Question

Consider a buffered solution containing \(\mathrm{CH}_{3} \mathrm{NH}_{3} \mathrm{Cl}\) and \(\mathrm{CH}_{3} \mathrm{NH}_{2}\). Which of the following statements concerning this solution is(are) true? \(\left(K_{\mathrm{a}}\right.\) for \(\mathrm{CH}_{3} \mathrm{NH}_{3}^{+}=2.3 \times 10^{-11}\).) a. A solution consisting of \(0.10 \mathrm{M} \mathrm{CH}_{3} \mathrm{NH}_{3} \mathrm{Cl}\) and \(0.10 \mathrm{M}\) \(\mathrm{CH}_{3} \mathrm{NH}_{2}\) would have a higher buffering capacity than one containing \(1.0 \mathrm{M} \mathrm{CH}_{3} \mathrm{NH}_{3} \mathrm{Cl}\) and \(1.0 \mathrm{M} \mathrm{CH}_{3} \mathrm{NH}_{2}\). b. If \(\left[\mathrm{CH}_{3} \mathrm{NH}_{2}\right]>\left[\mathrm{CH}_{3} \mathrm{NH}_{3}{ }^{+}\right]\), then the \(\mathrm{pH}\) is larger than the \(\mathrm{p} K_{\mathrm{a}}\) value. c. Adding more \(\left[\mathrm{CH}_{3} \mathrm{NH}_{3} \mathrm{Cl}\right]\) to the initial buffer solution will decrease the \(\mathrm{pH}\). d. If \(\left[\mathrm{CH}_{3} \mathrm{NH}_{2}\right]<\left[\mathrm{CH}_{3} \mathrm{NH}_{3}{ }^{+}\right]\), then \(\mathrm{pH}<3.36\). e. If \(\left[\mathrm{CH}_{3} \mathrm{NH}_{2}\right]=\left[\mathrm{CH}_{3} \mathrm{NH}_{3}{ }^{+}\right]\), then \(\mathrm{pH}=10.64\).

Short answer

The true statements concerning the given buffered solution are: b. If [\(\mathrm{CH_{3}NH_{2}\)] > [\(\mathrm{CH_{3}NH_{3}^{+}}\)], then the pH is larger than the pKa value. c. Adding more [\(\mathrm{CH_{3}NH_{3}Cl}\)] to the initial buffer solution will decrease the pH. e. If [\(\mathrm{CH_{3}NH_{2}\)] = [\(\mathrm{CH_{3}NH_{3}^{+}}\)], then pH = 10.64.

Step-by-step solution

Analyze the given Ka value

: Firstly, let's take a look at the given information. The Ka value for CH3NH3+ is given as \(2.3 \times 10^{-11}\). This value indicates that CH3NH3+ is a weak acid and will not dissociate completely in water.

Calculate the pKa and pH

: We are trying to find out if the provided statements are true or false by comparing the pH or buffering capacity. For this, we need to find the pKa of CH3NH3+ and the pH of the buffered solution. The pKa for CH3NH3+ can be calculated as \(pKa = -\log(K_a) = -\log(2.3 \times 10^{-11}) = 10.64 \). The pH of the buffered solution can be found using the Henderson-Hasselbalch equation as follows: \(pH = pKa + \log{\frac{[\mathrm{A^-}]}{[\mathrm{HA}]}}\), where [\(\mathrm{A^-}\)] is the concentration of CH3NH2 and [HA] is the concentration of CH3NH3+. Now, let's examine each provided statement.

Evaluate Statement a

: A solution with 0.10 M CH3NH3Cl and 0.10 M CH3NH2 would have a higher buffering capacity than one with 1.0 M CH3NH3Cl and 1.0 M CH3NH2. False. The buffering capacity increases as the concentration of both the acid and conjugate base increases. Therefore, the solution with 1.0 M CH3NH3Cl and 1.0 M CH3NH2 would have a higher buffering capacity.

Evaluate Statement b

: If [\(\mathrm{CH_3NH_2}\)] > [\(\mathrm{CH_3NH_{3}^{+}}\)], then the pH is larger than the pKa value. True. Based on the Henderson-Hasselbalch equation, if the concentration of CH3NH2 is greater than the concentration of CH3NH3+, the term \(\frac{[\mathrm{A^-}]}{[\mathrm{HA}]}\) will be greater than 1, and the pH would be larger than the pKa value.

Evaluate Statement c

: Adding more CH3NH3Cl to the initial buffer solution will decrease the pH. True. Increasing the concentration of CH3NH3Cl increases the concentration of CH3NH3+, hence the numerator in the Henderson-Hasselbalch equation will increase relative to the denominator and result in a decrease in pH.

Evaluate Statement d

: If [\(\mathrm{CH_3NH_2}\)] < [\(\mathrm{CH_3NH_{3}^{+}}\)], then pH < 3.36. False. If the concentration of CH3NH2 is less than the concentration of CH3NH3+, the pH value might be lower than the pKa value (10.64) according to the Henderson-Hasselbalch equation, but it doesn’t necessarily have to be less than 3.36.

Evaluate Statement e

: If [\(\mathrm{CH_3NH_2}\)] = [\(\mathrm{CH_3NH_{3}^{+}}\)], then pH = 10.64. True. When the concentration of CH3NH2 is equal to the concentration of CH3NH3+, the term \(\frac{[\mathrm{A^-}]}{[\mathrm{HA}]}\) becomes 1 and pH is equal to pKa value, which is 10.64 in this case. In conclusion, the statements b, c, and e are true, while statements a and d are false.

Key concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a fundamental relationship used in buffer solution chemistry. It is represented as:
\(pH = pKa + \log{\frac{[\mathrm{A^-}]}{[\mathrm{HA}]}}\).
This equation relates the pH of a buffer to the pKa (the negative log of the acid dissociation constant, Ka) and the ratio of the concentrations of the conjugate base (\(A^-\)) to the weak acid (HA). Essentially, the equation provides a method to calculate the pH of a solution when you know the concentrations of the weak acid and its conjugate base. Understanding this equation is crucial in predicting the effect of changing the concentration of either component of the buffer on the pH. In practice, adding more conjugate base will increase the pH, while adding more weak acid will decrease it.

Buffering Capacity

Buffering capacity refers to the ability of a buffered solution to resist changes in pH upon the addition of small amounts of an acid or a base. It is a measure of the efficiency of a buffer in maintaining a stable pH.

Factors Affecting Buffering Capacity

• The relative concentrations of the weak acid and its conjugate base: Optimal buffering occurs when these concentrations are equal.
• The absolute concentrations of the acid-base pair: Higher concentrations provide a greater capacity to neutralize added acids or bases.
• The pKa of the acid: A buffer is most effective when the pKa is close to the desired pH.

Thus, a buffer will have a higher buffering capacity if both the weak acid and the conjugate base are present in higher concentrations. This is why in the original exercise, statement a is incorrect, since a buffer with 1.0 M concentrations of both components will have a higher buffering capacity than one with 0.10 M.

Weak Acid and Conjugate Base

In the context of buffered solutions, the weak acid and its conjugate base are two components that work in tandem to resist changes in the pH of the solution. A weak acid (HA) partially dissociates in water to form its conjugate base (\(A^-\)) and a hydrogen ion (H+).
The relationship between the weak acid and its conjugate base is important for buffer solutions. The buffer works by using the weak acid to neutralize any added bases and the conjugate base to neutralize any added acids. This dynamic equilibrium is what provides the buffer’s ability to maintain a relatively constant pH. For example, in the buffered solution in the exercise, CH3NH3+ (methylammonium) acts as the weak acid, and CH3NH2 (methylamine) as the conjugate base.

pH and pKa Relationship

The pH and pKa relationship is a core concept in understanding the behavior of buffer solutions. The pH is a measure of the acidity or basicity of a solution, while the pKa is a measure of how strongly an acid dissociates in solution. When the pH is equal to the pKa, the concentrations of the weak acid and its conjugate base are equal, as is shown in statement e of the original exercise.

pH Greater than pKa

When the pH of the solution is greater than the pKa, the conjugate base form (\(A^-\)) is more prevalent than the weak acid (HA). This is depicted in statement b, which is true, as having a greater concentration of conjugate base results in a pH higher than pKa.

pH Less than pKa

Conversely, if the pH is less than pKa, the weak acid is the predominant form. This relationship is crucial for designing buffer systems for desired pH levels and is applied when determining the proper ratio of acid to base for a buffer solution.