Engineering Chemistry Module 1 – Thermodynamics, Chemical Kinetics & Catalysis

Thermodynamic Fundamentals

• System & Surroundings
  • Thermodynamic system = specific matter or region in space under study.
  • Surroundings = everything external to the system.

• System types
  • Open (exchange $q$ & matter)
  • Closed (exchange $q$ but not matter)
  • Isolated (no exchange of $q$ or matter)

• Properties
  • Intensive: independent of mass (T, P, density).
  • Extensive: proportional to mass (V, $U$, $H$, $S$, $n$).

• State vs. Path functions
  • State: depend only on initial & final states (T, P, V, $U$, $H$, $S$).
  • Path: depend on path (work $W$, heat $q$).

• Internal Energy
  $U = E<em>k + E</em>p$
  • Extensive & state function.
  • Changes through transfer of $q$ and/or $W$.

Thermodynamic Processes & Work

• Special processes
  • Adiabatic: $q = 0$
  • Isothermal: $\Delta T = 0$
  • Isobaric: $\Delta P = 0$
  • Isochoric: $\Delta V = 0$

• Mechanical work for expansion/compression
  $dW = -P<em>{ex} dV$ Total work: $W = - \int</em>{V<em>i}^{V</em>f} P_{ex} dV$

• Reversible vs. Irreversible
  • Reversible: can be inverted by infinitesimal change; $P<em>{ex}=P</em>{sys}$ at every step—gives maximum work.
  • Irreversible: finite driving force; $|W<em>{rev}| > |W</em>{irrev}|$.
  • True perfect reversibility impossible in nature—conceptual ideal.

Zeroth Law & Temperature

• Zeroth Law: If A $\rightleftharpoons$ C and B $\rightleftharpoons$ C are each in thermal equilibrium, then A $\rightleftharpoons$ B are in equilibrium.
• Basis for empirical temperature scale.

First Law of Thermodynamics

• Statement: Energy is conserved.
  $\Delta U = q + w$

• Sign convention (chemistry): q>0 or w>0 when energy enters system.
  Alternative form: $\Delta U = q - w_{by}$.

• Enthalpy: $H = U + PV$ (state, extensive). At constant P:
  $\Delta H = q_p$.

• Heat Capacity
  $C = \dfrac{q}{T<em>2-T</em>1}$;
  • Specific (per g) or molar.
  • $C<em>v = \left(\dfrac{\partial U}{\partial T}\right)</em>V$, $C<em>p = \left(\dfrac{\partial H}{\partial T}\right)</em>P$.
  • For ideal gas (1 mol): $C<em>p - C</em>v = R$.
  • Mono-atomic ideal gas: $C<em>v= \tfrac{3}{2}R$, $C</em>p=\tfrac{5}{2}R$, $\gamma = \dfrac{C<em>p}{C</em>v}=\tfrac{5}{3}$.

Work in Specific Paths (Ideal Gas Examples)

• Isothermal (\Delta U=0)
  • Reversible: $W = nRT \ln \dfrac{V<em>f}{V</em>i} = nRT \ln \dfrac{P<em>i}{P</em>f}$.
  • Free expansion in vacuum: $W = 0$.
  • Constant $P<em>{ex}$: $W = -P</em>{ex}(V<em>f - V</em>i)$.

• Adiabatic (q=0)
  • Reversible ideal gas relations:
  $TV^{\gamma-1}=\text{const},\; PV^{\gamma}=\text{const}$
  $W = \dfrac{C<em>v (T</em>i-T<em>f)}{1-\gamma} = C</em>v( T<em>i - T</em>f)$ (signs!)
  • Irreversible free expansion: $\Delta T=0, W=0$.

• Isochoric: $W=0, q = \Delta U$.

• Isobaric: $W = -P (V<em>f - V</em>i), q = \Delta H$.

Illustrative Calculations (First-Law Numericals)

• Vaporisation of $\text{H}_2\text{O}$ at $100^\circ\text{C}$: Work $\approx -3.1$ kJ mol$^{-1}$, $\Delta U\approx 37.6$ kJ mol$^{-1}$.
• Expansion of 1 mol gas from 10 dm$^3$ to 30 dm$^3$ at 1 atm: $W=-2.03$ kJ.
• Reversible adiabatic doubling of volume (monatomic): compute $q=0, W=\Delta U$ with $C_v=\tfrac32R$ → $\Delta U=-1.38$ kJ.
• Reversible isothermal expansion (10 mol, 300 K, 10→2 atm): $|W|=40.1$ kJ.

Second Law of Thermodynamics & Entropy

• Directionality of spontaneous change: total entropy of an isolated system increases: dS_{tot} > 0.
• Differential definition: $dS = \dfrac{dq_{rev}}{T}$ (state function).
• Isothermal reversible expansion of ideal gas:
  $\Delta S = nR \ln \dfrac{V<em>f}{V</em>i}$.
• Irreversible free expansion: \Delta S{sys}=nR \ln \dfrac{Vf}{Vi},\; \Delta S{surr}=0 \Rightarrow \Delta S_{tot}>0.
• Entropy change formulas (ideal gas)
  • General $T,V$ change: $\Delta S = C<em>v \ln \dfrac{T</em>f}{T<em>i} + R \ln \dfrac{V</em>f}{V<em>i}$ • $T,P$ change: $\Delta S = C</em>p \ln \dfrac{T<em>f}{T</em>i} - R \ln \dfrac{P<em>f}{P</em>i}$
  • Isochoric: $C<em>v \ln \dfrac{T</em>f}{T<em>i}$; Isobaric: $C</em>p \ln \dfrac{T<em>f}{T</em>i}$.

Heat Engines & Carnot Cycle

• Heat engine: converts heat $q<em>H$ absorbed at $T</em>H$ into work $W$, rejecting $q<em>C$ at $T</em>C$.

• Carnot cycle (reversible ideal-gas model)

  1. Isothermal expansion at $T<em>H$: $q</em>H = -W<em>1 = R T</em>H \ln \dfrac{V<em>B}{V</em>A}$.
  2. Adiabatic expansion to $T_C$.
  3. Isothermal compression at $T<em>C$: $q</em>C = -R T<em>C \ln \dfrac{V</em>B}{V_A}$ (released).
  4. Adiabatic compression back to $T<em>H$. • Net work: $W = q</em>H + q<em>C = R (T</em>H - T<em>C) \ln \dfrac{V</em>B}{V<em>A}$. • Thermal efficiency: $\eta = 1 - \dfrac{T</em>C}{T_H}$ (max possible).
     • No real engine can exceed Carnot efficiency (Kelvin–Planck statement).

• Practical implications
  • Power-plant efficiencies limited by exhaust temperature & irreversibilities; e.g., $T<em>H=573$ K, $T</em>C=293$ K → ideal $\eta$ = 0.489 (48.9 %).
  • Refrigerators/heat pumps = Carnot cycle run backward.

Free Energy Criteria

• Helmholtz energy: $A = U - TS$; at constant $V, T$: process spontaneous if $dA \le 0$.
• Gibbs energy: $G = H - TS$; at constant $P, T$: spontaneity if $dG \le 0$.
• Links to 2nd law via Clausius inequality: $dS_{tot} = dS - \dfrac{dq}{T} \ge 0$.
• Endothermic reactions (dH>0) can be spontaneous when accompanied by sufficient $+TdS$.

Third Law of Thermodynamics

• Entropy of a perfect crystal at $T=0$ K is zero: $S=0$.
• Statistical interpretation: $S = k \ln W$; at 0 K, $W=1$.
• Basis for tabulating absolute entropies and evaluating $\Delta S^\circ$ values.

Chemical Kinetics – Overview

• Scope: rates of chemical change, influencing factors (T, P, catalysts), and reaction mechanisms.
• Distinct from thermodynamics (which addresses direction & equilibria, not speed).

Rate Expressions & Differential Rate Laws

• For $aA + bB \to cC + dD$:
  $\text{rate} = -\dfrac1a \dfrac{d[A]}{dt} = -\dfrac1b \dfrac{d[B]}{dt} = \dfrac1c \dfrac{d[C]}{dt} = \dfrac1d \dfrac{d[D]}{dt}$.

• Rate law (empirical): $\text{rate}=k[A]^m[B]^n$.
  • $m,n$ = partial orders; $m+n$ = overall order.
  • Determined experimentally—cannot be deduced from stoichiometry except for elementary steps.
  • $k$ depends on T (Arrhenius), independent of concentration; units vary with order.

Integrated Rate Laws & Half-Life

• Zero-order: $d[A]/dt=-k$ ⇒ $[A]<em>t=[A]</em>0 - kt$. Half-life $t<em>{1/2}=\dfrac{[A]</em>0}{2k}$.

• First-order: $\ln\dfrac{[A]<em>t}{[A]</em>0}=-kt$ ⇒ straight line $\ln[A]$ vs $t$.
  Half-life independent of $[A]<em>0$: $t</em>{1/2}=\dfrac{0.693}{k}$.

• Second-order (single reactant): $\dfrac1{[A]<em>t}-\dfrac1{[A]</em>0}=kt$; $t<em>{1/2}=\dfrac{1}{k[A]</em>0}$.

• Pseudo-first-order: reaction with two reactants where one is in large excess so its concentration appears constant; effective rate law becomes first-order. Example: hydrolysis of ethyl acetate in aqueous acid/base.

Arrhenius Equation & Collision Theory

• Arrhenius: $k = A e^{-E<em>a/RT}$; logarithmic: $\ln k = \ln A - \dfrac{E</em>a}{RT}$.
  • Plot $\ln k$ vs $1/T$ → slope $-E<em>a/R$, intercept $\ln A$. • Higher $E</em>a$ ⇒ stronger T-dependence.

• Physical meaning
  • $A$: frequency factor (collision frequency × orientation probability).
  • $E_a$: minimum kinetic energy along reaction coordinate to reach transition state.

• Collision theory postulates: (1) collisions per unit time ∝ rate; (2) proper orientation; (3) energy ≥ $E<em>a$. • Only fraction $e^{-E</em>a/RT}$ are energetic enough.

• Potential-energy profile shows reactants → activated complex (transition state, maximum) → products. Catalyst lowers $E_a$ → larger $k$.

Catalysis – General Concepts

• Definition: substance that alters reaction rate, emerges unchanged in mass & chemical composition after reaction; phenomenon = catalysis.
• Features:
  • Not consumed; small amounts suffice.
  • Do not change equilibrium constant; simply reach equilibrium faster.
  • Provide alternative pathway with lower $E_a$ (or sometimes higher, for inhibitors).

Heterogeneous Catalysis

• Catalyst phase ≠ reactant phase (often solid with gaseous/liquid reactants).
• Occurs at interface; involves adsorption → surface reaction → desorption.
• Industrial examples:
  • Haber–Bosch $\text{N}<em>2 + 3\text{H}</em>2 \xrightarrow{Fe} 2\text{NH}<em>3$. • Contact process $2\text{SO}</em>2 + \tfrac12\text{O}<em>2 \xrightarrow{V</em>2O<em>5/Pt} 2\text{SO}</em>3$.
  • Catalytic cracking, polymerisation (Zeolites, Ziegler–Natta).
  • Decomposition of $\text{H}<em>2\text{O}</em>2$ by $\text{MnO}_2$.

Homogeneous Catalysis

• Catalyst & reactants in same phase (gas or solution).
• Gas-phase examples:
  • $2\text{SO}<em>2+O</em>2 \xrightarrow{NO} 2\text{SO}<em>3$ (lead-chamber process). • Photochemical $2\text{N}</em>2\text{O} \xrightarrow{Cl<em>2,h\nu} 2\text{N}</em>2+O_2$ via radical mechanism.
• Solution examples:
  • Acid/base-catalysed ester hydrolysis; sucrose inversion by $H^+$.

Enzyme Catalysis & Michaelis–Menten Framework

• Enzymes: biological protein catalysts; active site specific to substrate (lock-and-key or induced fit).
• Mechanistic steps: $E + S \rightleftharpoons ES \rightarrow E + P$.
• Drastically lower $E<em>a$ (e.g., catalase: $E</em>a$ reduced to 8 kJ mol$^{-1}$ for $\text{H}<em>2\text{O}</em>2$ decomposition; rate ↑ by $10^{15}$(!!)).
• Characteristics: high specificity, mild conditions, subject to inhibition/activation, saturable kinetics (Michaelis constant $K_M$).
• Practical relevance: metabolism, pharmaceuticals, biosensors.

Economic & Societal Implications

• Catalysis underpins ~35 % of global GDP; essential to energy, petrochemicals, food processing, environmental remediation (e.g., automotive catalytic converters).
• Thermodynamic efficiency limits inform sustainable energy technologies; drive for higher $T<em>H$ or lower $T</em>C$ to improve power-plant performance; motivate waste-heat recovery and advanced materials.
• Entropy principle frames discussions on irreversibility & environmental impact—continuous entropy production aligns with resource degradation; necessitates efficient processes.

Quick Reference – Key Equations

• $dW = -P_{ex} dV$; $\Delta U = q + w$; $H = U + PV$.
• $C<em>p - C</em>v = R$ (ideal gas); $\gamma = C<em>p/C</em>v$.
• $\Delta S = \int<em>i^f \dfrac{dq</em>{rev}}{T}$; ideal gas forms above.
• Carnot efficiency: $\eta<em>{max}=1-\dfrac{T</em>C}{T_H}$.
• Helmholtz: $A = U - TS$; Gibbs: $G = H - TS$.
• Rate laws: $\text{rate}=k[A]^m[B]^n$.
• First-order integrated: $\ln \dfrac{[A]<em>t}{[A]</em>0}=-kt$; half-life $0.693/k$.
• Arrhenius: $k = A e^{-E<em>a/RT}$; two-point form $\ln(k</em>2/k<em>1)= -E</em>a/R\,(1/T<em>2 - 1/T</em>1)$.

Problem-Solving Pointers

• Identify process constraints (P, V, T, q) then choose proper formula (isothermal, adiabatic, etc.).
• For entropy or free energy, ensure path is reversible in calculation even if actual process is not.
• Kinetic data: plot appropriate linear form to extract $k$ and order.
• Use Arrhenius plot for $E<em>a$; slope uncertainty → propagate to $E</em>a$.
• Confirm catalysis type via phase comparison; articulate mechanism when possible.