October 21th Chem Lecture Equilibrium Approximations, Spectrophotometry Principles, and Salt Hydrolysis Equilibria

The Small-x Approximation and the 5% Rule of Thumb

• The small-$x$ approximation is an algebraic simplification used in equilibrium calculations to determine whether the change in concentration ($x$) is negligible relative to the initial concentration of a reactant.

• A mechanical model illustrates the fractional consumption and percentage check:

  • Consider a pile containing an initial quantity of $100$ apples.

  • Consuming $1$ apple from the pile leaves the system with a consumed quantity of $1$.

  • The consumed fraction is calculated by dividing the consumed quantity by the initial total quantity:     $\text{Fraction} = \frac{1}{100} = 0.01$

  • Converting this fraction to a percentage requires multiplying by $100\%$:     $\text{Percentage} = 0.01 \times 100\% = 1\%$

  • Consuming $1$ apple out of $100$ corresponds to consuming exactly $1\%$ of the initial amount.

• Chemical equilibrium context for an acid ionization reaction:   $\text{HA} + \text{H}_2\text{O} \rightleftharpoons \text{A}^- + \text{H}_3\text{O}^+$

  • The initial concentration of the weak acid $[\text{HA}]_0$ corresponds to the initial total quantity ($100$ apples).

  • The concentration of conjugate base produced ($[\text{A}^-]$) corresponds to the quantity of reactant consumed.

  • When the acid ionization constant ($K_a$) is significantly large (e.g., $K_a \gg 1$), the reaction proceeds heavily toward products, consuming a substantial amount of reactant and generating a high concentration of $[\text{A}^-]$.

  • When $K_a$ is small (e.g., on the order of $10^{-5}$), the reaction consumes very little reactant; the vast majority of molecules remain on the reactant side, meaning $[\text{HA}]$ does not change significantly, yielding only a minor concentration of $[\text{A}^-]$.

• ICE Table setup for acid ionization:

  • Initial: $[\text{HA}]_0 = 100$, $[\text{A}^-]_0 = 0$, $[\text{H}_3\text{O}^+]_0 = 0$.

  • Change: $-x$ for the acid, $+x$ for each generated product.

  • Equilibrium: $[\text{HA}] = 100 - x$, $[\text{A}^-] = x$, $[\text{H}_3\text{O}^+] = x$.

  • The equilibrium constant expression is:     $K_a = \frac{x \times x}{100 - x}$

• Validity criteria for dropping $x$ in the denominator ($100 - x \approx 100$):

  • Compute the ratio of the change ($x$) to the initial concentration:     $\text{Fraction} = \frac{x}{[\text{HA}]_0}$

  • Multiply the resulting fraction by $100\%$ to determine the percent ionization:     $\text{Percent Check} = \frac{x}{[\text{HA}]_0} \times 100\%$

  • Under the standard rule of thumb, if the resulting value is less than $5\%$ ($< 5\%$), the small-$x$ approximation is fully valid, and the $x$ term in the denominator may be safely neglected.

Spectrophotometry, Light Intensity, and the Beer-Lambert Law

• Spectrophotometer instrumentation and measurement principles:

  • Basic laboratory spectrophotometers measure absorbance at single, discrete wavelengths rather than acquiring continuous spectral scans.

  • Research-grade spectrophotometers hold samples in standard cuvettes with an optical path length of $1 \times 1\,\text{cm}$.

  • Incident light with an initial intensity ($I_0$) enters the cuvette, traverses the sample solution, and exits with a transmitted intensity ($I$).

  • If a solution exhibits zero absorbance, all entering light exits without attenuation, resulting in:     $I = I_0$

  • If a solution absorbs light, the transmitted light intensity is reduced relative to the incident intensity, giving:     $I < I_0$

  • Consequently, the transmitted light fraction (transmittance ratio) is less than one:     $\frac{I}{I_0} < 1$

• Definition and dimensional analysis of absorbance ($A$):

  • Absorbance is defined logarithmically as:     $A = -\log\left(\frac{I}{I_0}\right)$

  • Because the ratio $\frac{I}{I_0} < 1$, its logarithm is negative; the leading negative sign guarantees that absorbance is reported as a positive number.

  • Detectors such as photomultiplier tubes (PMTs) or charge-coupled devices (CCDs) measure light intensity as electrical signals possessing physical units (e.g., $\text{V}$ or $\text{mV}$).

  • Taking the ratio of intensities allows the physical units to cancel out completely:     $\frac{[\text{Intensity Units}]}{[\text{Intensity Units}]} = 1$

  • Mathematical transcendental functions (including logarithmic functions) cannot take arguments that carry physical units; the argument must be strictly dimensionless.

  • Because the argument to the logarithm is dimensionless, absorbance ($A$) is an entirely unitless quantity.

• The Beer-Lambert Law formulation:   $A = \epsilon \cdot c \cdot l$

  • $A$ = Absorbance (dimensionless).

  • $l$ = Optical path length through the cuvette ($\text{cm}$), typically $1\,\text{cm}$.

  • $c$ = Molar concentration of absorbing solute ($\text{mol}\,\text{L}^{-1}$ or $\text{mol}\,\text{dm}^{-3}$).

  • $\epsilon$ = Molar extinction coefficient (molar absorptivity).

  • Unit balance of the Beer-Lambert equation:     $\text{Units of } \epsilon = \frac{1}{\left(\text{mol}\,\text{L}^{-1}\right) \times \text{cm}} = \text{L}\,\text{mol}^{-1}\,\text{cm}^{-1}$

  • Multiplying $\text{L}\,\text{mol}^{-1}\,\text{cm}^{-1}$ by $\text{mol}\,\text{L}^{-1}$ and $\text{cm}$ cancels every unit, producing a unitless absorbance value.

General Principles of Salt Dissociation and Hydrolysis

• Aqueous salt behavior is analyzed via a systematic two-step procedure:

  • Step 1: Identify the exact cation and anion species produced upon complete dissociation of the salt in water.

  • Step 2: Evaluate each ion individually to determine whether it behaves as a conjugate acid, a conjugate base, or a spectator ion with respect to water.

• Classification rules for ionic components:

  • Metal cations derived from strong bases (e.g., $\text{Na}^+$ from $\text{NaOH}$) do not accept or donate protons in aqueous media and act as neutral spectator ions.

  • Anions derived from strong monoprotic acids (e.g., $\text{Cl}^-$ from $\text{HCl}$) have negligible base strength, do not hydrolyze water, and act as neutral spectator ions.

  • Cations containing acidic protons derived from weak bases (e.g., $\text{NH}_4^+$) act as weak acids and hydrolyze water to yield hydronium ions ($\text{H}_3\text{O}^+$).

  • Anions derived from weak acids (e.g., $\text{CH}_3\text{CO}_2^-$) act as weak conjugate bases and hydrolyze water to yield hydroxide ions ($\text{OH}^-$).

Acid-Base Neutrality of Sodium Chloride Solutions

• Solid sodium chloride ($\text{NaCl}$) dissociates completely upon introduction to an aqueous environment (such as plain water or soup):   $\text{NaCl}_{(s)} \rightarrow \text{Na}^+_{(aq)} + \text{Cl}^-_{(aq)}$

• Chemical evaluation of the sodium cation ($\text{Na}^+$):

  • The hypothetical conjugate base pair for sodium is sodium hydroxide ($\text{NaOH}$).

  • Sodium hydroxide is a strong base that dissociates completely into $\text{Na}^+$ and $\text{OH}^-$; undissociated $\text{NaOH}$ does not exist in aqueous solution.

  • Setting up the equilibrium reaction with water:     $\text{Na}^+ + 2\,\text{H}_2\text{O} \rightleftharpoons \text{NaOH} + \text{H}_3\text{O}^+$

  • Substituting $\text{NaOH} \rightarrow \text{Na}^+ + \text{OH}^-$ causes $\text{Na}^+$ to appear on both sides of the expression, algebraically canceling out.

  • The net process collapses to the autoionization of water:     $2\,\text{H}_2\text{O} \rightleftharpoons \text{H}_3\text{O}^+ + \text{OH}^-$

  • For the reverse association reaction, the equilibrium constant is the reciprocal of the water autoionization constant ($K_w$):     $K = \frac{1}{K_w} = \frac{1}{10^{-14}} = 1 \times 10^{14}$

  • Because this constant is extraordinarily large ($1 \times 10^{14}$), the equilibrium lies entirely toward the product water molecules.

  • Free aqueous $\text{Na}^+$ ions do not react with water or alter hydronium and hydroxide concentrations.

• Chemical evaluation of the chloride anion ($\text{Cl}^-$):

  • Chloride represents the conjugate base of hydrochloric acid ($\text{HCl}$).

  • Hydrochloric acid is a strong acid that ionizes completely in water; undissociated $\text{HCl}$ does not exist in aqueous solution:     $\text{HCl} + \text{H}_2\text{O} \rightarrow \text{Cl}^- + \text{H}_3\text{O}^+$

  • The forward ionization constant ($K_a$) for $\text{HCl}$ is exceptionally large ($K_a \gg 1$).

  • Formulating the base hydrolysis of chloride with water:     $\text{Cl}^- + \text{H}_2\text{O} \rightleftharpoons \text{HCl} + \text{OH}^-$

  • The base equilibrium constant ($K_b$) for $\text{Cl}^-$ is related to the conjugate acid constant ($K_a$) via $K_w$:     $K_b = \frac{K_w}{K_a}$

  • Given the enormous magnitude of $K_a$ for $\text{HCl}$, the resulting base constant is approximately:     $K_b \approx 10^{-20}$

  • An equilibrium constant of $10^{-20}$ is so infinitesimally small that no measurable products ($\text{HCl}$ or $\text{OH}^-$) are generated.

• Solution impact:

  • Neither $\text{Na}^+$ nor $\text{Cl}^-$ undergoes hydrolysis in water.

  • Adding $\text{NaCl}$ to an aqueous system leaves the concentrations of $\text{H}_3\text{O}^+$ and $\text{OH}^-$ entirely unchanged at $10^{-7}\,\text{mol}\,\text{dm}^{-3}$.

  • The pH of water or soup remains fixed at neutral ($\text{pH} = 7$), regardless of the amount of $\text{NaCl}$ added.

Hydrolysis of Ammonium Chloride and Acidic Salt Behavior

• Complete dissociation of solid ammonium chloride in water:   \text{NH}_4\text{Cl}_{(s)} \rightarrow \text{NH}_4^+_{(aq)} + \text{Cl}^-_{(aq)}

• Evaluation of the constituent ions:

  • As established, $\text{Cl}^-$ is the conjugate base of a strong acid ($\text{HCl}$), undergoes no hydrolysis, and contributes nothing to the pH.

  • The ammonium ion ($\text{NH}_4^+$) possesses an acidic proton and functions as a weak Brønsted-Lowry acid in aqueous solution by donating a proton to water:     $\text{NH}_4^+ + \text{H}_2\text{O} \rightleftharpoons \text{NH}_3 + \text{H}_3\text{O}^+$

• Determining the acid ionization constant ($K_a$) of $\text{NH}_4^+$:

  • Standard reference tables in chemistry textbooks generally do not report $K_a$ values directly for $\text{NH}_4^+$.

  • Reference tables do list the base ionization constant ($K_b$) for the conjugate base ammonia ($\text{NH}_3$).

  • The $K_a$ of $\text{NH}_4^+$ is calculated using the conjugate relationship:     $K_a = \frac{K_w}{K_b}$

• Impact on solution pH:

  • Although the numerical value of $K_a$ for $\text{NH}_4^+$ is relatively small, it is sufficiently large to establish an active equilibrium that produces hydronium ions ($\text{H}_3\text{O}^+$).

  • The net concentration of $[\text{H}_3\text{O}^+]$ in the solution increases ($[\text{H}_3\text{O}^+] > 10^{-7}\,\text{mol}\,\text{dm}^{-3}$).

  • As hydronium concentration increases, the solution becomes acidic, and the pH drops below $7$ ($\text{pH} < 7$).

Hydrolysis of Sodium Acetate and Basic Salt Calculations

• Dissociation behavior of sodium acetate:

  • Sodium acetate ($\text{NaCH}_3\text{CO}_2$) dissociates completely upon dissolution:     \text{NaCH}_3\text{CO}_2{(s)} \rightarrow \text{Na}^+_{(aq)} + \text{CH}_3\text{CO}_2^-_{(aq)}

  • The sodium cation ($\text{Na}^+$) is an unreactive spectator ion that does not participate in acid-base equilibria.

  • The acetate anion ($\text{CH}_3\text{CO}_2^-$) is the conjugate base of acetic acid ($\text{CH}_3\text{COOH}$) and acts as a weak base by accepting a proton from water:     $\text{CH}_3\text{CO}_2^- + \text{H}_2\text{O} \rightleftharpoons \text{CH}_3\text{COOH} + \text{OH}^-$

• Derivation of the base ionization constant ($K_b$):

  • Textbooks tabulate the acid dissociation constant for acetic acid ($K_a$).

  • The base constant ($K_b$) for acetate is computed via:     $K_b = \frac{K_w}{K_a}$

  • Evaluating this expression yields:     $K_b = 5.6 \times 10^{-10}$

• Quantitative equilibrium determination of pH for a $0.01\,\text{M}$ $\text{NaCH}_3\text{CO}_2$ solution:

  • Complete stoichiometric dissociation of a $0.01\,\text{M}$ solution yields an initial acetate concentration of:     $[\text{CH}_3\text{CO}_2^-]_0 = 0.01\,\text{M}$

  • ICE table setup for acetate hydrolysis:

  • Initial concentrations: $[\text{CH}_3\text{CO}_2^-]_0 = 0.01\,\text{M}$, $[\text{CH}_3\text{COOH}]_0 = 0$, $[\text{OH}^-]_0 \approx 0$.

  • Changes in concentration: $-x$ for $\text{CH}_3\text{CO}_2^-$, $+x$ for $\text{CH}_3\text{COOH}$, $+x$ for $\text{OH}^-$.

  • Equilibrium concentrations: $[\text{CH}_3\text{CO}_2^-] = 0.01 - x$, $[\text{CH}_3\text{COOH}] = x$, $[\text{OH}^-] = x$.

  • The equilibrium constant expression is:     $K_b = \frac{x \times x}{0.01 - x} = 5.6 \times 10^{-10}$

  • Applying the small-$x$ approximation assumes $x \ll 0.01$, simplifying the denominator to $0.01$:     $\frac{x^2}{0.01} = 5.6 \times 10^{-10}$

  • Solving for $x$ directly yields the equilibrium hydroxide ion concentration:     $x^2 = 5.6 \times 10^{-12}$     $x = [\text{OH}^-] = 2.37 \times 10^{-6}\,\text{M}$

  • Computing the pOH of the basic solution:     $\text{pOH} = -\log([\text{OH}^-])$

  • Computing pH from pOH:     $\text{pH} = 14 - \text{pOH} = 8.4$

  • The formation of excess hydroxide ions ($\text{OH}^-$) shifts the aqueous solution from neutral ($\text{pH} = 7.0$) to basic ($\text{pH} = 8.4$).

Questions & Discussion

• Question / Prompt: How do you verify whether the small-$x$ approximation check is mathematically valid?

  • Clarification: The small-$x$ approximation check represents a percentage calculation. One takes the calculated change value ($x$), divides it by the initial starting concentration of the substance, and multiplies the resulting decimal by $100\%$. If that calculated percentage is strictly less than $5\%$, dropping $x$ from the denominator is valid.

• Question / Prompt: Does dissolving sodium chloride ($\text{NaCl}$) into a bowl of soup change its pH?

  • Student Response: A student suggested that the pH would probably increase.

  • Follow-up Inquiry / Prompt: A request was made for students to defend the opposing position (that pH does not change) or provide reasoning.

  • Student Response / Hypothesis: A student suggested that upon dissolving $\text{NaCl}$, the $\text{Cl}^-$ ion would react with water to form $\text{HCl}$, but because $\text{HCl}$ is a strong acid, it dissociates.

  • Resolution / Clarification: While the hypothetical base reaction $\text{Cl}^- + \text{H}_2\text{O} \rightleftharpoons \text{HCl} + \text{OH}^-$ can be formally written, its equilibrium constant is $K_b \approx 10^{-20}$. This value is far too tiny for any products to form. Because neither $\text{Na}^+$ nor $\text{Cl}^-$ reacts with water, adding an entire package of salt to soup will make it inedible, but the pH remains entirely unchanged.