Question

Which of the following has a higher \(\mathrm{pH}\) : (a) \(1.0 \mathrm{M} \mathrm{NH}_{3}\), (b) \(0.20 \mathrm{M} \mathrm{NaOH}\left(K_{\mathrm{b}}\right.\) for \(\left.\mathrm{NH}_{3}=1.8 \times 10^{-5}\right)\) ?

Short answer

The NaOH solution has a higher pH value of 13.30 compared to NH₃'s 11.63.

Step-by-step solution

Understand the Nature of Each Solution

Recognize that both solutions are basic. Ammonia (NH₃) is a weak base, and sodium hydroxide (NaOH) is a strong base. This difference fundamentally affects their pH values.

Calculate pOH of NaOH Solution

Since NaOH is a strong base, it completely dissociates in water. Thus, the concentration of OH⁻ is equal to the concentration of NaOH. Calculate the pOH: \[pOH = - ext{log}[OH^-] = - ext{log}(0.20) = 0.70\]

Calculate pH of NaOH Solution

Use the relationship between pH and pOH: \[pH = 14 - pOH = 14 - 0.70 = 13.30\]

Determine Concentration of OH- in NH3 Solution

Use the equation for the base dissociation constant to find [OH⁻]: \[ ext{Kb} = \frac{[NH_4^+][OH^-]}{[NH₃]}\]Assume \([NH_4^+] = [OH^-] = x\) and solve for x in \[1.8 \times 10^{-5} = \frac{x^2}{1.0 - x} \approx \frac{x^2}{1.0}\] This simplifies to \[x^2 = 1.8 \times 10^{-5}\], so \[x = \sqrt{1.8 \times 10^{-5}} = 4.24 \times 10^{-3} \].

Calculate pOH of NH3 Solution

Use the concentration of OH⁻ found from the previous step:\[pOH = - ext{log}(4.24 \times 10^{-3}) = 2.37\]

Calculate pH of NH3 Solution

Convert the pOH to pH using the same relationship:\[pH = 14 - 2.37 = 11.63\]

Compare pH Values

The pH of the NaOH solution is 13.30, while the pH of the NH₃ solution is 11.63. Higher pH indicates a solution is more basic.

Key concepts

Understanding Strong Bases

Strong bases, like sodium hydroxide (NaOH), dissociate completely in water. This means that every molecule of NaOH separates into sodium ions (Na⁺) and hydroxide ions (OH⁻).
This complete dissociation results in a high concentration of OH⁻ ions in the solution.

• NaOH fully dissociates: \[\text{NaOH} \rightarrow \text{Na}^+ + \text{OH}^-\]
• High OH⁻ concentration leads to a higher pH.

Strong bases are typically found at the top of the pH scale, often ranging from pH 12 to 14. This high pH level indicates a very basic or alkaline solution. Because of this, NaOH is commonly used in cleaning products and industrial applications for its strong, basic properties.

Exploring Weak Bases

Weak bases, such as ammonia (NH₃), only partially dissociate in water. Unlike strong bases, not every NH₃ molecule contributes to the formation of hydroxide ions.

• In a weak base equilibrium: \[\text{NH}_3 + \text{H}_2\text{O} \rightleftharpoons \text{NH}_4^+ + \text{OH}^-\]
• Small amount of NH₄⁺ and OH⁻ is formed.

The dissociation is influenced by the base dissociation constant, \(K_b\). This constant gives an idea of how far the equilibrium shifts to the right, producing more OH⁻ ions.
Because weak bases do not fully dissociate, their pH values are lower than those of strong bases.

The Art of pH Calculation

Calculating pH involves understanding the relationship between pH and pOH. pH is a measure of hydrogen ion concentration, while pOH is a measure of hydroxide ion concentration.
- Formula for calculating pOH:\[pOH = -\log[\text{OH}^-]\]- Conversion to pH:\[pH = 14 - pOH\]In the case of a strong base like NaOH, the pOH is easily calculated using the concentration of the base since it fully dissociates.
For weak bases like NH₃, you must account for the equilibrium state using \(K_b\). This involves solving an equilibrium expression to find OH⁻ concentration before determining pOH and eventually pH.

Understanding Acid-Base Equilibrium

Acid-base equilibrium involves the balance between acids and bases in a solution. This balance is described by equilibrium constants, which help us predict the concentrations of ions in solution.

• For bases, the equilibrium constant is \(K_b\).
• It represents the degree of ionization of the base in water.

The equilibrium adjustments in weak bases like ammonia occur because not all molecules convert to products, leading to a measurable equilibrium state.
This concept is crucial when calculating pH, as it requires an understanding of reversible reactions and equilibrium in order to accurately determine ion concentrations in solutions.