To determine the relationship between the amounts of acetic acid added to each container, we need to analyze how the pH changes in each solution when acetic acid is introduced. The pH scale is logarithmic, meaning that each whole number change on the scale represents a tenfold change in acidity. In this case, we are lowering the pH from 6.5 to 5.5, which indicates an increase in acidity.
Understanding pH and Acetic Acid
The pH of a solution is a measure of its hydrogen ion concentration. A lower pH means a higher concentration of hydrogen ions. Acetic acid (CH₃COOH) is a weak acid that partially dissociates in water to release hydrogen ions (H⁺). When we add acetic acid to our containers, we are increasing the concentration of H⁺ ions, which lowers the pH.
Initial Conditions of the Containers
- Container X: 10 ml of water (pH 6.5)
- Container Y: 5 ml of milk + 5 ml of water (pH 6.5)
- Container Z: 10 ml of milk (pH 6.5)
Calculating the Change in pH
To lower the pH from 6.5 to 5.5, we need to understand the relationship between the amount of acetic acid added and the resulting pH. The pH change can be calculated using the formula:
pH = -log[H⁺]
At pH 6.5, the concentration of hydrogen ions is approximately 3.16 x 10⁻⁷ M, and at pH 5.5, it is about 3.16 x 10⁻⁶ M. This means that to achieve a pH of 5.5, we need to increase the concentration of H⁺ ions by a factor of 10.
Comparing the Containers
Now, let’s consider how the volume and composition of each container affect the amount of acetic acid needed:
- Container X: Contains only water, so it will require a certain amount of acetic acid to achieve the desired pH.
- Container Y: Contains a mixture of milk and water. The presence of milk, which contains proteins and other compounds, may buffer the solution, requiring more acetic acid to achieve the same pH change.
- Container Z: Contains only milk, which also has buffering capacity. Similar to container Y, it may require a different amount of acetic acid compared to container X.
Determining the Relationship Between P, Q, and R
Given that all three containers reach the same final pH of 5.5, we can infer that:
- Container X (P) will likely require the least amount of acetic acid because it has no buffering agents.
- Container Y (Q) will require more acetic acid than X due to the buffering effects of the milk and water mixture.
- Container Z (R) may require an amount similar to or slightly more than Q, depending on the specific buffering capacity of the milk.
In conclusion, the relationship between the amounts of acetic acid added can be summarized as follows:
P < Q < R
This means that the amount of acetic acid added to container X (P) is less than that added to container Y (Q), which in turn is less than that added to container Z (R). This hierarchy is due to the varying buffering capacities of the solutions in each container.
