Question

A solution with a pH of 9.22 is prepared by adding water to \(0.413 \mathrm{~mol}\) of KX to make \(2.00 \mathrm{~L}\) of solution. What is the \(\mathrm{pH}\) of the solution after \(0.368 \mathrm{~mol}\) of \(\mathrm{HX}\) is added?

Short answer

Answer: After adding 0.368 moles of the acid HX, the approximate pH of the solution is 13.27.

Step-by-step solution

Determine the initial concentration of KX

To find the initial concentration of KX, we can divide the moles of KX by the volume of the solution: Initial concentration of KX = \(\frac{0.413\:mol}{2.00\:L}\) Initial concentration of KX = \(0.2065 \: mol/L\)

Determine the initial concentration of OH- from pH

The given pH is 9.22. Use the relation: \(pH + pOH = 14\) Therefore, \(pOH = 14 - pH = 14 - 9.22 = 4.78\) Now, we can determine the concentration of OH-: \([OH^{-}] = 10^{-pOH} = 10^{-4.78} = 1.658 \times 10^{-5} \: mol/L\)

Relate the initial concentration of KX and OH-

KX is a strong base, so the number of OH- ions produced is equal to the concentration of KX. Hence, Initial concentration of KX = \([OH^{-}] = 1.658 \times 10^{-5} \: mol/L\)

Calculate moles of OH- initially present

Moles of OH- initially present = Initial concentration of OH- × volume of solution Moles of OH- initially = \(1.658 \times 10^{-5} \: mol/L \times 2.00 \: L = 3.316 \times 10^{-5} \: mol\)

React HX with OH-

Now, react 0.368 moles of HX with the initial moles of OH- Reaction: HX + OH- \(\rightarrow\) H2O + X- Moles of OH- left after reaction = moles of OH- initially - moles of HX added \(= 3.316 \times 10^{-5} \: mol - 0.368 \: mol\) Since the added moles of HX are much greater, all the OH- will react with HX. Therefore, moles of OH- left after reaction will be zero, and moles of HX left = \(0.368 \: mol - 3.316 \times 10^{-5} \: mol\).

Calculate concentration of HX left

Concentration of HX left after the reaction = \(\frac{moles\:of\:HX\:left}{total\:volume}\) Total volume remains the same, so: Concentration of HX left = \(\frac{0.368 \: mol - 3.316 \times 10^{-5} \: mol}{2.00 \: L} =0.183997 \: mol/L \approx 0.184 \: mol/L\)

Calculate the pH of the solution after the addition of HX

At this point, we have a solution containing mostly HX (about 0.184 mol/L). As HX is an acid, to find the pH, first find the pX: \(pX = -\log_{10}[X^{-}] = -\log_{10}(0.184)\) \(pX = 0.735\) Next, we can use the relation \(pX + pH = 14\) to find the pH: \(\mathrm{pH} = 14 - pX = 14 - 0.735 = 13.265\) Thus, the pH of the solution after adding 0.368 mol of HX is approximately 13.27.

Key concepts

Understanding Acid-Base Reactions

Acid-base reactions are fundamental processes in chemistry where an acid donates a proton (H+) to a base. In the context of the exercise provided, when you add hydrochloric acid (HX) to a solution containing hydroxide ions (OH-), a neutralization reaction occurs. In this reaction, the HX acts as the acid and OH- as the base, resulting in the formation of water (H2O) and an ionic compound, X-.

This type of reaction is crucial in understanding how the pH of a solution can change drastically upon the addition of an acid or a base. The pH scale, which ranges from 0 to 14, measures how acidic or basic a solution is. A pH less than 7 is acidic, while a pH greater than 7 is basic. When HX is added to the basic KX solution, we expect the pH to decrease, indicating that the solution is becoming more acidic as HX neutralizes OH- ions. Nonetheless, in our exercise, the large excess of HX ensures the solution remains basic.

Molar Concentration and Its Impact on Reactions

Molar concentration, or molarity, is a measure of the concentration of a solute in a solution, denoted as moles per liter ((mol/L)). It plays a crucial role in predicting the outcome of chemical reactions in solution. In the exercise, we first calculated the initial molarity of KX to be 0.2065 mol/L.

Molarity becomes especially important when reacting substances in a solution, as it allows for the determination of the quantity of reactants and the extent to which a reaction can occur. The stoichiometry of the acid-base reaction, based on the molarity, helps us predict the amount of acid required to neutralize the base. When you added 0.368 moles of HX to the solution, knowing the molarity of the base helped us understand that all hydroxide ions would be neutralized, effectively predicting the pH change.

The Inverse pOH and pH Relationship

The pH and pOH of a solution are inversely related and add up to 14. This means that as the concentration of hydrogen ions (H+) in a solution increases, pH decreases, making the solution more acidic. Conversely, as hydroxide ions (OH-) increase, pOH decreases while pH increases, indicating a more basic solution.

In our exercise, after determining the initial pOH from the given pH, we used this relationship to find the initial concentration of OH- ions. Then, the subsequent addition of acid (HX) should have reduced the pH, but the excess of added acid ensured it remained basic. The final calculation of pH after the addition of HX was simplified by using the inverse relationship between pH and pX (which represents the concentration of the conjugate base of the acid). This relation underlies all pH calculations and is a crucial concept for understanding how solutions respond to additions of acids or bases.