Problem 75 - Chemistry: Principles and Reactions

Question

The \(\mathrm{pH}\) of a household ammonia cleaning solution is \(11.68\). How many grams of ammonia are needed in a 1.25-L solution to give the same pH?

Step-by-Step Solution

Answer

Answer: Approximately 0.102 grams of ammonia are needed to prepare a 1.25 L solution with a pH of 11.68.

Step 1: Calculate the concentration of \(\mathrm{OH^-}\) ions from the given pH value

Using the relationship between pH, pOH, and the ion product constant for water (\(K_\mathrm{w}\)), we can find the \(\mathrm{OH^-}\) concentration. \(pH + pOH = 14\) To find the pOH, subtract the given pH value from 14: \(pOH = 14 - pH = 14 - 11.68 = 2.32\) Now, we can determine the concentration of \(\mathrm{OH^-}\): \([\mathrm{OH^{-}}] = 10^{-pOH} = 10^{-2.32} \approx 4.79 \times 10^{-3} \mathrm{M}\)

Step 2: Find the concentration of ammonia \(\mathrm{NH_3}\)

The ammonia cleaning solution contains a weak base, ammonium hydroxide (\(\mathrm{NH_4OH}\)), that ionizes as follows: \(\mathrm{NH_4OH} \rightarrow \mathrm{NH_3} + \mathrm{OH^-}\) The concentration of \(\mathrm{NH_3}\) can be approximated as the concentration of \(\mathrm{OH^-}\), after considering the ionization reaction. So, \([\mathrm{NH_{3}}] \approx 4.79 \times 10^{-3} \mathrm{M}\)

Step 3: Calculate the mass of ammonia required for 1.25 L of the solution

To find the mass of ammonia needed, we will use the molarity formula: mass = molarity \(\times\) volume \(\times\) molar mass \(\text{mass }(\mathrm{NH_3}) = [\mathrm{NH_3}] \times V \times \text{ Molar Mass }(\mathrm{NH_3})\) \(\text{mass }(\mathrm{NH_3}) = (4.79 \times 10^{-3}\mathrm{M})(1.25 \mathrm{L})(17.03 \frac{\mathrm{g}}{\mathrm{mol}})\) \(\text{mass }(\mathrm{NH_3}) \approx 0.102 \mathrm{g}\) Therefore, approximately 0.102 grams of ammonia are needed to prepare 1.25 L of a solution with a pH of 11.68.

Key Concepts

Understanding Ammonia Concentration

Ammonia is a common ingredient found in many household cleaning products, known for its effectiveness in removing grime. Typically, ammonia solutions are assessed based on their concentration, a crucial factor that determines their cleaning efficiency. In the context of pH calculation, knowing the concentration of ammonia is vital since it allows us to discern how it behaves in different solutions. For instance, in a weak base solution like ammonium hydroxide, ammonia (\(\mathrm{NH_3}\)) dissociates partially to form hydroxide ions (\(\mathrm{OH^-}\)). The concentration of \(\mathrm{OH^-}\) is therefore instrumental in approximating the concentration of ammonia when considering ionic equilibrium. By understanding that the ammonia concentration aligns closely with the \(\mathrm{OH^-}\) concentration, we can determine the solution's pH and overall effectiveness. This relationship is key to calculating how much ammonia is required to reach a desired pH level in a solution.

Ionic Equilibrium in Ammonia Solutions

Ionic equilibrium is an essential concept in chemistry, especially when dealing with weak base solutions like those involving ammonia. When a weak base such as ammonium hydroxide (\(\mathrm{NH_4OH}\)) is dissolved in water, it establishes an equilibrium between its ionized and unionized forms.

• The ionization reaction of \(\mathrm{NH_4OH}\) can be represented as:
• \(\mathrm{NH_4OH} \rightarrow \mathrm{NH_3} + \mathrm{OH^-}\)

This equation shows the dissociation of ammonium hydroxide into ammonia and hydroxide ions. In equilibrium conditions, weak bases do not fully dissociate. Hence, the concentration of \(\mathrm{NH_3}\) at equilibrium is closely related to the concentration of \(\mathrm{OH^-}\). Knowing the ionic equilibrium helps in predicting the concentration of each species in the solution simply by measuring one. For example, once we have the concentration of \(\mathrm{OH^-}\), we can infer the ammonia concentration. This insight is critical when adjusting ammonia concentrations to achieve specific pH levels.

Molarity Calculation and Its Role

Molarity is a measure of the concentration of a solute in a solution, expressed in moles per liter (M). In the case of calculating pH levels, molarity plays a key role, especially when determining how much of a substance is needed for a solution.To calculate molarity, we need to know:

• The volume of the solution
• The amount of solute or substance present

In the scenario of ammonia and its impact on pH, the molarity of \(\mathrm{NH_3}\) roughly equals the concentration of \(\mathrm{OH^-}\) derived from pH values. Thus, by establishing this molar concentration, we can further deduce the mass of ammonia required. Applying the formula: \[mass = \text{molarity} \times \text{volume} \times \text{molar mass}\]allows us to convert this concentration to a physical mass of ammonia needed. In the problem set, this calculation determines the mass of \(\mathrm{NH_3}\) necessary for 1.25 L of solution to achieve a pH of 11.68, cementing the fundamental connection between molarity and practical chemical applications.