A buffer solution is prepared in which the concentration of \(\mathrm{NH}_{3}\) is \(0.30 \mathrm{M}\) and the concentration of \(\mathrm{NH}_{4}{ }^{*}\) is \(0.20 \mathrm{M}\). If the equilibrium constant, \(\mathrm{K}_{\mathrm{b}}\) for \(\mathrm{NH}_{3}\) equals \(1.8 \times 10^{-5}\), what is the \(\mathrm{pH}\) of this solution ? ( \(\log 2.7=0.433\) ). (1) \(9.08\) (2) \(9.43\) (3) \(11.72\) (4) \(8.73\)

The Henderson-Hasselbalch equation is a valuable formula in understanding buffer systems. It allows you to calculate the pH (or pOH) of a buffer solution based on the concentrations of its components. For a basic buffer, like the one involving ammonia (\text{NH}_3) and ammonium ion (\text{NH}_4^+), the equation is: \ \( \text{pOH} = \text{p}K_b + \text{log} \frac{[\text{NH}_4^+]}{[\text{NH}_3]} \)
The equation includes \text{p}K_b, which is the negative logarithm of the base dissociation constant (\text{K}_b). Here's why this is useful:

• It simplifies complex equilibrium expressions.
• By using it, you can quickly find pH if you know the concentrations of species in your buffer.

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Using given values, you can calculate the pOH of a solution and subsequently the pH, which is crucial for understanding the acidity/basicity of the solution.

Ammonia (\text{NH}_3) acts as a weak base in water and establishes an equilibrium with ammonium ion (\text{NH}_4^+). This can be written as: \[ \text{NH}_3 + H_2O \rightleftharpoons \text{NH}_4^+ + OH^- \]
The equilibrium constant for this reaction is the base dissociation constant, \text{K}_b. For ammonia, given \text{K}_b is 1.8 \times 10^{-5}, which indicates ammonia's tendency to accept a proton and form \text{NH}_4^+. To solve pH problems involving ammonia:

• Identify the concentrations of \text{NH}_3 and \text{NH}_4^+ available.
• Calculate the ratio \( \frac{[\text{NH}_4^+]}{[\text{NH}_3]} \). In our exercise, it was 0.20 M / 0.30 M = 0.667.
• Insert these values into the Henderson-Hasselbalch equation to determine pOH.

Understanding ammonia's equilibrium helps in understanding how ammonia solutions resist changes in pH, thus proving their buffer properties.

pOH and pH are interconnected measures of a solution's acidity and basicity. The relationship is defined by the equation: \[ \text{pH} + \text{pOH} = 14 \]
This is due to the self-ionization of water which maintains a constant concentration product of hydrogen (\text{H}^+) and hydroxide (\text{OH}^-) ions. Knowing one measure can help calculate the other:

• If you know pOH, subtract it from 14 to get pH.
• This relationship is vital when you use the Henderson-Hasselbalch equation to find pOH first and then translate it to pH, as shown in our step-by-step solution.

For instance, once you have calculated pOH to be 4.312, you can easily find pH: \( \text{pH} = 14 - 4.312 = 9.688 \). This method ensures accurate and easy determination of the pH of the solution.