Problem 84 - General Chemistry: Principles and Modern Applications

Question

Show that when \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) is reduced to half its original value, the \(\mathrm{pH}\) of a solution increases by 0.30 unit, regardless ofthe initial \(p H .\) Is it also true that when any solution is diluted to half its original concentration, the pH increases by 0.30 unit? Explain.

Step-by-Step Solution

Answer

Yes, when \(\mathrm{H}_{3} \mathrm{O}^{+}\) concentration of a solution is reduced to half, the pH increases by 0.30 unit regardless of the initial value. However, halving the concentration of any solution does not always result in a 0.30 unit increase in pH. This principle only applies to solutions where the concentrations of \(\mathrm{H}_{3} \mathrm{O}^{+}\) or \(\mathrm{OH}^{-}\) are altered.

Step 1: Relating change in Hydronium ion concentration to pH

The pH of a solution is defined as the negative of the base 10 logarithm of the hydronium ion (\(\mathrm{H}_{3} \mathrm{O}^{+}\)) concentration. The equation for pH can be written as: \[ \mathrm{pH} = - \log[\mathrm{H}_{3} \mathrm{O}^{+}]\]Now, if the concentration of \( \mathrm{H}_{3} \mathrm{O}^{+}\) is halved, for example from an initial concentration C to C/2, the change in pH, ΔpH, can be evaluated from:\[ Δ\mathrm{pH} = -\log ([\mathrm{H}_{3} \mathrm{O}^{+}]_{final}) + \log ([\mathrm{H}_{3} \mathrm{O}^{+}]_{initial}) \] \[ Δ\mathrm{pH} = -\log (C/2) + \log (C) \]

Step 2: Calculation of pH change

Rewrite the previous equation using the properties of logarithm (log(X/Y) = log(X) - log(Y)):\[ Δ\mathrm{pH} = \log (2) \] Applying the log base 10 value of 2, which is approximated to 0.30, we obtain:\[ Δ\mathrm{pH} = 0.30 \]This shows that, when the hydronium ion concentration is halved, the pH of a solution increases by 0.30 unit, regardless of the initial pH, meaning, it's independent of what the starting concentration (C) is.

Step 3: Applicability of similar change for any solution concentration

The second part of the problem questions whether the same rules apply when any solution is diluted to half of its original concentration. It is important to remember that the pH is only determined by the concentration of \(\mathrm{H}_{3} \mathrm{O}^{+}\) (in acidic solutions) or \(\mathrm{OH}^{-}\) (base in basic solutions). These are the only ions that contribute to pH or pOH, respectively. Hence, if a solution of a substance that does not contribute to hydrogen or hydroxide ions is diluted, the pH will remain the same. Therefore, it is not universally true that diluting any solution to half its initial concentration will result in an increase in pH by 0.30 units.

Key Concepts

Hydronium Ion Concentration

Understanding hydronium ion concentration is essential for grasping pH changes. Hydronium ions, represented as \( \mathrm{H}_{3} \mathrm{O}^{+} \), form when water molecules combine with hydrogen ions (\( \mathrm{H}^{+} \)). This reaction mainly occurs in acidic solutions and plays a crucial role in determining acidity levels.

pH measures how acidic or basic a solution is, and it is directly linked to these hydronium ions. The formula for calculating pH is \( \mathrm{pH} = - \log[\mathrm{H}_{3} \mathrm{O}^{+}] \). Here, the concentration of \( \mathrm{H}_{3} \mathrm{O}^{+} \) is expressed in moles per liter (M).

• More hydronium ions mean a lower pH and more acidity.
• Fewer hydronium ions result in a higher pH and less acidity.

When you change the concentration, for example, by halving it, you directly affect the pH value.

Logarithms in Chemistry

Logarithms simplify the complexity of chemical equations, especially in pH calculations. The logarithmic scale used in pH helps in expressing large variations in hydronium ion concentrations in a manageable way.

Using the base 10 logarithm, a large change in concentration represents a more straightforward and smaller numerical shift. The equation \( \mathrm{pH} = - \log[\mathrm{H}_{3} \mathrm{O}^{+}] \) is vital for this understanding.

• Halving the concentration \( [\mathrm{H}_{3} \mathrm{O}^{+}] \) means calculating \( \log(2) \), which simplifies to approximately 0.30.

This decrease of 0.30 in pH points out how essential logarithms are for perceiving changes in chemical concentrations.

Solution Dilution Impact

Dilution of a solution impacts its concentration and, subsequently, its pH. When an acidic solution is diluted, the hydronium ion concentration decreases, leading to a change in pH.

However, not all solutions behave identically. The key lies in whether the solution contributes to \( \mathrm{H}_{3} \mathrm{O}^{+} \) ions:

• In solutions that actively contribute hydronium ions, halving the concentration means a specific pH change of 0.30 units.
• In other solutions, like neutral or certain buffered solutions, diluting may not alter the pH as expected.

Understanding different solution behaviors helps in predicting changes and controlling chemical reactions effectively.