pH & [H⁺]

• The pH Definition: The term pH stands for 'potential of hydrogen' and is mathematically defined as the negative base-10 logarithm of the hydrogen ion concentration, expressed as $pH = -\log_{10}[H^+]$. This definition means that as the concentration of hydrogen ions increases, the pH value decreases, indicating a more acidic solution.

• Hydrogen Ion Concentration: The notation $[H^+]$ represents the molar concentration of hydrogen ions (or hydronium ions, $H_3O^+$) in units of $mol \cdot dm^{-3}$. To find the concentration when the pH is known, the equation is rearranged to the exponential form $[H^+] = 10^{-pH}$.

• The Logarithmic Scale: Because the scale is logarithmic, each whole pH unit represents a ten-fold difference in acidity. For example, a solution with a pH of 4 has ten times the concentration of $H^+$ ions as a solution with a pH of 5, and one hundred times the concentration of a solution with a pH of 6.

• Self-Ionization of Water: In any aqueous solution, water molecules exist in equilibrium with hydrogen and hydroxide ions: $H_2O(l) \rightleftharpoons H^+(aq) + OH^-(aq)$. This equilibrium is fundamental to understanding how pH relates to alkalinity.

• The Kw Constant: The equilibrium constant for this process is known as the ionic product of water, $K_w = [H^+,OH^-]$. At a standard temperature of 298 K, $K_w$ is constant at $1.00 \times 10^{-14} \, mol^2 \cdot dm^{-6}$, meaning the product of the two ion concentrations always equals this value.

• Temperature Dependence: The dissociation of water is an endothermic process, so increasing the temperature shifts the equilibrium to the right. This increases the concentration of both $H^+$ and $OH^-$ ions, which causes the $K_w$ value to increase and the pH of pure water to decrease, even though the water remains neutral ($[H^+] = [OH^-]$).

• Strong Monoprotic Acids: Strong acids like $HCl$ are assumed to ionize completely in solution. For a monoprotic acid, the concentration of hydrogen ions $[H^+]$ is equal to the initial concentration of the acid $[HA]$, allowing for a direct calculation of pH using the $-\log$ formula.

• Strong Bases and Kw: Strong bases like $NaOH$ also ionize completely, but they release hydroxide ions ($OH^-$) rather than hydrogen ions. To find the pH of a base, one must first determine $[OH^-]$, then use $K_w$ to find $[H^+]$ via the relationship $[H^+] = \frac{K_w}{[OH^-]}$, and finally calculate the pH.

• Dilution Effects: When an acidic solution is diluted by a factor of 10, the $[H^+]$ decreases by a factor of 10, resulting in a pH increase of exactly 1 unit. Diluting by a factor of 100 (e.g., adding 990 $cm^3$ of water to 10 $cm^3$ of acid) increases the pH by 2 units.

Comparison of Acid Types

• Diprotic Nuances: While diprotic acids have two replaceable protons, the second ionization step is often an equilibrium rather than a complete dissociation. In many exam contexts, the first step is treated as complete, while the second is suppressed by the high concentration of $H^+$ already present, meaning the actual $[H^+]$ is slightly less than double the acid concentration.

• The Negative Sign Error: A frequent mistake is forgetting the negative sign in the formula $pH = -\log[H^+]$. This leads to negative pH values for standard concentrations, which should serve as an immediate red flag to the student.

• Confusing [Acid] with [H+]: Students often assume that the concentration of the acid is always the concentration of $H^+$. This is only true for strong monoprotic acids; for diprotic acids or weak acids, this assumption leads to incorrect pH values.

• The Neutrality Myth: A common misconception is that pH 7 is always neutral. Neutrality is defined by $[H^+] = [OH^-]$, not by a specific number on the scale; as temperature changes, the pH of a neutral solution changes accordingly.