Engineering Chemistry Module 1: Thermodynamics, Chemical Kinetics, and Catalysis

Introduction to Thermodynamics and Basic Terminologies

• Thermodynamics is defined as the study of the transformations of energy. It allows for the quantitative discussion of energy changes and the making of useful predictions.

• The term originates from "Thermo" meaning heat and "Dynamics" meaning power. In physical science, it is defined as the science of energy and the interconversion of energy between various forms.

• Fundamental laws of nature govern these transformations. Real-world applications include:

  • Burning fuel in a furnace to provide heat.

  • Burning fuel in an engine to produce mechanical work.

  • Chemical reactions in batteries pumping electrons through a circuit to generate electrical work (chemical energy to electrical energy).

  • Electrical fans converting electrical energy to mechanical energy.

Thermodynamic Systems and Boundaries

• The Universe is divided into two primary parts: the system and the surroundings.

• The System is the specific part of the world under investigation. Examples include a reaction vessel, an engine, an electrochemical cell, or a biological cell.

• The Surroundings comprise everything outside the system; this is the region where measurements are typically made.

• The Universe is defined by the equation: $\text{Universe} = \text{System} + \text{Surrounds}$.

• A Wall or boundary separates the system from the surroundings. The nature of this boundary determines the transfer of matter and energy.

• System classifications based on boundary characteristics:

  • Open System: Matter and energy can both be transferred through the boundary.

  • Closed System: Matter cannot pass through the boundary, but energy can be exchanged with the surroundings (e.g., via heat transfer or mechanical work like expansion).

  • Isolated System: A closed system that has neither mechanical nor thermal contact with its surroundings; it exchanges neither matter nor energy.

Walls and System Properties

• Wall Types:

  • Permeable and diathermal: Allows matter and heat transfer.

  • Impermeable and diathermal: Prevents matter transfer but allows heat transfer.

  • Impermeable and adiabatic: Prevents both matter and heat transfer.

• System Properties:

  • Intensive Properties: Characteristics independent of the mass or size of the system, such as pressure ($P$), temperature ($T$), and density ($d$). If a system is split in half, these values remain the same.

  • Extensive Properties: Characteristics that depend on the mass of the system, such as mass ($m$), volume ($V$), momentum, and internal energy ($U$).

  • Specific Intensive Properties: These are extensive properties per unit mass. Examples include specific volume ($v = \frac{V}{m}$) and specific energy ($e = \frac{E}{m}$).

States and Thermodynamic Processes

• The State of the System is defined by a set of properties (variables like pressure, temperature, volume, and composition) that completely describe the system at a specific time. In a given state, all properties have fixed values and are uniform throughout the system.

• A Process occurs when one or more variables undergo change, moving the system from one equilibrium state to another.

• Path refers to the series of states through which a system passes during a process.

• Cycle: A process where the system returns to its initial state, meaning the initial and final states are identical.

• Specific Thermodynamic Processes:

  • Adiabatic: No heat is transferred ($q = 0$).

  • Isothermal: Temperature remains constant ($\Delta T = 0$).

  • Isobaric: Pressure remains constant ($\Delta P = 0$).

  • Isochoric: Volume remains constant ($\Delta V = 0$).

  • Isentropic: Entropy remains constant.

  • Isenthalpic: Enthalpy remains constant.

Functions of State vs. Path

• State Function: A property whose value depends solely on the current state of the system, independent of the path taken to reach that state. Examples include volume ($V$), pressure ($P$), temperature ($T$), internal energy ($U$), enthalpy ($H$), and entropy ($S$).

• Path Function: A property whose value depends on the specific path taken between the initial and final states. Examples include work ($w$) and heat ($q$).

Internal Energy, Heat, and Work

• Internal Energy ($U$): The sum of kinetic energy and potential energy within a system. It is an extensive property and a state function. Changes occur when energy is transferred as heat or work.

• Work ($w$): Defined as motion against an opposing force. Doing work is equivalent to raising a weight in the surroundings. Examples include gas expansion pushing a piston or a chemical reaction driving an electric current through a resistance.

• Energy: The capacity of a system to do work. Work done on an isolated system (e.g., compressing gas) increases its energy. When a system does work (e.g., piston moving out), its energy decreases.

• Heat ($q$): The transfer of energy as a result of a temperature difference between the system and surroundings. In molecular terms, heat transfer stimulates random motion (thermal motion) of atoms.

• Exothermic Process: A process that releases energy as heat into the surroundings (e.g., combustion). In a diathermic container, this results in heat release; in an adiabatic container, it results in a temperature rise within the system.

• Endothermic Process: A process that acquires energy as heat from the surroundings (e.g., vaporization of water). In a diathermic container, energy flows into the system; in an adiabatic container, the system's temperature falls.

Laws of Thermodynamics

• Zeroth Law: If two physical systems are in thermal equilibrium with a third system, they are in thermal equilibrium with each other. This law defines thermal equilibrium and forms the basis for the definition of temperature.

• First Law: The law of conservation of energy. It states that the energy of an isolated system remains constant. Any disappearing energy in one form must appear in another form.

  • Mathematical expression: $\Delta U = q + w$

  • Sign convention: $+ve$ if energy enters the system as heat or work; $-ve$ if energy leaves the system.

• Second Law: Defines the direction of spontaneous change. It states that the entropy ($S$) of an isolated system increases in the course of a spontaneous change.

  • Entropy is a state function: $dS = \frac{dq_{rev}}{T}$.

  • Total entropy change for the universe: $\Delta S_{tot} = \Delta S_{sys} + \Delta S_{sur} > 0$ for spontaneous processes.

• Third Law: The entropy of all perfect crystalline substances is zero at absolute zero ($T = 0\,K$).

  • Statistical definition: $S = k \ln(W)$, where $k$ is the Boltzmann constant and $W$ is the number of microstates. At $T = 0$, $W = 1$, thus $S = 0$.

Expansion Work and Enthalpy

• Work of Expansion: Calculated as $dw = -P_{ex}dV$.

  • Free Expansion: Expansion against zero opposing force ($P_{ex} = 0$), thus $w = 0$.

  • Expansion against Constant Pressure (Isobaric): $w = -P_{ex}(V_f - V_i) = -P_{ex}\Delta V$. This is represented graphically on an indicator diagram (p-V graph) as the area under the horizontal line at $P = P_{ex}$.

  • Isothermal Reversible Expansion: For $n$ moles of an ideal gas, $w = -nRT \ln\left(\frac{V_f}{V_i}\right)$. Reversible work is the maximum work available for a process.

• Enthalpy ($H$): Defined to quantify heat transfer at constant pressure. $H = U + PV$.

  • Change in Enthalpy: $\Delta H = \Delta U + P\Delta V$.

  • At constant pressure, $\Delta H = q_p$.

Heat Capacity

• Heat Capacity ($C$): The quantity of heat required to raise the temperature divided by the temperature difference: $C = \frac{q}{\Delta T}$.

• Specific Heat: Heat capacity for $1\,g$ of substance.

• Molar Heat Capacity: Heat capacity for $1\,mol$ of substance. This varies with temperature.

• At Constant Volume ($C_v$): $C_v = \left(\frac{dU}{dT}\right)_v$. Since $w = 0$, $\Delta U = q_v$.

• At Constant Pressure ($C_p$): $C_p = \left(\frac{dH}{dT}\right)_p$.

• Relationship for Ideal Gas: $C_p - C_v = R$ (for $1\,mol$). Heat capacity ratio $\gamma = \frac{C_p}{C_v}$.

Reversible and Adiabatic Processes

• Reversible Process: A change that can be reversed by an infinitesimal modification of a variable. The system remains in equilibrium with surroundings throughout.

• Reversible Adiabatic Expansion: Characterized by no heat transfer ($q = 0$), so $\Delta U = w_{ad}$. For a perfect gas, $w_{ad} = C_v \Delta T$.

  • Temperature-Volume Relationship: $V_i T_i^{c} = V_f T_f^{c}$, where $c = \frac{C_{v,m}}{R}$.

  • Other relationships: $P V^{\gamma} = \text{constant}$; $T V^{\gamma - 1} = \text{constant}$.

Carnot Cycle and Heat Engines

• A Heat Engine transforms heat into work in a cyclic process, requiring a hot reservoir ($Q_H$ at $T_h$) and a cold reservoir ($Q_C$ at $T_c$).

• Carnot Cycle Stages:

  1. Isothermal Expansion: Reversible expansion at $T_h$; system in contact with hot source; $\Delta S = \frac{q_h}{T_h}$.

  2. Adiabatic Expansion: Reversible expansion where temperature falls from $T_h$ to $T_c$; $\Delta S = 0$.

  3. Isothermal Compression: Reversible compression at $T_c$; heat released to cold sink; $\Delta S = \frac{q_c}{T_c}$.

  4. Adiabatic Compression: Reversible compression back to initial state at $T_h$; $\Delta S = 0$.

• Efficiency ($\eta$): The fraction of heat converted to work. $\eta = \frac{w}{q_h} = 1 - \frac{T_c}{T_h}$.

• Second Law Statements:

  • Kelvin-Planck: It is impossible for any process to have as its sole result the transfer of heat from a cooler object to a warmer object.

  • Clausius: The energy of the universe remains constant; the entropy of the universe tends toward a maximum.

Free Energy and Spontaneity

• Helmholtz Free Energy ($A$): $A = U - TS$. At constant temperature and volume, spontaneity requires $dA \le 0$.

• Gibbs Free Energy ($G$): $G = H - TS$. At constant temperature and pressure, spontaneity requires $dG \le 0$.

  • Equation: $\Delta G = \Delta H - T\Delta S$.

  • Endothermic Reactions ($\Delta H > 0$): Can be spontaneous if the increase in entropy ($\Delta S$) is large enough such that $T\Delta S$ outweighs $\Delta H$, making $\Delta G$ negative.

  • Exothermic Reactions ($\Delta H < 0$): Commonly spontaneous unless the decrease in entropy is extremely negative.

  • At chemical equilibrium: $dG = 0$.

Chemical Kinetics: Reaction Rates

• Chemical Kinetics deals with the study of reaction rates, factors affecting them (pressure, temperature, catalyst), and mechanisms.

• Rate of Reaction: The change in concentration of reactants or products per unit time ($mol\,dm^{-3}\,s^{-1}$).

  • For $A + B \rightarrow C + D$:

  • Rate = $-\frac{d[A]}{dt} = -\frac{d[B]}{dt} = +\frac{d[C]}{dt} = +\frac{d[D]}{dt}$.

  • For general stoichiometry $aA + bB \rightarrow cC + dD$:

  • Rate = $-\frac{1}{a}\frac{d[A]}{dt} = -\frac{1}{b}\frac{d[B]}{dt} = +\frac{1}{c}\frac{d[C]}{dt} = +\frac{1}{d}\frac{d[D]}{dt}$.

Rate Laws and Order

• Rate Law: The experimental relationship between rate and concentration: $\text{rate} = k[A]^m[B]^n$.

• Rate Constant ($k$): Proportionality constant independent of concentration but dependent on temperature.

• Order of Reaction: The exponents ($m, n$) in the rate law. The overall order is the sum $m + n$.

  • Zero Order: Rate is independent of concentration ($\text{rate} = k$).

  • First Order: Rate depends on one concentration to the first power ($\text{rate} = k[A]$).

  • Second Order: Rate depends on the square of one concentration or the product of two ($\text{rate} = k[A]^2$ or $k[A][B]$).

• Pseudo-First Order: A second-order reaction behaving like first-order because one reactant (e.g., water in hydrolysis) is in great excess.

Integrated Rate Laws and Half-Life

• First Order Integration:

  • $\ln[A]_t = \ln[A]_0 - kt$ or $[A]_t = [A]_0 e^{-kt}$.

  • A plot of $\ln[A]$ vs. $t$ yields a straight line with slope $-k$.

  • Half-Life ($t_{1/2}$): Time for concentration to reduce by half. For first order: $t_{1/2} = \frac{\ln(2)}{k} \approx \frac{0.693}{k}$. It is independent of initial concentration.

  • Time Constant (\tau): Time for concentration to fall to $\frac{1}{e}$ of its initial value; $\tau = \frac{1}{k}$.

• Zero Order Integration: $[A]_t - [A]_0 = -kt$.

• Second Order Integration: $\frac{1}{[A]_t} - \frac{1}{[A]_0} = kt$.

Temperature Dependence and Collision Theory

• Arrhenius Equation: $k = A e^{-\frac{E_a}{RT}}$.

  • $A$: Pre-exponential factor (frequency factor), measures collision rate.

  • $E_a$: Activation energy, the minimum energy required to form products.

  • A plot of $\ln(k)$ vs. $\frac{1}{T}$ allows determination of $E_a$ (from slope) and $A$ (from intercept).

• Collision Theory Postulates:

  1. Reaction rate is proportional to reactant collision rate.

  2. Species must collide in the correct orientation.

  3. Collisions must have sufficient energy ($E \ge E_a$) to rearrange valence shells.

• Potential Energy Profile:

  • Reactants must cross an energy barrier ($E_a$).

  • The peak corresponds to the Activated Complex or Transition State.

  • High $E_a$ results in slow reactions; low $E_a$ allows for rapid reactions.

Catalysis

• Catalyst: A chemical entity that increases or decreases the reaction rate without being changed in chemical properties or mass. It lowers the activation energy by providing an alternative mechanism with a lower energy transition state.

• Characteristics:

  • Remains unchanged in mass/composition.

  • Only small quantities are required.

  • Does not change the equilibrium constant, just speeds the attainment of equilibrium.

• Types of Catalysis:

  • Heterogeneous: Catalyst in a different phase than reactants (e.g., solid iron in gaseous Haber process for $NH_3$; $V_2O_5$ in Contact process for $H_2SO_4$).

  • Homogeneous: Catalyst in the same phase (e.g., $NO$ gas in oxidation of $SO_2$; acid/alkali in ester hydrolysis).

  • Enzyme: Protein-based biological catalysts.

Enzyme Kinetics: Michaelis-Menten Mechanism

• Enzymes work in a "lock and key" fashion at an active site. The Substrate (S) fits into the Enzyme (E) to form an Enzyme-Substrate Complex (ES).

• Michaelis-Menten Mechanism Steps:

  1. $E + S \xrightarrow{k_a} ES$

  2. $ES \xrightarrow{k_a'} E + S$

  3. $ES \xrightarrow{k_b} P + E$

• Michaelis-Menten Rate Law: $R_0 = \frac{k_2[S]_0[E]_0}{[S]_0 + K_m}$, where $K_m$ is the Michaelis constant.

• Saturation: When $[S]_0 \gg K_m$, the rate reaches a maximum $R_{max} = k_2[E]_0$.

• Lineweaver-Burk Equation: Inverts the rate law to create a linear plot: $\frac{1}{R_0} = \frac{1}{R_{max}} + \frac{K_m}{R_{max}[S]_0}$. Slope = $\frac{K_m}{R_{max}}$, Y-intercept = $\frac{1}{R_{max}}$.

• Turnover Number: Defined by $k_2$, representing the maximum number of substrate molecules processed per unit time.

Numerical Examples

• Example (Expansion Work): 1 mole of an ideal gas expands from $10\,dm^3$ to $30\,dm^3$ against 1 atm persistent pressure. $w = -P_{ex} \Delta V = -1\,atm \times (30 - 10)\,dm^3 = -20\,atm\,dm^3$. (Conversion: $1\,atm\,dm^3 \approx 101.325\,J$).

• Example (Efficiency): A Carnot engine operates between $100\,^oC$ ($373\,K$) and $25\,^oC$ ($298\,K$). $\eta = 1 - \frac{298}{373} = 0.201$ or $20.1\%$.

• Example (Kinetics): If for a first-order reaction a substance decomposes such that $3.0\,g$ goes to $0.375\,g$ in $36\,min$. Since $0.375 = \frac{3.0}{8}$ (which is 3 half-lives), $3t_{1/2} = 36$, so $t_{1/2} = 12\,min$.