Problem 29 - Chemistry Matter and Change

Question

Challenge Calculate pH and pOH for an aqueous solution containing \(1.0 \times 10^{-3}\) mol of HCl dissolved in 5.0 \(\mathrm{L}\) of solution.

Step-by-Step Solution

Answer

The concentration of HCl in the solution is \(\frac{1.0 \times 10^{-3} \text{mol}}{5.0 \text{L}} = 2.0 \times 10^{-4} \text{M}\). Since HCl is a strong acid, the concentration of H+ ions is the same: \([H+] = 2.0 \times 10^{-4} \text{M}\). The pH of the solution is \(pH = -\log{(2.0 \times 10^{-4})} \approx 3.70\). To find the pOH, use the relationship \(pH + pOH = 14\), so \(pOH = 14 - 3.70 = 10.30\). The pH of the solution is 3.70 and the pOH is 10.30.

Step 1: Determine the concentration of HCl in the solution

We are given that the solution has 1.0 x 10⁻³ mol of HCl dissolved in 5.0 L of solution. We can calculate the concentration of HCl (in moles per liter) by dividing the number of moles by the volume of the solution as follows: Concentration of HCl = (moles of HCl) / (volume of solution)

Step 2: Calculate the concentration of H+ ions

Since HCl is a strong acid, it will completely dissociate in the solution. Therefore, the concentration of H+ ions will be equal to the concentration of HCl in the solution. Let's represent this concentration as [H+]. [H+] = Concentration of HCl

Step 3: Calculate the pH of the solution

Now that we have the concentration of H+ ions, we can calculate the pH using the following formula: pH = -log([H+]) Plug in the concentration [H+] into the formula and solve for pH.

Step 4: Calculate the pOH of the solution

Knowing the pH of the solution, we can determine the pOH using the relationship between pH and pOH: pH + pOH = 14 Rearrange the formula to solve for pOH: pOH = 14 - pH Plug in the pH value calculated in step 3 and solve for pOH. We have now calculated both the pH and pOH of the aqueous solution containing 1.0 x 10⁻³ mol of HCl dissolved in 5.0 L of solution.

Key Concepts

HCl Solution Concentration

When we talk about the concentration of an HCl solution, we’re referring to the amount of HCl present in a given volume of solution. It's important for students to understand how to find this concentration as it is a starting point for many acid-base calculations.

For our example, the concentration is found by dividing the number of moles of HCl by the volume of the solution. Imagine you have a small vial with a certain number of marbles, each marble representing a mole of HCl. The size of the vial is analogous to the volume of the solution. The more marbles you can fit in the vial (moles in a volume), the higher the concentration of your solution will be.

In mathematical terms, given that the number of moles is 1.0 x 10⁻³ and the volume is 5.0 L, the concentration ([C]) is calculated as:
\[[C] = \frac{1.0 \times 10^{-3}\, \text{mol}}{5.0\, \text{L}}\]
This would yield the molar concentration of HCl in the solution, which is crucial for the subsequent calculations of pH and pOH.

pH Calculation

The pH of a solution tells you how acidic or basic that solution is. The 'p' in pH stands for 'power' or 'potenz' in German, meaning 'power of hydrogen', while the 'H' refers to the hydrogen ion concentration. A low pH indicates high acidity, while a high pH indicates high basicity.

The strength of an acid or a base is typically expressed by its pH value, which is calculated using the negative logarithm of the hydrogen ion concentration. If you're unsure what a logarithm is, think of it like the opposite of raising a number to a power. The formula for pH is as follows:
\[\text{pH} = -\log([H^+])\]
The square brackets around \(H^+\) denote the molar concentration of hydrogen ions. In our example, this concentration is equal to the molarity of HCl, because HCl is a strong acid and dissociates completely. So if we have our calculated concentration of HCl, we simply use it to find the pH.

pOH Calculation

Once the pH has been determined, calculating the pOH is a straightforward process due to their inverse relationship. The pOH measures the concentration of hydroxide ions (OH⁻) in a solution. When acids dissolve in water, they produce hydrogen ions, hence affecting the pH, while bases produce hydroxide ions, which influence the pOH.

Connect pH and pOH using the fundamental relation in water at 25°C:
\[\text{pH} + \text{pOH} = 14\]
To find the pOH from the pH, you simply subtract the pH from 14. Picture it as a seesaw; as one side (pH) goes up, the other side (pOH) must come down to balance to a constant point, which in this case is 14.

Knowing the pH helps us understand the acidity, while knowing the pOH helps us gauge the basicity. This balance is very important in many chemical processes, including biological ones within our bodies.

Acid Dissociation

Acid dissociation is a crucial concept for understanding acid strength and for performing pH and pOH calculations. It refers to the separation of an acid into ions when it dissolves in water. For a strong acid like HCl, this process is considered to be complete, meaning all the HCl molecules will separate into hydrogen (H⁺) and chloride (Cl⁻) ions.

Imagine dropping a sugar cube into a hot cup of tea; it dissolves and the particles disperse evenly throughout the tea. Similarly, when HCl is added to water, it dissociates entirely into H⁺ and Cl⁻. Since the hydrogen ions are what affect pH, knowing that a strong acid like HCl dissociates completely allows us to directly relate its initial concentration to the concentration of hydrogen ions:
\[[H^+] = [HCl]\]
This relationship simplifies the calculations, as we can assume that each HCl molecule contributes one hydrogen ion to the solution.