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Calculate the \(\mathrm{pH}\) and \(\left[\mathrm{S}^{2-}\right]\) in a \(0.10 \mathrm{M} \mathrm{H}_{2} \mathrm{~S}\) solution. Assume \(K_{\mathrm{a}_{1}}=1.0 \times 10^{-7} ; K_{\mathrm{a}_{2}}=1.0 \times 10^{-19}\)

Short Answer

Expert verified
In summary, for a 0.10 M H鈧係 solution with Ka鈧 = 1.0 x 10鈦烩伔 and Ka鈧 = 1.0 x 10鈦宦光伖, the pH is 4, and the concentration of S虏鈦 ions is 1.0 x 10鈦宦光伖 M.

Step by step solution

01

Write the ionization reactions and dissociation constants

For the two-step ionization of H鈧係, we have the following reactions and dissociation constants: 1. First ionization: \(H_2S \rightleftharpoons H^+ + HS^-\) Dissociation constant Ka鈧: \[K_{a_{1}} = \frac{[H^+][HS^-]}{[H_{2}S]} = 1.0脳10^{-7}\] 2. Second ionization: \(HS^- \rightleftharpoons H^+ + S^{2-}\) Dissociation constant Ka鈧: \[K_{a_{2}} = \frac{[H^+][S^{2-}]}{[HS^-]} = 1.0脳10^{-19}\]
02

Write the ionization expressions

Let x mol of H鈧係 ionizes. So, the concentration of each species after the first ionization can be expressed as follows: \[ [H^+] = x \] \[ [HS^-] = x \] \[ [H_2S] = 0.10 - x \] Now, substitute these expressions in Ka鈧: \[1.0脳10^{-7} = \frac{x \cdot x}{0.10 - x}\]
03

Solve for x and calculate the pH

As Ka鈧 is a weak acid dissociation constant, the value of x will be significantly smaller than the initial concentration (0.10) of H鈧係. Therefore, we can assume \(x 鈮 0.10\). This allows us to simplify the equation for x: \[1.0脳10^{-7} 鈮 \frac{x^2}{0.10}\] Solve for x: \[x 鈮 \sqrt{1.0脳10^{-7} \cdot 0.10} = 1.0脳10^{-4}\] So, the concentration of H鈦 ions is approximately 1.0 x 10鈦烩伌 M. Now, the pH can be calculated: \[pH = -\log{[H^+]} = -\log{(1.0脳10^{-4})} = 4\]
04

Calculate the concentration of S虏鈦 using Ka鈧

Knowing the concentration of the H鈦 ion and the first ionization product (HS鈦), we can now proceed to calculate the concentration of the sulfide ion (S虏鈦) using Ka鈧: \[1.0脳10^{-19} = \frac{(1.0脳10^{-4}) [S^{2-}]}{x}\] Substitute x = 1.0 x 10鈦烩伌 \[1.0脳10^{-19} = \frac{(1.0脳10^{-4}) [S^{2-}]}{1.0脳10^{-4}}\] Solve for the concentration of S虏鈦: \[[S^{2-}] = 1.0脳10^{-19}\] So, the concentration of sulfide ions (S虏鈦) is 1.0 x 10鈦宦光伖 M. To summarize, the pH of the 0.10 M H鈧係 solution is 4, and the concentration of S虏鈦 ions is 1.0 x 10鈦宦光伖 M.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ionization Reactions
Ionization reactions occur when a molecule dissociates into ions. This is a common process in solutions containing acids and bases. For our exercise, the ionization of hydrosulfuric acid is analyzed in two steps. In the first ionization reaction, H鈧係 dissociates into H鈦 ions and HS鈦 ions. The relevant reaction is:
- \( H_2S \rightleftharpoons H^+ + HS^- \) Each dissociation reaction has an associated equilibrium constant, known as the dissociation constant.
The second ionization is:
- \( HS^- \rightleftharpoons H^+ + S^{2-} \) Understanding these ionization reactions helps us calculate the concentrations of ions in a solution, which in turn allows us to determine the pH and other properties of the solution.
Dissociation Constants
Dissociation constants, represented by \( K_a \), are crucial for understanding how well an acid dissociates in water. They give us a measure of the acid's strength and the equilibrium position of its ionization reaction. For our problem, there are two dissociation constants given for H鈧係.
* The first dissociation constant (\( K_{a1} \)) is \( 1.0 \times 10^{-7} \).* It relates to the equilibrium of the reaction \( H_2S \rightleftharpoons H^+ + HS^- \).* The second dissociation constant (\( K_{a2} \)) is \( 1.0 \times 10^{-19} \).* This constant applies to the next equilibrium: \( HS^- \rightleftharpoons H^+ + S^{2-} \).
These constants allow us to write expressions that predict the concentration of products and reactants at equilibrium. Using these expressions, we can solve for unknowns such as ion concentrations and eventually the pH of the solution.
Weak Acid Dissociation
Weak acids, like H鈧係, do not completely ionize in solution. This partial ionization is what we are dealing with when calculating pH in this exercise. In the case of H鈧係, the first dissociation has a low \( K_a \) value of \( 1.0 \times 10^{-7} \), indicating modest ionization. The second dissociation is even weaker with a \( K_a \) of \( 1.0 \times 10^{-19} \), signifying negligible ionization under normal conditions. This means:
- Most H鈧係 remains as un-ionized molecules in solution.- Only a small portion produces H鈦 and HS鈦 ions.- Even fewer HS鈦 ions produce S虏鈦 in the second dissociation.
Understanding weak acid behavior is essential to correctly determine pH and concentrations of different ions in solutions.
Sulfide Ion Concentration
The sulfide ion (S虏鈦) emerges from the second step of H鈧係 ionization. Due to the extremely small \( K_{a2} \) for the second dissociation, the concentration of S虏鈦 ions in solution is very low. Here鈥檚 why:
- The ionization equilibrium strongly favors the reactants, meaning very few S虏鈦 are produced.- In our solution, even after calculating, the concentration of \([S^{2-}]\) is observed to be \( 1.0 \times 10^{-19} \) M.
While the concentration of sulfide ions is generally negligible, it is still essential to consider it in calculations for thoroughness and accuracy, particularly in chemical environments where trace ions can have significant effects.

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