Question

For each solution, calculate the initial and final \(\mathrm{pH}\) after adding \(0.010 \mathrm{~mol}\) of HCl. a. \(500.0 \mathrm{~mL}\) of pure water b. \(500.0 \mathrm{~mL}\) of a buffer solution that is \(0.125 \mathrm{M}\) in \(\mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}\) and \(0.115 \mathrm{M}\) in \(\mathrm{NaC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}\) c. \(500.0 \mathrm{~mL}\) of a buffer solution that is \(0.155 \mathrm{M}\) in \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{2}\) and \(0.145 \mathrm{M}\) in \(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{NH}_{3} \mathrm{Cl}\)

Short answer

a. Initial pH of pure water is 7.00. Final pH after HCl addition is around 1.70. b. Initial pH of acetic acid buffer solution is approximately 4.76. Final pH after HCl addition will decrease slightly. c. Initial pH of amine buffer is based on the pKb of C2H5NH2 and the given concentrations. Final pH after HCl addition will decrease.

Step-by-step solution

Part a (Initial pH of Pure Water)

To find the initial pH of pure water, we use the fact that the concentration of hydronium ions, [H3O+], is equal to 1 x 10^-7 M at 25°C (neutral pH). The pH is calculated using the formula pH = -log[H3O+], where [H3O+] is the concentration of H3O+ ions in moles per liter.

Part a (Final pH after HCl Addition)

Adding 0.010 mol of HCl to 500.0 mL of water means we have 0.010 mol / 0.500 L = 0.020 M of additional H+ in the solution. The pH of the solution is the negative logarithm of the hydronium ion concentration. The total [H3O+] after addition is the sum of the initial [H3O+] and [HCl]. Thus, pH = -log(0.020 M).

Part b (Initial pH of Buffer Solution)

A buffer solution's pH is given by the Henderson-Hasselbalch equation: pH = pKa + log([A-]/[HA]). For acetic acid \(HC2H3O2\), the pKa is approximately 4.75. The concentration ratio is [NaC2H3O2] (conjugate base) to [HC2H3O2] (acid), which is 0.115 M / 0.125 M. Plugging these into the Henderson-Hasselbalch equation yields the initial pH of the buffer.

Part b (Final pH after HCl Addition)

The added H+ from HCl will react with the [A-] from the buffer. The moles of HCl added is 0.010 mol, and the total volume doesn't change significantly, so we subtract this from [A-] and add it to [HA] to get new concentrations, and then use the Henderson-Hasselbalch equation again.

Part c (Initial pH of Buffer Solution)

For the buffer composed of a weak base (C2H5NH2) and its conjugate acid (C2H5NH3+), we use the Henderson-Hasselbalch equation in the following form: pOH = pKb + log([Base]/[Acid]). The pKb for C2H5NH2 is found by taking the negative logarithm of its Kb, and the concentration ratio is [C2H5NH2] to [C2H5NH3Cl]. After finding pOH, convert to pH by using pH = 14 - pOH.

Part c (Final pH after HCl Addition)

When HCl is added to this buffer solution, it reacts with the weak base C2H5NH2, turning it into the conjugate acid C2H5NH3+. Calculate the changes in the base and acid concentrations after adding HCl, then apply the modified Henderson-Hasselbalch equation to find the final pH.

Key concepts

Henderson-Hasselbalch Equation

Understanding the Henderson-Hasselbalch equation is crucial for students studying buffer systems in chemistry. A buffer solution is a special chemical solution that resists changes in pH when small amounts of an acid or base are added. This equation provides a straightforward way to calculate the pH of a buffer solution.
The equation is expressed as
\[\text{pH} = \text{pKa} + \log\left(\frac{[\text{A}^-]}{[\text{HA}]}\right)\]
where pKa is the acid dissociation constant, [A^-] is the concentration of the conjugate base, and [HA] is the concentration of the weak acid. When you have a weak base buffer, you would use the analogous equation for pOH, involving the pKb and the concentrations of the weak base and its conjugate acid:
\[\text{pOH} = \text{pKb} + \log\left(\frac{[\text{Base}]}{[\text{Acid}]}\right)\]
Then you convert pOH to pH using the formula \[\text{pH} = 14 - \text{pOH}\]. In our exercise with buffer solutions, we use the Henderson-Hasselbalch equation to understand how the pH is initially established and how it changes after adding hydrochloric acid (HCl).

pH and pOH Calculations

Calculating pH and pOH is a foundational skill in chemistry that provides insight into the acidic or basic nature of a solution. The pH scale typically ranges from 0 to 14. A pH less than 7 indicates an acidic solution, 7 is a neutral solution, and greater than 7 is a basic solution. The pH is calculated by taking the negative logarithm (base 10) of the hydronium ion concentration:
\[\text{pH} = -\log[H_3O^+]\]
The pOH, on the other hand, is calculated similarly but based on the concentration of hydroxide ions (OH^-):
\[\text{pOH} = -\log[OH^-]\]
For a neutral solution at 25°C, the concentrations of H3O+ and OH^- are equal, and the pH and pOH both equal 7. It is also important to remember that in any aqueous solution at 25°C, the sum of pH and pOH will always equal 14:\[\text{pH} + \text{pOH} = 14\]. In our exercise, we used these calculations to determine the initial pH of pure water and the pH changes upon adding an acid.

Acid-Base Titration

In chemistry, acid-base titration is a technique used to find the concentration of an unknown acid or base by reacting it with a known volume and concentration of an acid or base (titrant). A pH indicator or pH meter is commonly used to monitor the pH levels as the titrant is added. The point at which the moles of titrant equal the moles of substance being titrated is known as the equivalence point, and this often corresponds to a significant change in pH.
Within the context of our exercise, when HCl is added to the buffer solutions, we're essentially performing a mini-titration. This HCl reacts with the components of the buffer to change its composition. In buffer solution b, HCl reacts with the conjugate base (sodium acetate, NaC2H3O2), while in buffer solution c, it reacts with the weak base (ethylamine, C2H5NH2). Calculating the new concentrations of the species in solution after adding HCl allows us to use the Henderson-Hasselbalch equation to find the new pH.

Hydronium Ion Concentration

The hydronium ion (H3O+) concentration is a measure of the number of moles of hydrogen ions that have associated with water molecules per liter of solution. It is a key component in determining the pH of a solution, as seen in our exercise. Initially, for pure water, the hydronium concentration is known to be at a standard level of \[1 \times 10^{-7} \text{ M}\] at 25°C.
When an acid like HCl is added to water, it dissociates completely, releasing H+ ions which combine with water to form more H3O+ ions. The new hydronium ion concentration influences the pH value, which can be calculated using the negative logarithm of the H3O+ concentration. For a buffer solution, it is crucial to account for how many hydronium ions are used up in the reactions with the buffer components, as this determines how much the pH changes when an acid like HCl is added.