Problem 60 - Chemistry The Science in Context

Question

Calculate the \(\mathrm{pH}\) of a \(1.00 \times 10^{-3} M\) solution of cysteine \(\left(\mathrm{p} K_{\mathrm{a}_{1}}=1.7, \mathrm{p} K_{22}=8.3\right) .\) Is selenocysteine a stronger acid than cysteine?

Step-by-Step Solution

Answer

Answer: The pH of the given cysteine solution is 2.7. We cannot determine if selenocysteine is a stronger acid than cysteine based on the provided information as the pKa values for selenocysteine are not given.

Step 1: Identify the species involved in the reaction and write the expressions for the pKa1 and pKa2 equilibria

In this case, the cysteine has two acidic groups with pKa values of 1.7 and 8.3. The dissociation reactions for cysteine can be represented as follows: For the first ionization: \(H_{2}Cys \rightleftharpoons HCys^{-} + H^{+}\) with a pKa1 of 1.7 For the second ionization: \(HCys^{-} \rightleftharpoons Cys^{2-} + H^{+}\) with a pKa2 of 8.3

Step 2: Calculate the fraction of each species at equilibrium

To determine the fractions of each species at equilibrium, we will make use of the Henderson-Hasselbalch equation: \(pH = pKa + \log_{10} \left(\frac{[A^-]}{[HA]}\right)\) Using pKa1 (1.7) to determine the fraction of \(H_{2}Cys\) and \(HCys^{-}\): \(-1.7 = \log_{10} \left(\frac{[HCys^{-}]}{[H_{2}Cys]}\right)\) \(log_{10} \left(\frac{[HCys^{-}]}{[H_{2}Cys]}\right) = -1.7\) \(\frac{[HCys^{-}]}{[H_{2}Cys]} = 10^{-1.7}\) Using pKa2 (8.3) to determine the fraction of \(HCys^{-}\) and \(Cys^{2-}\): \(-8.3 = \log_{10} \left(\frac{[Cys^{2-}]}{[HCys^{-}]}\right)\) \(\frac{[Cys^{2-}]}{[HCys^{-}]} = 10^{-8.3}\)

Step 3: Calculate the concentration of each species

We are given the initial concentration of cysteine (\(1.00 \times 10^{-3} M\)). Then, using the equilibrium fractions calculated in Step 2, we can find each species' concentration: \([H_{2}Cys] = [1.00 \times 10^{-3}] (1 - 10^{-1.7})\) \([HCys^{-}] = [1.00 \times 10^{-3}] 10^{-1.7}\) \([Cys^{2-}] = [1.00 \times 10^{-3}] 10^{-1.7} 10^{-8.3}\)

Step 4: Calculate the pH of the solution

Since the \(HCys^{-}\) form is the most dominant of the acidic species, we will use its concentration to calculate the pH of the solution. \(pH = -\log_{10}([H^{+}]) = -\log_{10}([HCys^{-}]) = -\log_{10}([1.00 \times 10^{-3}]10^{-1.7})\) Calculating the pH, we get: \(pH = 2.7\)

Step 5: Determine if selenocysteine is a stronger acid than cysteine

Comparing the acidic strength between two acids depends on their pKa values. The acid with the lower pKa value is considered a stronger acid. In this exercise, we are not given the pKa values for selenocysteine. Therefore, it's not possible to determine if selenocysteine is a stronger acid than cysteine based on the available information. In conclusion, the pH of the given cysteine solution is 2.7. As we do not have the pKa values for selenocysteine, we cannot determine if it is a stronger acid than cysteine.

Key Concepts

pH Calculation

Calculating the pH of a solution is a fundamental skill in acid-base chemistry. In this exercise, we deal with cysteine, an amino acid with two acidic groups, each having its own dissociation constant (pKa value). The pH of a solution essentially tells us how acidic or basic the solution is. For acidic solutions, the pH is less than 7.
To calculate the pH, we generally use the formula:

• \( pH = -\log_{10}([H^+]) \)

This formula is applied after determining the concentration of hydrogen ions \([H^+]\) in the solution. In this exercise, since cysteine has two ionizable groups, we begin by understanding the dissociation process and calculating the equilibrium concentrations of various ionic forms, such as \( H_2Cys \), \( HCys^- \), and \( Cys^{2-} \). Once we know the dominant species responsible for contributing \([H^+]\), we use its concentration to calculate the pH.
For example, after performing an analysis using the Henderson-Hasselbalch equation (discussed later), it was found that \( HCys^- \) significantly affects the pH in this scenario. Once the concentration of \( HCys^- \) is known, we plug it into the pH formula to find that the pH equals 2.7, suggesting the solution is acidic.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a crucial tool in acid-base chemistry for calculating the pH of buffer solutions. It relates the pH, pKa, and the ratio of the concentrations of the deprotonated form \([A^-]\) and the protonated form \([HA]\) of the acid:

• \( pH = pKa + \log_{10} \left(\frac{[A^-]}{[HA]}\right) \)

This equation derived from the acid dissociation equilibrium expression, simplifies the calculation by applying the logarithmic form.
In this exercise, the Henderson-Hasselbalch equation is used to calculate the concentrations of different ionic forms at equilibrium. By substituting the pKa values for the dissociation steps of cysteine, and knowing the total concentration, it helps in determining the fractions of ionized species. For instance, using the given pKa values (1.7 for \(H_2Cys\) to \(HCys^-\)), we can find the equilibrium ratio of \([HCys^-]\) to \([H_2Cys]\). This allows for an easier understanding of the dominant ionic species in a solution, related to the observed pH.

pKa Values

Understanding pKa values is fundamental in grasping acid-base equilibria in solutions. The pKa value essentially represents the strength of an acid; the lower the pKa, the stronger the acid. It is the negative logarithm of the acid dissociation constant (Ka):

• \( pKa = -\log_{10}(Ka) \)

These values help predict the behavior of acids in solutions, particularly in mixtures where multiple acidic groups compete for dissociation.
In our exercise, cysteine's pKa values of 1.7 and 8.3 indicate two titration steps for every ionizable hydrogen ion. The first value is significantly lower, indicating a more robust initial ionization compared to the second step. Hence, at a particular pH, these pKa values tell us which ionic form is predominant. When calculating the pH, understanding where our solution stands relative to these pKa values is vital, as it provides insight into which species contributes most to the equilibrium hydrogen ion concentration.
Thus, knowing pKa values not only helps in ranking acidic strength but is also instrumental in using the Henderson-Hasselbalch equation, facilitating accurate pH calculations and reinforcing the understanding of acid-base reactions.