Question

Calculate the ratio \(\left[\mathrm{NH}_{3}\right] /\left[\mathrm{NH}_{4}{ }^{+}\right]\) in ammonia/ammonium chloride buffered solutions with the following \(\mathrm{pH}\) values: a. \(\mathrm{pH}=9.00\) b. \(\mathrm{pH}=8.80\) c. \(\mathrm{pH}=10.00\) d. \(\mathrm{pH}=9.60\)

Short answer

The ratio of the concentrations of ammonia to ammonium ion for each given pH value is approximately: a. pH = 9.00: \(\frac{[\mathrm{NH}_{3}]}{[\mathrm{NH}_{4}^+]} \approx 0.56\) b. pH = 8.80: \(\frac{[\mathrm{NH}_{3}]}{[\mathrm{NH}_{4}^+]} \approx 0.36\) c. pH = 10.00: \(\frac{[\mathrm{NH}_{3}]}{[\mathrm{NH}_{4}^+]} \approx 5.62\) d. pH = 9.60: \(\frac{[\mathrm{NH}_{3}]}{[\mathrm{NH}_{4}^+]} \approx 2.24\)

Step-by-step solution

Write down the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is given by: \[ \mathrm{pH} = \mathrm{p}K_\mathrm{a} + \log \frac{[\mathrm{A}^-]}{[\mathrm{HX}]} \] where pH is the pH of the solution, pKa is the acid dissociation constant of the acidic species (NH4+), [A-] is the concentration of the conjugate base (NH3), and [HX] is the concentration of the conjugate acid (NH4+).

Calculate the ratio for each pH value

To calculate the ratio \(\frac{\left[\mathrm{NH}_{3}\right]}{\left[\mathrm{NH}_{4}^+\right]}\) for each pH value, we will rearrange the Henderson-Hasselbalch equation to isolate the ratio and then substitute the given pH values and the known pKa value (9.25) for the NH3/NH4+ system. \[ \frac{[\mathrm{NH}_{3}]}{[\mathrm{NH}_{4}^+]} = 10^{\mathrm{pH} - \mathrm{p}K_\mathrm{a}} \] a. For pH = 9.00: \[ \frac{[\mathrm{NH}_{3}]}{[\mathrm{NH}_{4}^+]} = 10^{9.00 - 9.25} = 10^{-0.25} \approx 0.56 \] b. For pH = 8.80: \[ \frac{[\mathrm{NH}_{3}]}{[\mathrm{NH}_{4}^+]} = 10^{8.80 - 9.25} = 10^{-0.45} \approx 0.36 \] c. For pH = 10.00: \[ \frac{[\mathrm{NH}_{3}]}{[\mathrm{NH}_{4}^+]} = 10^{10.00 - 9.25} = 10^{0.75} \approx 5.62 \] d. For pH = 9.60: \[ \frac{[\mathrm{NH}_{3}]}{[\mathrm{NH}_{4}^+]} = 10^{9.60 - 9.25} = 10^{0.35} \approx 2.24 \] So, the ratio of the concentrations of ammonia to ammonium ion for each given pH value is approximately: a. pH = 9.00: \(\frac{\left[\mathrm{NH}_{3}\right]}{\left[\mathrm{NH}_{4}^+\right]} \approx 0.56\) b. pH = 8.80: \(\frac{\left[\mathrm{NH}_{3}\right]}{\left[\mathrm{NH}_{4}^+\right]} \approx 0.36\) c. pH = 10.00: \(\frac{\left[\mathrm{NH}_{3}\right]}{\left[\mathrm{NH}_{4}^+\right]} \approx 5.62\) d. pH = 9.60: \(\frac{\left[\mathrm{NH}_{3}\right]}{\left[\mathrm{NH}_{4}^+\right]} \approx 2.24\)

Key concepts

pH Calculation

Understanding how to calculate the pH of a solution is crucial in acid-base chemistry. The pH is a measure of the hydrogen ion concentration in a solution, expressed on a logarithmic scale. The formula for pH is given by:\[pH = -\log[\text{H}^+]\]where \([\text{H}^+]\) is the molarity of hydrogen ions in the solution. The pH scale ranges from 0 to 14:- A pH less than 7 indicates an acidic solution- A pH of 7 is considered neutral- A pH greater than 7 indicates a basic (or alkaline) solutionIn the exercise, we use the pH values to find the ratio of concentrations in a buffered ammonia/ammonium system. This is essential in determining how well the solution can resist changes in pH when small amounts of acid or base are added.

Acid-Base Chemistry

Acid-base chemistry focuses on reactions involving proton transfer. Acids are proton donors, while bases are proton acceptors. This interaction is central to many chemical processes. In the case of ammonia \(\text{NH}_3\) and ammonium chloride \(\text{NH}_4^+\), ammonia acts as a weak base, and ammonium is its conjugate acid. When these two are in solution, they establish an equilibrium:\[\text{NH}_4^+ \rightleftharpoons \text{NH}_3 + \text{H}^+\]This equilibrium setup is crucial as it dictates how the solution behaves in terms of pH stability. The acid dissociation constant \(\text{K}_a\) for ammonium reflects how readily it donates protons to water, affecting the pH of the solution. Through the Henderson-Hasselbalch equation, this relationship allows us to understand the changes in the concentrations of the acidic and basic components of the buffer.Acid-base chemistry is not just theoretical; it has practical applications such as in pharmaceuticals, agriculture, and even food industry, where controlling the pH of a solution is necessary for product stability.

Buffer Solutions

Buffer solutions are special because they maintain a relatively stable pH when small amounts of acid or base are added. This property is due to the presence of both a weak acid and its conjugate base in the solution, which react to minimize pH change.In our original exercise, we deal with an ammonia/ammonium chloride buffer. Such buffers work efficiently near the pKa of the acid part of the buffer system, which for ammonium chloride is 9.25. When the solution's pH is near the pKa, both the acids and bases in the buffer can effectively neutralize any added acid (by employing the base, \(\text{NH}_3\)) or base (by employing the acid, \(\text{NH}_4^+\)).This makes buffers important in biological systems, where enzymes require specific pH ranges to function. They are also used in industrial processes where chemical reactions must remain steady over time. Understanding buffers are fundamental in creating solutions in which biological and chemical reactions can proceed optimally.