pH

• Self-Ionization of Water: Water molecules undergo a slight reversible reaction to form hydrogen and hydroxide ions: $H_2O(l) \rightleftharpoons H^+(aq) + OH^-(aq)$. This equilibrium exists in all aqueous solutions, meaning both ions are always present regardless of whether the solution is acidic or basic.

• Ionic Product of Water ($K_w$): The equilibrium constant for this process is known as $K_w$, and at $298K$, its value is constant at $1.0 \times 10^{-14} \, mol^2 \, dm^{-6}$. The relationship is defined as $K_w = [H^+][OH^-]$, which allows for the calculation of one ion's concentration if the other is known.

• Neutrality Condition: A solution is defined as neutral when the concentration of hydrogen ions exactly equals the concentration of hydroxide ions. At $298K$, this occurs when $[H^+] = \sqrt{K_w} = 1.0 \times 10^{-7} \, mol \, dm^{-3}$, resulting in a pH of 7.00.

Strong Acids and Bases

• Strong Acids: These species dissociate completely in water. For a monoprotic strong acid, the concentration of hydrogen ions is assumed to be equal to the initial concentration of the acid, allowing for direct calculation using $pH = -\log_{10}[Acid]$.
• Strong Bases: These also ionize fully, but they produce hydroxide ions ($OH^-$). To find the pH, one must first use $K_w$ to find the hydrogen ion concentration: $[H^+] = \frac{K_w}{[OH^-]}$, then apply the pH formula.

Weak Acids

• Partial Dissociation: Weak acids only partially ionize, establishing an equilibrium. The extent of this ionization is quantified by the acid dissociation constant, $K_a = \frac{[H^+][A^-]}{[HA]}$.
• Simplified Calculation: By assuming that the concentration of $H^+$ equals the concentration of the conjugate base $A^-$, and that the initial acid concentration remains roughly constant, the formula simplifies to $[H^+] = \sqrt{K_a \times [HA]}$. The pH is then found by taking the negative log of this result.

Key Formula for Weak Acids: $pH = -\log_{10}\sqrt{K_a \times [HA]}$

• Strong Acid Dilution: When a strong acid is diluted by a factor of 10, the concentration of $H^+$ decreases by a factor of 10, which results in an increase of exactly 1.0 pH unit. This linear relationship holds until the solution becomes so dilute that the ionization of water itself must be considered.

• Weak Acid Dilution: Diluting a weak acid by a factor of 10 does not result in a 1.0 unit pH increase; instead, the pH typically increases by approximately 0.5 units. This occurs because dilution shifts the equilibrium position to favor more dissociation (Le Chatelier's Principle), partially offsetting the decrease in concentration.

• Significant Figures: In pH calculations, the number of decimal places in the pH value should match the number of significant figures in the concentration value. For example, a concentration of $1.5 \times 10^{-3}$ (2 sig figs) should result in a pH given to 2 decimal places (e.g., 2.82).

• Sanity Checks: Always verify if your calculated pH matches the nature of the substance. If you are calculating the pH of a base and get a value below 7, you likely forgot to convert $[OH^-]$ to $[H^+]$ using $K_w$.

• Temperature Sensitivity: Remember that $K_w$ is temperature-dependent. If an exam question specifies a temperature other than $298K$, the neutral pH may not be 7.00, though the method of calculation remains the same.

• The 'Negative' Sign: A common error is forgetting the negative sign in the formula $pH = -\log[H^+]$. This leads to negative pH values for dilute acids, which is a clear indicator of a calculation mistake.

• Weak Acid Approximation: Students often forget to take the square root when calculating $[H^+]$ for weak acids. The $K_a$ expression relates to $[H^+]^2$, so the square root is essential before taking the log.

• Dilution Limits: It is a misconception that infinite dilution of an acid will lead to an infinitely high pH. As the acid becomes extremely dilute, the pH will asymptotically approach 7.00 but never exceed it, as the water's own $H^+$ ions become the dominant source.