🧪Advanced Chemical Engineering Science Unit 2 – Advanced Thermodynamics

Fundamental Concepts Recap

• Thermodynamics studies the interrelationships between heat, work, and energy in a system
• Zeroth Law of Thermodynamics establishes the concept of thermal equilibrium between two systems in contact
• First Law of Thermodynamics states that energy cannot be created or destroyed, only converted from one form to another (conservation of energy)
• Second Law of Thermodynamics introduces the concept of entropy and states that the total entropy of an isolated system always increases over time
• Enthalpy ($H$) represents the total heat content of a system at constant pressure
• Gibbs free energy ($G$) predicts the spontaneity of a process at constant temperature and pressure
  • Processes with $\Delta G < 0$ are spontaneous
  • Processes with $\Delta G > 0$ are non-spontaneous
• Helmholtz free energy ($A$) describes the maximum amount of work a system can perform at constant temperature and volume

Laws of Thermodynamics in Depth

• Zeroth Law of Thermodynamics
  • If two systems are in thermal equilibrium with a third system, they are also in thermal equilibrium with each other
  • Allows for the definition of temperature and the concept of a thermometer
• First Law of Thermodynamics
  • Expressed as $\Delta U = Q - W$, where $\Delta U$ is the change in internal energy, $Q$ is the heat added to the system, and $W$ is the work done by the system
  • For a cyclic process, $\oint \delta Q = \oint \delta W$, meaning the net heat added equals the net work done
• Second Law of Thermodynamics
  • Entropy ($S$) is a measure of the disorder or randomness of a system
  • For a reversible process, $\Delta S = \int \frac{\delta Q_{rev}}{T}$
  • For an irreversible process, $\Delta S > \int \frac{\delta Q}{T}$
  • The Clausius inequality states that for any cyclic process, $\oint \frac{\delta Q}{T} \leq 0$
• Third Law of Thermodynamics
  • As the temperature of a system approaches absolute zero, its entropy approaches a constant minimum value
  • It is impossible to reach absolute zero in a finite number of steps

Advanced Equations of State

• Ideal Gas Law: $PV = nRT$, where $P$ is pressure, $V$ is volume, $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is temperature
• van der Waals Equation: $\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT$, accounts for intermolecular forces and molecular size
  • $a$ represents the attraction between molecules
  • $b$ represents the volume occupied by the molecules
• Redlich-Kwong Equation: $P = \frac{RT}{V_m - b} - \frac{a}{\sqrt{T}V_m(V_m + b)}$, improves upon the van der Waals equation by better describing the behavior of gases at high pressures
• Peng-Robinson Equation: $P = \frac{RT}{V_m - b} - \frac{a\alpha}{V_m(V_m + b) + b(V_m - b)}$, further improves the accuracy of the Redlich-Kwong equation, especially for polar and associating fluids
  • $\alpha$ is a temperature-dependent parameter that accounts for the shape of the molecule
• Virial Equation: $\frac{PV}{nRT} = 1 + \frac{B}{V} + \frac{C}{V^2} + \cdots$, expresses the compressibility factor as a power series in inverse volume
  • $B$ and $C$ are virial coefficients that depend on temperature and the nature of the gas

Thermodynamic Cycles and Efficiency

• Carnot Cycle: An idealized thermodynamic cycle consisting of two isothermal and two adiabatic processes
  • Represents the most efficient heat engine operating between two thermal reservoirs
  • Efficiency: $\eta = 1 - \frac{T_C}{T_H}$, where $T_C$ and $T_H$ are the temperatures of the cold and hot reservoirs, respectively
• Rankine Cycle: A practical vapor power cycle used in steam power plants
  • Consists of four processes: isentropic compression (pump), isobaric heat addition (boiler), isentropic expansion (turbine), and isobaric heat rejection (condenser)
  • Efficiency: $\eta = \frac{W_{net}}{Q_{in}} = \frac{W_t - W_p}{Q_{in}}$, where $W_t$ is the work done by the turbine, $W_p$ is the work done by the pump, and $Q_{in}$ is the heat input
• Brayton Cycle: A gas power cycle used in gas turbines and jet engines
  • Consists of four processes: isentropic compression (compressor), isobaric heat addition (combustion chamber), isentropic expansion (turbine), and isobaric heat rejection (exhaust)
  • Efficiency: $\eta = 1 - \frac{T_1}{T_2}$, where $T_1$ and $T_2$ are the temperatures at the beginning and end of the isentropic compression process, respectively
• Otto Cycle: An idealized thermodynamic cycle representing the operation of spark-ignition internal combustion engines
  • Consists of four processes: isentropic compression, isochoric heat addition, isentropic expansion, and isochoric heat rejection
  • Efficiency: $\eta = 1 - \frac{1}{r^{\gamma - 1}}$, where $r$ is the compression ratio and $\gamma$ is the specific heat ratio of the working fluid

Non-Ideal Systems and Mixtures

• Fugacity ($f$) is a measure of the effective pressure of a real gas, accounting for non-ideal behavior
  • For an ideal gas, fugacity equals pressure
  • Fugacity coefficient ($\phi$) relates fugacity to pressure: $f = \phi P$
• Activity ($a$) is a measure of the effective concentration of a component in a non-ideal mixture
  • For an ideal mixture, activity equals mole fraction
  • Activity coefficient ($\gamma$) relates activity to mole fraction: $a = \gamma x$
• Excess properties describe the deviation of a mixture's properties from those of an ideal mixture
  • Excess Gibbs free energy: $G^E = G - G^{id}$, where $G^{id}$ is the Gibbs free energy of an ideal mixture
  • Excess enthalpy: $H^E = H - H^{id}$, where $H^{id}$ is the enthalpy of an ideal mixture
  • Excess entropy: $S^E = S - S^{id}$, where $S^{id}$ is the entropy of an ideal mixture
• Partial molar properties represent the contribution of each component to the total property of a mixture
  • Partial molar Gibbs free energy (chemical potential): $\mu_i = \left(\frac{\partial G}{\partial n_i}\right)_{T,P,n_j}$
  • Partial molar enthalpy: $\bar{H}_i = \left(\frac{\partial H}{\partial n_i}\right)_{T,P,n_j}$
  • Partial molar entropy: $\bar{S}_i = \left(\frac{\partial S}{\partial n_i}\right)_{T,P,n_j}$

Chemical Equilibrium and Reaction Thermodynamics

• Chemical equilibrium is a dynamic state where the forward and reverse reaction rates are equal
• Equilibrium constant ($K$) relates the concentrations of reactants and products at equilibrium
  • For a general reaction $aA + bB \rightleftharpoons cC + dD$, $K = \frac{[C]^c[D]^d}{[A]^a[B]^b}$, where $[X]$ represents the concentration of species $X$
• Gibbs free energy of reaction ($\Delta G_r$) predicts the spontaneity of a chemical reaction
  • $\Delta G_r = \Delta G_r^{\circ} + RT \ln Q$, where $\Delta G_r^{\circ}$ is the standard Gibbs free energy of reaction and $Q$ is the reaction quotient
  • At equilibrium, $\Delta G_r = 0$ and $Q = K$
• van't Hoff equation relates the equilibrium constant to temperature: $\frac{d \ln K}{dT} = \frac{\Delta H_r^{\circ}}{RT^2}$
  • Integrating the van't Hoff equation yields: $\ln \frac{K_2}{K_1} = -\frac{\Delta H_r^{\circ}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$
• Le Chatelier's principle states that a system at equilibrium will shift to counteract any imposed change in conditions
  • Increasing temperature favors the endothermic direction
  • Increasing pressure favors the direction with fewer moles of gas
  • Adding a reactant or removing a product shifts the equilibrium to the right (towards products)

Statistical Thermodynamics Introduction

• Statistical thermodynamics relates the microscopic properties of a system to its macroscopic thermodynamic properties
• Microstate is a specific configuration of a system, describing the positions and momenta of all particles
• Macrostate is a set of microstates that share the same macroscopic properties (e.g., temperature, pressure, volume)
• Boltzmann distribution describes the probability of a system being in a particular microstate with energy $E_i$: $P_i = \frac{e^{-E_i/kT}}{\sum_j e^{-E_j/kT}}$, where $k$ is the Boltzmann constant
• Partition function ($Z$) is the sum of the Boltzmann factors for all microstates: $Z = \sum_i e^{-E_i/kT}$
  • Relates microscopic properties to macroscopic thermodynamic functions
  • Helmholtz free energy: $A = -kT \ln Z$
  • Internal energy: $U = kT^2 \left(\frac{\partial \ln Z}{\partial T}\right)_{V,N}$
  • Entropy: $S = k \ln Z + kT \left(\frac{\partial \ln Z}{\partial T}\right)_{V,N}$
• Maxwell-Boltzmann statistics describe the distribution of particles over energy levels in a classical system
  • Assumes distinguishable particles and neglects quantum effects
• Fermi-Dirac statistics describe the distribution of fermions (particles with half-integer spin) over energy levels
  • Accounts for the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state
• Bose-Einstein statistics describe the distribution of bosons (particles with integer spin) over energy levels
  • Allows for multiple bosons to occupy the same quantum state, leading to phenomena such as Bose-Einstein condensation

Real-World Applications and Case Studies

• Power generation: Thermodynamic principles are used to design and optimize power plants (coal, natural gas, nuclear) for efficient energy conversion
  • Rankine cycle is used in steam power plants
  • Brayton cycle is used in gas turbines and combined cycle power plants
• Refrigeration and air conditioning: Thermodynamic cycles (vapor-compression, absorption) are used to transfer heat from a low-temperature reservoir to a high-temperature reservoir
  • Coefficient of Performance (COP) measures the efficiency of refrigeration systems: $COP = \frac{Q_L}{W}$, where $Q_L$ is the heat removed from the low-temperature reservoir and $W$ is the work input
• Automotive engines: Internal combustion engines (spark-ignition, diesel) rely on thermodynamic principles for efficient operation
  • Otto cycle represents the ideal operation of spark-ignition engines
  • Diesel cycle represents the ideal operation of diesel engines
• Chemical and petrochemical industries: Thermodynamics plays a crucial role in the design and optimization of chemical processes
  • Equilibrium constants and Gibbs free energy are used to predict the feasibility and extent of chemical reactions
  • Equations of state are used to model the behavior of fluids in process equipment (reactors, distillation columns, heat exchangers)
• Materials science and engineering: Thermodynamic principles are applied to understand and predict the behavior of materials
  • Phase diagrams represent the equilibrium states of a system as a function of temperature, pressure, and composition
  • Gibbs free energy minimization is used to predict the stability of phases and the driving force for phase transformations
• Environmental science and climate change: Thermodynamics is essential for understanding the Earth's energy balance and the impact of human activities on the climate
  • Greenhouse effect is driven by the absorption and emission of thermal radiation by atmospheric gases
  • Entropy and the Second Law of Thermodynamics provide insights into the irreversibility of climate change and the need for sustainable energy practices