Comprehensive Guide to Brønsted-Lowry Acids, Bases, pH Scales, and Autoionization Equilibrium

Brønsted-Lowry Acid-Base Theory and Conjugate Species

• Brønsted-Lowry Acid Definition: A reactant that donates a proton ($H^+$) in a chemical reaction.

• Brønsted-Lowry Base Definition: A reactant that accepts a proton ($H^+$) in a chemical reaction.

• Mechanics of Conjugate Pairs:

  • When an acid donates a proton, the resulting species possesses a negative charge (or a decrease in charge by $+1$), which naturally attracts protons; consequently, this deprotonated species acts as a base and is designated as the conjugate base.

  • When a base accepts a proton, the resulting species acquires an additional proton to give away; consequently, this protonated species acts as an acid and is designated as the conjugate acid.

• Reaction-by-Reaction Analysis of Acid-Base Pairs:

  • Reaction (a): Reaction of aqueous ammonia with nitric acid:     $NH_{3\,(aq)} + HNO_{3\,(aq)} \rightarrow NH_{4\,(aq)}^+ + NO_{3\,(aq)}^-$

  • Base ($\text{B}$): $NH_{3\,(aq)}$

  • Acid ($\text{A}$): $HNO_{3\,(aq)}$

  • Conjugate Acid ($\text{CA}$): $NH_{4\,(aq)}^+$

  • Conjugate Base ($\text{CB}$): $NO_{3\,(aq)}^-$

  • Reaction (b): Reaction of aqueous propanoic acid with potassium hydroxide:     $CH_3CH_2CO_2H_{(aq)} + KOH_{(aq)} \rightarrow H_2O_{(l)} + KCH_3CH_2CO_{2\,(aq)}$

  • Acid ($\text{A}$): $CH_3CH_2CO_2H_{(aq)}$

  • Base ($\text{B}$): $KOH_{(aq)}$

  • Conjugate Acid ($\text{CA}$): $H_2O_{(l)}$

  • Conjugate Base ($\text{CB}$): $KCH_3CH_2CO_{2\,(aq)}$

  • Reaction (c): Reaction of aqueous carbonic acid with disodium hydrogen phosphate:     $H_2CO_{3\,(aq)} + Na_2HPO_{4\,(aq)} \rightarrow NaHCO_{3\,(aq)} + NaH_2PO_{4\,(aq)}$

  • Acid ($\text{A}$): $H_2CO_{3\,(aq)}$

  • Base ($\text{B}$): $Na_2HPO_{4\,(aq)}$

  • Conjugate Base ($\text{CB}$): $NaHCO_{3\,(aq)}$

  • Conjugate Acid ($\text{CA}$): $NaH_2PO_{4\,(aq)}$

  • Reaction (d): Reaction of liquid water with sulfuric acid:     $H_2O_{(l)} + H_2SO_{4\,(aq)} \rightarrow H_3O_{(aq)}^+ + HSO_{4\,(aq)}^-$

  • Base ($\text{B}$): $H_2O_{(l)}$

  • Acid ($\text{A}$): $H_2SO_{4\,(aq)}$

  • Conjugate Acid ($\text{CA}$): $H_3O_{(aq)}^+$

  • Conjugate Base ($\text{CB}$): $HSO_{4\,(aq)}^-$

  • Reaction (e): Reaction of liquid water with aniline:     $H_2O_{(l)} + C_6H_5NH_{2\,(aq)} \rightarrow OH_{(aq)}^- + C_6H_5NH_{3\,(aq)}^-$

  • Acid ($\text{A}$): $H_2O_{(l)}$

  • Base ($\text{B}$): $C_6H_5NH_{2\,(aq)}$

  • Conjugate Base ($\text{CB}$): $OH_{(aq)}^-$

  • Conjugate Acid ($\text{CA}$): $C_6H_5NH_{3\,(aq)}^+$

Amphiprotic Substances and the Autoionization Equilibrium of Water

• Amphiprotic Compounds: Chemical species capable of behaving as both an acid (proton donor) and a base (proton acceptor) depending on the reacting partner. In the set of reactions examined, $H_2O$ is the only species exhibiting this dual behavior.

• Autoionization of Water: Water exhibits amphiprotic behavior because it can autoionize through self-reaction:   $H_2O + H_2O \rightleftharpoons H_3O^+ + OH^-$

• Water-Dissociation Equilibrium Constant ($K_w$):

  • The equilibrium constant expression for the autoionization of water is represented as:     $K_w = [H_3O^+][OH^-]$

  • At a standard temperature of $25\,^{\circ}\text{C}$, the numerical value of this equilibrium constant is:     $K_w = 1.0 \times 10^{-14}$

Logarithmic Scales, Significant Figures, and Mathematical Relationships

• The General Logarithmic Operator ($pX$):

  • In acid-base chemistry, the operator $p$ represents the negative base-10 logarithm:     $pX = -\log(X)$

  • For concentration-based expressions, concentrations are evaluated in units of $mol\,dm^{-3}$, though the units are omitted during logarithmic computation, rendering logarithmic parameters strictly dimensionless/unitless.

• Operational Definitions:

  • Hydronium ion potential:     $pH = -\log([H_3O^+])$

  • Hydroxide ion potential:     $pOH = -\log([OH^-])$

  • Autoionization constant potential:     $pK_w = -\log(K_w)$

• Significant Figure Rules for Logarithmic Quantities:

  • The number of significant figures in the original concentration corresponds directly to the number of digits situated to the right of the decimal point in the calculated logarithm (termed the mantissa).

  • The digits to the left of the decimal point represent the magnitude order (the characteristic) and do not count toward significant figures.

  • Example: For a hydronium concentration of $[H_3O^+] = 7.1 \times 10^{-3}\,mol\,dm^{-3}$ (which contains 2 significant figures), the corresponding pH is $2.15$, displaying exactly 2 decimal places to reflect the 2 significant figures.

• Derivation of the Fundamental Relation ($pK_w = pH + pOH$):

  • Using the logarithmic product identity:     $\log(A \times B) = \log(A) + \log(B)$

  • Taking the negative logarithm of both sides of the autoionization equilibrium expression:     $-\log(K_w) = -\log([H_3O^+] \times [OH^-])$     $-\log(K_w) = -\log([H_3O^+]) + (-\log([OH^-]))$

  • Substituting defined parameters yields:     $pK_w = pH + pOH$

  • At $25\,^{\circ}\text{C}$:     $pK_w = -\log(1.0 \times 10^{-14}) = 14.00$     $14.00 = pH + pOH$

• Solution Classification Benchmarks at $25\,^{\circ}\text{C}$:

  • Neutral Solution:     $pH = 7$     $pOH = 7$

  • Acidic Solution:     $pH < 7$

  • Basic Solution:     $pH > 7$

Acidity, Basicity, and Ion Calculations in Chemical Solutions

• Determination of pH, pOH, and Solution Nature from Known Concentration:

  • Milk ($[H_3O^+] = 3.2 \times 10^{-7}\,mol\,dm^{-3}$):     $pH = -\log(3.2 \times 10^{-7}) = 6.49$     $pOH = 14.00 - 6.49 = 7.51$

  • Classification: Characterized as approximately neutral ($\sim\text{neutral}$) / slightly acidic.

  • Pickle Juice ($[H_3O^+] = 2.0 \times 10^{-4}\,mol\,dm^{-3}$):     $pH = -\log(2.0 \times 10^{-4}) = 3.70$     $pOH = 14.00 - 3.70 = 10.30$

  • Classification: Strongly acidic.

  • Beer ($[H_3O^+] = 3.2 \times 10^{-5}\,mol\,dm^{-3}$):     $pH = -\log(3.2 \times 10^{-5}) = 4.49$     $pOH = 14.00 - 4.49 = 9.51$

  • Classification: Acidic.

  • Blood ($[H_3O^+] = 4.0 \times 10^{-8}\,mol\,dm^{-3}$):     $pH = -\log(4.0 \times 10^{-8}) = 7.40$     $pOH = 14.00 - 7.40 = 6.60$

  • Classification: Characterized as approximately neutral ($\sim\text{neutral}$) / slightly basic.

• Complete Dissociation Calculation for a Strong Monoprotic Acid:

  • Solution of $0.125\,mol\,dm^{-3}\,HCl$:     $HCl_{(aq)} \rightarrow H_{(aq)}^+ + Cl_{(aq)}^- = 0.125\,mol\,dm^{-3}\,H^+ + 0.125\,mol\,dm^{-3}\,Cl^-由此可知$     $[H_3O^+] = 0.125\,mol\,dm^{-3}$     $pH = -\log(0.125) = 0.903$     $pOH = 14.00 - 0.903 = 13.097$

• Determination of Ion Concentrations from Known pH Values:

  • Lime Juice ($pH = 1.9$):     $[H_3O^+] = 10^{-1.9} = 0.013\,mol\,dm^{-3}$     $[OH^-] = \frac{1.0 \times 10^{-14}}{0.013} = 7.9 \times 10^{-13}\,mol\,dm^{-3}$

  • Classification: Acidic.

  • Tomato Juice ($pH = 4.2$):     $[H_3O^+] = 10^{-4.2} = 6.3 \times 10^{-5}\,mol\,dm^{-3}$     $[OH^-] = \frac{1.0 \times 10^{-14}}{6.3 \times 10^{-5}} = 1.6 \times 10^{-10}\,mol\,dm^{-3}$

  • Classification: Acidic.

  • Saliva ($pH = 7.0$):     $[H_3O^+] = 10^{-7.0} = 1.0 \times 10^{-7}\,mol\,dm^{-3}$     $[OH^-] = \frac{1.0 \times 10^{-14}}{1.0 \times 10^{-7}} = 1.0 \times 10^{-7}\,mol\,dm^{-3}$

  • Classification: Neutral.

  • Kitchen Cleanser ($pH = 9.3$):     $[H_3O^+] = 10^{-9.3} = 5.0 \times 10^{-10}\,mol\,dm^{-3}$     $[OH^-] = \frac{1.0 \times 10^{-14}}{5.0 \times 10^{-10}} = 2.0 \times 10^{-5}\,mol\,dm^{-3}$

  • Classification: Basic.

Ion Concentration Calculations in Strong Base Solutions

• Molarity Determination of Hydroxide Ions:

  • For a solution containing $0.100\,mol$ of $NaOH$ dissolved in a total volume of $125\,cm^3$ ($0.125\,dm^3$):     $[OH^-] = \frac{0.100\,mol}{0.125\,dm^3} = 0.800\,mol\,dm^{-3}$

• Coupled Hydronium Concentration via Autoionization Constant:

  • Applying the equilibrium expression for water dissociation:     $[H_3O^+] = \frac{K_w}{[OH^-]}$     $[H_3O^+] = \frac{1.0 \times 10^{-14}}{0.800\,mol\,dm^{-3}} = 1.25 \times 10^{-14}\,mol\,dm^{-3}$

Dilute Strong Acid Solutions and Autoionization Corrections

• The Apparent Paradox of Extremely Dilute Strong Acids:

  • When adding enough $HCl$ to pure water to create an acid concentration of $[HCl] = 1.0 \times 10^{-8}\,mol\,dm^{-3}$, a naive single-step logarithmic calculation yields:     $pH = -\log(1.0 \times 10^{-8}) = 8.00$

  • Physical Inconsistency: This result is physically impossible because adding an acid to pure water cannot yield a basic pH ($pH > 7$).

• Accounting for the Autoionization of Pure Water:

  • Pure water contains an inherent concentration of $[H^+]$ due to continuous autoionization.

  • Given that in pure water, $[H^+] = [OH^-]$:     $K_w = [H^+]^2$     $[H^+] = \sqrt{1.0 \times 10^{-14}} = 1.0 \times 10^{-7}\,mol\,dm^{-3}$

• Total Hydronium Ion Concentration Summation:

  • The total concentration of protons in solution must account for both the water solvent contribution and the dissolved acid contribution:     $[H^+]_{\text{total}} = [H^+]_{\text{water}} + [H^+]_{\text{acid}}$     $[H^+]_{\text{total}} = 1.0 \times 10^{-7}\,mol\,dm^{-3} + 1.0 \times 10^{-8}\,mol\,dm^{-3} = 1.1 \times 10^{-7}\,mol\,dm^{-3}$

• Corrected pH Determination:

  • Calculating pH from the total proton concentration yields:     $pH = -\log(1.1 \times 10^{-7}) = 6.96$

  • This corrected value properly reflects a solution that is slightly acidic ($pH < 7$), reconciling the physical expectation with quantitative equilibrium behavior.