5.1 Stoichiometry, Analytical Chemistry, and Chemical Equilibrium

Stoichiometric calculations form the bedrock of chemical engineering material balances. When chemical reactions occur, species are consumed or generated in fixed molar ratios determined by the balanced equation. The limiting reactant is completely consumed first if a reaction goes to completion. Reactants present in quantities greater than their stoichiometric requirements are in excess; the fractional excess is the moles of reactant fed minus the stoichiometric moles required, divided by the stoichiometric moles required. For reactant A, the fractional conversion (f_A) is: ( f_A = \frac{n_{A,0} - n_A}{n_{A,0}} ) where (n_{A,0}) and (n_A) are the initial and final molar flow rates. To generalize reacting systems, the extent of reaction (\xi) is used. The molar flow rate of any species (i) is: ( n_i = n_{i,0} + \nu_i \xi ) where (\nu_i) is the stoichiometric coefficient (negative for reactants, positive for products). Yield is the moles of desired product formed divided by the moles of limiting reactant fed, while selectivity is the ratio of desired to undesired product.

Worked Example: Methane Combustion

Consider the combustion of methane:

[ \text{CH}_4 + 2\text{O}_2 \rightarrow \text{CO}_2 + 2\text{H}_2\text{O} ]

If 10 mol/s of (\text{CH}_4) and 30 mol/s of (\text{O}_2) are fed to a reactor, and the conversion of methane is 0.80, calculate the leaving molar flow rates. The stoichiometric ratio of (\text{O}_2) to (\text{CH}_4) is 2:1. The feed ratio is 30/10 = 3:1. Since the feed contains more (\text{O}_2) than required, (\text{CH}_4) is the limiting reactant and (\text{O}_2) is in excess. The stoichiometric amount of (\text{O}_2) required is 20 mol/s. The excess flow rate of (\text{O}2) is 10 mol/s (50% excess). At 80% conversion, the extent of reaction is (\xi = 8) mol/s. The outlet flow rates are: ( n{\text{CH}4} = 10 - 8 = 2 ) mol/s, ( n{\text{O}2} = 30 - 2(8) = 14 ) mol/s, ( n{\text{CO}2} = 8 ) mol/s, and ( n{\text{H}_2\text{O}} = 16 ) mol/s. The total outlet flow rate is 40 mol/s.

The table below summarizes the material balance for this methane combustion system:

Chemical Equilibrium

Many chemical reactions do not proceed to 100% completion; instead, they reach a state of dynamic chemical equilibrium where the forward and reverse reaction rates are equal. For a general gas-phase reaction:

[ a\text{A} + b\text{B} \rightleftharpoons c\text{C} + d\text{D} ]

The equilibrium constant in terms of concentrations is K_c:

[ K_c = \frac{[\text{C}]^c [\text{D}]^d}{[\text{A}]^a [\text{B}]^b} ]

For gas-phase systems, using partial pressures defines K_p:

[ K_p = \frac{P_{\text{C}}^c P_{\text{D}}^d}{P_{\text{A}}^a P_{\text{B}}^b} ]

The relationship between K_p and K_c is: ( K_p = K_c (RT)^{\Delta n} ), where (\Delta n = (c + d) - (a + b)) is the change in gas moles, R is the universal gas constant, and T is temperature. According to Le Chatelier's principle, if a system at equilibrium is disturbed, it shifts to counteract the disturbance. Increasing pressure shifts the equilibrium toward the side with fewer moles of gas. The effect of temperature depends on the sign of the standard enthalpy of reaction. For endothermic reactions, adding heat shifts the equilibrium to the right, increasing K. For exothermic reactions, adding heat shifts the equilibrium to the left, decreasing K. Temperature dependence is given by the van 't Hoff equation:

[ \ln\left(\frac{K_2}{K_1}\right) = -\frac{\Delta H^0}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right) ]

Acid-Base Chemistry and pH

In aqueous solutions, the autoionization of water establishes a fundamental equilibrium: ( \text{H}2\text{O} \rightleftharpoons \text{H}^+ + \text{OH}^- ). At 25°C, ( K_w = [\text{H}^+][\text{OH}^-] = 1.0 \times 10^{-14} ), leading to ( \text{pH} + \text{pOH} = 14.0 ) where ( \text{pH} = -\log{10}[\text{H}^+] ) and ( \text{pOH} = -\log_{10}[\text{OH}^-] ). Strong acids and bases dissociate completely in water. For a weak monoprotic acid HA, dissociation is incomplete: ( \text{HA} \rightleftharpoons \text{H}^+ + \text{A}^- ). The acid dissociation constant K_a is: ( K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]} ). For a weak acid solution of initial concentration C_0, if (x \ll C_0), the approximation (C_0 - x \approx C_0) yields (x = [\text{H}^+] \approx \sqrt{K_a C_0}). Buffer systems containing a weak acid and conjugate base resist pH changes; the pH is found using the Henderson-Hasselbalch equation: ( \text{pH} = \text{p}K_a + \log_{10}\left(\frac{[\text{A}^-]}{[\text{HA}]}\right) ).

The Carbonate System and pH/pC Diagrams

The carbonate system is the primary buffer in natural waters and wastewater treatment, consisting of dissolved carbon dioxide and carbonic acid (grouped as (\text{H}_2\text{CO}_3^*)), bicarbonate, and carbonate. The speciation is governed by two sequential dissociation equilibria:

[ \text{H}_2\text{CO}_3^* \rightleftharpoons \text{H}^+ + \text{HCO}_3^- \quad (\text{p}K_1 = 6.35) ]

[ \text{HCO}_3^- \rightleftharpoons \text{H}^+ + \text{CO}_3^{2-} \quad (\text{p}K_2 = 10.33) ]

A pH/pC diagram plots negative log concentrations of species against pH at fixed total carbonate. At pH < (\text{p}K_1), (\text{H}_2\text{CO}_3^*) is dominant; between (\text{p}K_1) and (\text{p}K_2), (\text{HCO}_3^-) dominates; at pH > (\text{p}K_2), (\text{CO}_3^{2-}) dominates. Line intersections correspond to (\text{p}K_a) values where conjugate species concentrations are equal.

Redox and Electrochemistry

Redox reactions involve electron transfer; oxidation is electron loss, while reduction is electron gain. Balancing redox reactions requires the half-reaction method, balancing mass and charge. Standard cell potential is: ( E^0_{\text{cell}} = E^0_{\text{reduction, cathode}} - E^0_{\text{reduction, anode}} ). The cell potential under non-standard conditions is determined by the Nernst equation: ( E = E^0 - \frac{RT}{nF}\ln Q ), where n is the moles of electrons transferred, F is Faraday's constant (96,485 C/mol), Q is the reaction quotient, R is 8.314 J/mol·K, and T is temperature. At 25°C, using base-10 logarithm, it simplifies to: ( E = E^0 - \frac{0.0592}{n}\log_{10} Q ). Faraday's Law of Electrolysis states that the mass of a substance deposited during electrolysis is: ( m = \frac{I \cdot t \cdot M}{n \cdot F} ), where I is current in amperes, t is time in seconds, M is the molar mass of the substance, and n is the valence of the ions.

Analytical Chemistry Techniques

Analytical chemistry provides quantitative and qualitative identification of chemical species and is tested through both wet (classical) and instrumental methods on the FE Chemical exam. Wet chemistry relies on stoichiometric reactions and gravimetric or volumetric measurement, while instrumental methods exploit physical properties (light absorption, electrical response, mass-to-charge ratio) for rapid, sensitive analysis.

Titration (Volumetric Analysis)

A titration determines an unknown analyte concentration by reacting it with a standard solution of known concentration (titrant) until the equivalence point is reached. For a strong acid-strong base titration the equivalence point occurs at pH 7; for a weak acid-strong base titration the equivalence point is above 7 due to conjugate-base hydrolysis; for a strong acid-weak base titration it is below 7. The half-equivalence point of a weak acid titration is diagnostic because $[\text{HA}] = [\text{A}^-]$ there, giving $\text{pH} = \text{p}K_a$. Indicators are chosen so their transition range brackets the equivalence-point pH; phenolphthalein (pH 8.2–10.0) is common for weak acid–strong base titrations, and methyl orange (pH 3.1–4.4) is used for strong acid–weak base titrations.

Gravimetric Analysis

Gravimetric analysis determines an analyte mass by precipitating it as a stable, low-solubility compound of known stoichiometry, filtering, drying, and weighing. The analyte mass is recovered from the precipitate mass using the gravimetric factor:

[ m_{\text{analyte}} = m_{\text{precipitate}} \times \frac{\text{MW}{\text{analyte}}}{\text{MW}{\text{precipitate}}} ]

For example, sulfate ($\text{SO}_4^{2-}$) is determined gravimetrically by precipitating $\text{BaSO}_4$ (MW = 233.4 g/mol) from a hot, acidic solution with $\text{BaCl}_2$; the gravimetric factor for $\text{SO}_4^{2-}$ is $96.06/233.4 = 0.4116$.

Chromatography

Chromatography separates analytes based on differential partitioning between a stationary phase and a mobile phase. The retention factor ($k$) characterizes analyte affinity for the stationary phase:

[ k = \frac{t_R - t_M}{t_M} ]

where $t_R$ is the analyte retention time and $t_M$ is the void (dead) time for an unretained species. In gas chromatography (GC), the mobile phase is an inert gas (He, $\text{N}_2$, or $\text{H}_2$) and separation is driven by analyte vapor pressure and interaction with the column coating. In high-performance liquid chromatography (HPLC), the mobile phase is a pressurized liquid and the stationary phase is typically bonded silica. Selectivity ($\alpha = k_2/k_1$) and resolution quantify peak separation:

[ R_s = \frac{2(t_{R,2} - t_{R,1})}{w_1 + w_2} ]

where $w_1, w_2$ are the baseline peak widths. Resolution $R_s \geq 1.5$ indicates baseline separation.

Spectroscopy and Instrumental Methods

Spectroscopic methods measure analyte interactions with electromagnetic radiation. UV-Vis absorption spectroscopy follows the Beer-Lambert law:

[ A = \varepsilon , b , C ]

where $A$ is absorbance (dimensionless), $\varepsilon$ is the molar absorptivity ($\text{L}/(\text{mol}\cdot\text{cm})$), $b$ is the path length (cm), and $C$ is the analyte concentration (mol/L). Absorbance is related to transmittance by $A = -\log_{10} T$. The linear $A$–$C$ relationship is the basis for quantitative calibration curves. Atomic absorption spectroscopy (AAS) measures trace metals by atomizing the sample and analyzing absorption of element-specific wavelengths. Infrared (IR) spectroscopy identifies functional groups from vibrational frequencies (carbonyl stretch near 1700 cm$^{-1}$, O–H stretch near 3300 cm$^{-1}$). Mass spectrometry (MS) quantifies the mass-to-charge ratio of ionized fragments and is often coupled to GC (GC-MS) for separation and identification.

Electroanalytical Methods

Potentiometry measures cell potential under zero-current conditions; the pH glass electrode is the most common potentiometric sensor, with its potential given by the Nernst equation applied at the membrane interface. Conductivity measurements detect total ionic strength by applying an AC voltage across electrodes of known cell constant. These instrumental methods extend wet chemistry to the trace (ppm/ppb) concentrations common in environmental and pharmaceutical analysis.