pH Scale — Revision Notes

⚡ 30-Second Revision

pH Definition: — $pH = -\log_{10}[H\textsuperscript{+}]$ \\ - pOH Definition: $pOH = -\log_{10}[OH\textsuperscript{-}]$ \\ - **Relationship at $25^\circ C$:** $pH + pOH = 14$ \\ - **Ionic Product of Water ($K_w$):** $K_w = [H\textsuperscript{+}][OH\textsuperscript{-}] = 1.0 \times 10^{-14}$ at $25^\circ C$ \\ - Neutral pH: 7 at $25^\circ C$ (where $[H\textsuperscript{+}] = [OH\textsuperscript{-}] = 10^{-7}$ M) \\ - Acidic: $pH < 7$, $[H\textsuperscript{+}] > 10^{-7}$ M \\ - Basic: $pH > 7$, $[H\textsuperscript{+}] < 10^{-7}$ M \\ - Strong Acids/Bases: Complete dissociation, $[H\textsuperscript{+}] \approx C_{acid}$ or $[OH\textsuperscript{-}] \approx C_{base}$ \\ - Weak Acids/Bases: Partial dissociation, use $K_a$ or $K_b$ (e.g., $[H\textsuperscript{+}] = \sqrt{K_a \cdot C_{acid}}$ for weak acid approximation) \\ - Logarithmic Scale: Each unit change in pH is a 10-fold change in $[H\textsuperscript{+}]$

2-Minute Revision

The pH scale is a concise way to express the acidity or basicity of aqueous solutions. It's defined as the negative logarithm of the hydrogen ion concentration, $pH = -\log[H\textsuperscript{+}]$. A complementary scale, pOH, uses the hydroxide ion concentration, $pOH = -\log[OH\textsuperscript{-}]$.

At $25^\circ C$, these scales are linked by the fundamental relationship $pH + pOH = 14$, derived from the ionic product of water, $K_w = [H\textsuperscript{+}][OH\textsuperscript{-}] = 10^{-14}$. Pure water is neutral with pH 7 at $25^\circ C$, indicating equal concentrations of $10^{-7}$ M for both H\textsuperscript{+} and OH\textsuperscript{-} ions.

Solutions with pH less than 7 are acidic, while those with pH greater than 7 are basic. Remember that the pH scale is logarithmic, meaning a change of one pH unit signifies a tenfold change in $[H\textsuperscript{+}]$.

For strong acids and bases, calculating pH is straightforward as they fully dissociate. For weak acids and bases, their partial dissociation requires using their respective dissociation constants, $K_a$ or $K_b$, often with approximations.

Always consider the autoionization of water for very dilute solutions and remember that neutral pH changes with temperature.

5-Minute Revision

The pH scale is your go-to tool for quantifying acidity and basicity in aqueous solutions. It's mathematically defined as $pH = -\log_{10}[H\textsuperscript{+}]$, where $[H\textsuperscript{+}]$ is the molar concentration of hydrogen ions.

Similarly, $pOH = -\log_{10}[OH\textsuperscript{-}]$ for hydroxide ions. These two are intrinsically linked by the autoionization of water, $H_2O \rightleftharpoons H\textsuperscript{+} + OH\textsuperscript{-}$, which has an ionic product $K_w = [H\textsuperscript{+}][OH\textsuperscript{-}]$.

At $25^\circ C$, $K_w = 1.0 \times 10^{-14}$, leading to the crucial relationship $pH + pOH = 14$. This also sets the neutral point at pH 7 for $25^\circ C$, where $[H\textsuperscript{+}] = [OH\textsuperscript{-}] = 10^{-7}$ M.

Acidic solutions have $pH < 7$ (higher $[H\textsuperscript{+}]$), while basic solutions have $pH > 7$ (higher $[OH\textsuperscript{-}]$).

Key Calculation Scenarios:

1
Strong Acids/Bases: — Assume complete dissociation. For $0.001$ M HCl, $[H\textsuperscript{+}] = 10^{-3}$ M, so $pH = 3$. For $0.001$ M NaOH, $[OH\textsuperscript{-}] = 10^{-3}$ M, so $pOH = 3$, and $pH = 14 - 3 = 11$.
2
Weak Acids/Bases: — Use $K_a$ or $K_b$. For a weak acid HA, $HA \rightleftharpoons H\textsuperscript{+} + A\textsuperscript{-}$. If initial $[HA] = C$ and $[H\textsuperscript{+}] = x$, then $K_a = \frac{x^2}{C-x}$. Often, $x \ll C$, so $x \approx \sqrt{K_a \cdot C}$. Example: $0.1$ M acetic acid ($K_a = 1.8 \times 10^{-5}$). $[H\textsuperscript{+}] = \sqrt{1.8 \times 10^{-5} \times 0.1} = \sqrt{1.8 \times 10^{-6}} \approx 1.34 \times 10^{-3}$ M. $pH \approx 2.87$.
3
Dilution: — A tenfold dilution changes pH by one unit. Be careful with very dilute solutions (e.g., $10^{-8}$ M HCl). Here, water's autoionization cannot be ignored. The total $[H\textsuperscript{+}]$ will be slightly less than 7, not 8. You must solve $x = C_{acid} + K_w/x$ for $[H\textsuperscript{+}]$.
4
Mixtures: — Calculate moles of H\textsuperscript{+} and OH\textsuperscript{-} separately. Determine the excess moles after neutralization. Divide by the total volume to get the final concentration of the excess ion, then calculate pH/pOH. Example: 50 mL of 0.1 M HCl (0.005 mol H\textsuperscript{+}) + 50 mL of 0.08 M NaOH (0.004 mol OH\textsuperscript{-}). Excess H\textsuperscript{+} = 0.001 mol. Total volume = 0.1 L. $[H\textsuperscript{+}] = 0.01$ M. $pH = 2$.

Remember, the pH scale is temperature-dependent; neutral pH is only 7 at $25^\circ C$. Also, distinguish between acid/base strength (extent of ionization) and concentration (amount of solute). Practice logarithmic calculations and be mindful of approximations.

Prelims Revision Notes

1
Definition of pH and pOH:

* $pH = -\log_{10}[H\textsuperscript{+}]$ (where $[H\textsuperscript{+}]$ is molar concentration of hydrogen ions or hydronium ions, $H_3O\textsuperscript{+}$). * $pOH = -\log_{10}[OH\textsuperscript{-}]$ (where $[OH\textsuperscript{-}]$ is molar concentration of hydroxide ions).

1
**Ionic Product of Water ($K_w$):**

* $H_2O(l) \rightleftharpoons H\textsuperscript{+}(aq) + OH\textsuperscript{-}(aq)$. * $K_w = [H\textsuperscript{+}][OH\textsuperscript{-}]$. * At $25^\circ C$, $K_w = 1.0 \times 10^{-14}$.

1
Relationship between pH and pOH:

* At $25^\circ C$, $pH + pOH = 14$. * This is derived from $pK_w = pH + pOH$, where $pK_w = -\log K_w$.

1
**pH Scale Interpretation (at $25^\circ C$):**

* Neutral: $pH = 7$, $[H\textsuperscript{+}] = [OH\textsuperscript{-}] = 10^{-7}$ M. * Acidic: $pH < 7$, $[H\textsuperscript{+}] > 10^{-7}$ M. * Basic (Alkaline): $pH > 7$, $[H\textsuperscript{+}] < 10^{-7}$ M.

1
Logarithmic Nature: — Each unit change in pH represents a tenfold change in $[H\textsuperscript{+}]$. For example, pH 2 is 100 times more acidic than pH 4.
2
Calculations for Strong Acids/Bases:

* Strong Acid (e.g., HCl): $[H\textsuperscript{+}] = C_{acid}$. Calculate pH directly. * Strong Base (e.g., NaOH): $[OH\textsuperscript{-}] = C_{base}$. Calculate pOH, then $pH = 14 - pOH$.

1
Calculations for Weak Acids/Bases:

* Weak Acid (HA): $HA \rightleftharpoons H\textsuperscript{+} + A\textsuperscript{-}$. Use $K_a = \frac{[H\textsuperscript{+}][A\textsuperscript{-}]}{[HA]}$. Often, $[H\textsuperscript{+}] \approx \sqrt{K_a \cdot C_{acid}}$ (approximation valid if $C_{acid}/K_a > 100$).

* Weak Base (B): $B + H_2O \rightleftharpoons BH\textsuperscript{+} + OH\textsuperscript{-}$. Use $K_b = \frac{[BH\textsuperscript{+}][OH\textsuperscript{-}]}{[B]}$. Often, $[OH\textsuperscript{-}] \approx \sqrt{K_b \cdot C_{base}}$.

Then find pH from pOH.

1
Effect of Dilution:

* Diluting an acid increases its pH; diluting a base decreases its pH. * A 10-fold dilution changes pH by 1 unit. * Crucial for very dilute solutions: For $[H\textsuperscript{+}]$ or $[OH\textsuperscript{-}]$ concentrations $\le 10^{-7}$ M, the autoionization of water must be considered. The total $[H\textsuperscript{+}]$ is the sum from the acid/base and water. For example, for $10^{-8}$ M HCl, pH is slightly less than 7 (approx. 6.97), not 8.

1
Mixtures of Strong Acid and Strong Base:

* Calculate moles of H\textsuperscript{+} and OH\textsuperscript{-} separately. * Determine the excess moles after neutralization. * Calculate the final concentration of the excess ion using the total volume. * Calculate pH/pOH from this final concentration.

1
Temperature Dependence: — $K_w$ and thus the neutral pH are temperature-dependent. Neutral pH is 7 only at $25^\circ C$. At higher temperatures, neutral pH is lower than 7.

Vyyuha Quick Recall

Pure Hydrogen Ions Count: Positive Hydrogen Ions Cause Acidity, Lower PH Means More Acidic. Positive OH Ions Cause Basicity, Lower POH Means More Basic. PH Plus POH Equals Fourteen.