable protons, there are two constants: Ka₁ and Ka₂ for the first and second dissociation steps, respectively. The formula to calculate Ka is:
\[ Ka = \frac{[H^+][A^-]}{[HA]} \]
During the first dissociation of a diprotic acid, Ka₁ can be determined by the concentration of hydrogen ions squared, divided by the initial concentration of the acid minus the ion concentration, as indicated in the exercise solution. These constants help chemists understand how the acid will behave in different environments.

Ka and pKa Relationship

The relationship between Ka and pKa is another fundamental aspect of acid-base chemistry. The pKa value is simply the negative logarithm of the Ka value, which provides a more convenient way to express acid strength. The formula to find pKa is:
\[ pKa = -\log(Ka) \]
As seen in the exercise, once the Ka₁ value is calculated, the corresponding pKa₁ can be determined. As the magnitude of Ka decreases, indicating a weaker acid, the pKa value increases. Knowing the pKa helps predict the extent of dissociation of an acid at various pH levels, which is vital for controlling reactions in a laboratory or industrial setting.

Acidity of Salt Solutions

The acidity of salt solutions is determined by the nature of the ions in the solution. Salts derived from a weak acid and a strong base, like NaHA, can often form solutions with acidic pH because the conjugate base (A⁻) from the weak acid has the ability to accept a proton from water and generate additional hydrogen ions. This effect is related to the hydrolysis of the anion.

In the provided problem, we infer that since solutions of NaHA are acidic, Ka₂ must be significantly smaller than Kw, the ion product constant for water, and the related pKa₂ must be greater than 7 to result in a pH less than 7. By extension, since pKa₁ is also fairly small, the acid's second dissociation step must be weaker than the first, as the given pKa₂ options also imply.