Sources and References

• J. M. Smith, H. C. Van Ness, M. M. Abbott, M. T. Swihart. Introduction to Chemical Engineering Thermodynamics, 9th ed. McGraw-Hill, 2022.
• M. D. Koretsky. Engineering and Chemical Thermodynamics, 2nd ed. Wiley, 2013.
• J. R. Elliott, C. T. Lira. Introductory Chemical Engineering Thermodynamics, 2nd ed. Prentice Hall, 2012.
• S. I. Sandler. Chemical, Biochemical, and Engineering Thermodynamics, 5th ed. Wiley, 2017.
• B. E. Poling, J. M. Prausnitz, J. P. O’Connell. The Properties of Gases and Liquids, 5th ed. McGraw-Hill, 2001.
• I. Prigogine, R. Defay. Chemical Thermodynamics. Longmans, 1954.
• K. Denbigh. The Principles of Chemical Equilibrium, 4th ed. Cambridge, 1981.
• A. Bejan. Advanced Engineering Thermodynamics, 4th ed. Wiley, 2016.
• D. G. Peng, D. B. Robinson. “A New Two-Constant Equation of State.” Ind. Eng. Chem. Fundam., 1976.
• G. Soave. “Equilibrium Constants from a Modified Redlich–Kwong Equation of State.” Chem. Eng. Sci., 1972.
• MIT OpenCourseWare, 10.40 Thermodynamics and Kinetics (course topic list and problem framing).
• Stanford CHEMENG 160 Chemical Engineering Thermodynamics (course topic list).
• D. L. Nelson, M. M. Cox. Lehninger Principles of Biochemistry, 8th ed. Macmillan, 2021 (for biological-thermodynamics chapter).
• A. J. Bard, L. R. Faulkner. Electrochemical Methods: Fundamentals and Applications, 2nd ed. Wiley, 2001 (for electrochemical chapter).

---

Chapter 1: First Law Review and the Control Volume Energy Balance

Chemical engineering thermodynamics begins with the accounting of energy across interfaces that are almost always open to mass flow. Unlike the closed systems of introductory physics, the reactors, columns, compressors and turbines of a chemical plant continuously exchange matter with their surroundings. Chapter 1 rebuilds the first law from that vantage point.

1.1 System, Surroundings, Property

A system is any region of space chosen for analysis, bounded by a real or imaginary surface called the control surface. Its complement is the surroundings. A property is any quantity whose change depends only on the initial and final states, not the path. Thermodynamic state is fixed, for a single-phase simple compressible substance, by any two independent intensive properties — a statement known as the state postulate.

Intensive properties (temperature \( T \), pressure \( P \), specific volume \( v \), specific internal energy \( u \), specific enthalpy \( h \), specific entropy \( s \)) do not scale with system size. Extensive properties (\( V \), \( U \), \( H \), \( S \), \( N \)) do. Molar and specific quantities are the extensive property divided by moles or mass, and they are intensive.

1.2 The Closed-System First Law

For a closed system undergoing a change of state between equilibria,

\[ \Delta U = Q - W, \]

where \( Q \) is heat added to the system and \( W \) is work done by the system on the surroundings. Sign conventions vary across textbooks; we keep \( Q \) positive into the system and \( W \) positive out of the system. For a quasi-static boundary-work process,

\[ W = \int_{V_1}^{V_2} P \, dV. \]

1.3 The Open-System Energy Balance

For a control volume with multiple streams, accounting for shaft work \( \dot W_s \), heat rate \( \dot Q \), and flow work associated with pushing fluid across the boundary gives

\[ \frac{dE_{cv}}{dt} = \dot Q - \dot W_s + \sum_{in} \dot m_i \left( h_i + \frac{V_i^2}{2} + g z_i \right) - \sum_{out} \dot m_e \left( h_e + \frac{V_e^2}{2} + g z_e \right). \]

Enthalpy \( h = u + Pv \) absorbs the flow-work term \( Pv \) so it need not be carried separately. The steady-state, steady-flow, single-inlet/single-outlet form reduces to

\[ \dot Q - \dot W_s = \dot m \left[ \Delta h + \frac{\Delta V^2}{2} + g \Delta z \right]. \]

1.4 Applications to Standard Unit Operations

Specializations of the steady-state balance to common devices are tabulated below.

Device	Simplifying assumptions	Reduced first law
Adiabatic turbine/compressor	\( \dot Q \approx 0 \), negligible KE/PE	\( \dot W_s = \dot m ( h_{in} - h_{out} ) \)
Throttle valve	adiabatic, no shaft work, negligible KE	\( h_{in} = h_{out} \)
Nozzle	adiabatic, \( \dot W_s = 0 \)	\( \tfrac{V_{out}^2 - V_{in}^2}{2} = h_{in} - h_{out} \)
Heat exchanger (one side)	\( \dot W_s = 0 \)	\( \dot Q = \dot m ( h_{out} - h_{in} ) \)
Adiabatic mixer	\( \dot Q = 0 \), \( \dot W_s = 0 \)	\( \sum_{in} \dot m_i h_i = \sum_{out} \dot m_e h_e \)

Chapter 2: The Second Law, Entropy, and Exergy

Energy is conserved; its quality is not. The second law formalizes the quality distinction and yields the entropy property, Clausius inequality, and the exergy (availability) function used throughout the course.

2.1 Statements of the Second Law

The Kelvin–Planck statement forbids a cycle that produces net work while exchanging heat with a single reservoir. The Clausius statement forbids a cycle whose sole effect is heat transfer from a cold to a hot reservoir. Both are equivalent; either implies the existence of an absolute temperature scale and the property entropy.

2.2 Entropy and the Clausius Inequality

For any cycle,

\[ \oint \frac{\delta Q}{T} \le 0, \]

with equality only for reversible cycles. Entropy is defined by

\[ dS = \left( \frac{\delta Q}{T} \right)_{rev}. \]

The entropy generation rate for a general control volume is

\[ \dot S_{gen} = \frac{dS_{cv}}{dt} + \sum_{out} \dot m_e s_e - \sum_{in} \dot m_i s_i - \sum_j \frac{\dot Q_j}{T_j} \ge 0. \]

Steady-state adiabatic single-stream devices give \( \dot S_{gen} = \dot m ( s_{out} - s_{in} ) \ge 0 \), which is the signature of real (irreversible) operation.

2.3 Isentropic Efficiency

Real turbines and compressors depart from reversible behaviour. Isentropic efficiency compares to the reversible adiabatic benchmark at the same inlet state and outlet pressure:

\[ \eta_{t} = \frac{h_{in} - h_{out,actual}}{h_{in} - h_{out,s}}, \qquad \eta_{c} = \frac{h_{out,s} - h_{in}}{h_{out,actual} - h_{in}}. \]

Typical industrial values range from 0.75 to 0.9 for large turbomachines.

2.4 Exergy (Availability)

Exergy is the maximum work obtainable as a system is brought to equilibrium with a specified dead state \( ( T_0, P_0 ) \). For a closed system,

\[ \Phi = ( U - U_0 ) + P_0 ( V - V_0 ) - T_0 ( S - S_0 ). \]

For a flowing stream, the flow exergy (availability) is

\[ \psi = ( h - h_0 ) - T_0 ( s - s_0 ) + \frac{V^2}{2} + g z. \]

The steady-state exergy balance reads

\[ 0 = \sum_j \left( 1 - \frac{T_0}{T_j} \right) \dot Q_j - \dot W_s + \sum_{in} \dot m_i \psi_i - \sum_{out} \dot m_e \psi_e - \dot I, \]

where \( \dot I = T_0 \dot S_{gen} \) is the irreversibility rate (Gouy–Stodola theorem). Exergy analysis localizes losses far more diagnostically than a first-law energy balance: a heat exchanger may have near-unity energy efficiency while destroying a large fraction of the stream’s work potential through finite-temperature heat transfer.

Chapter 3: Property Relations and Maxwell Equations

To apply the laws we need to compute changes in \( h \), \( s \), \( u \) from measurable quantities \( P \), \( V \), \( T \), \( c_P \), \( c_V \). The fundamental property relations furnish the required connections.

3.1 The Fundamental Equations

For a simple compressible pure substance in a closed system,

\[ dU = T \, dS - P \, dV. \]

Legendre transforms produce three further fundamental relations:

\[ dH = T \, dS + V \, dP, \]\[ dA = -S \, dT - P \, dV, \]\[ dG = -S \, dT + V \, dP, \]

where \( A = U - TS \) is the Helmholtz energy and \( G = H - TS \) is the Gibbs energy. Each equation identifies the natural variables of the corresponding potential: \( U(S,V) \), \( H(S,P) \), \( A(T,V) \), \( G(T,P) \).

3.2 Maxwell Relations

Because mixed second partials of a state function are equal, the four fundamental equations each yield a Maxwell relation:

\[ \left( \frac{\partial T}{\partial V} \right)_S = -\left( \frac{\partial P}{\partial S} \right)_V, \quad \left( \frac{\partial T}{\partial P} \right)_S = \left( \frac{\partial V}{\partial S} \right)_P, \]\[ \left( \frac{\partial S}{\partial V} \right)_T = \left( \frac{\partial P}{\partial T} \right)_V, \quad \left( \frac{\partial S}{\partial P} \right)_T = -\left( \frac{\partial V}{\partial T} \right)_P. \]

The last two are the workhorses because they convert entropy derivatives — not directly measurable — into PVT derivatives that any equation of state delivers.

3.3 Departure Functions and the Residual Concept

For a real fluid at \( (T,P) \) we split any property into an ideal-gas part at the same temperature and a residual (departure) part that measures non-ideality:

\[ M^R ( T,P ) = M(T,P) - M^{ig}(T,P). \]

Thus \( h = h^{ig}(T) + h^R(T,P) \) and \( s = s^{ig}(T,P) + s^R(T,P) \), with the residuals computed from a volume-explicit or pressure-explicit equation of state. For example,

\[ h^R = \int_0^P \left[ V - T \left( \frac{\partial V}{\partial T} \right)_P \right] dP \quad (T \text{ const}). \]

The ideal-gas contribution is tabulated via \( c_P^{ig}(T) \).

3.4 Useful Derivative Identities

The ratio of heat capacities \( \gamma = c_P / c_V \), the Joule–Thomson coefficient \( \mu_{JT} = (\partial T / \partial P)_h \), and the speed of sound \( a \) are all expressible via Maxwell substitutions. In particular,

\[ \mu_{JT} = \frac{1}{c_P} \left[ T \left( \frac{\partial V}{\partial T} \right)_P - V \right], \]

which is central to Chapter 10 (liquefaction).

Chapter 4: Equations of State

A PVT equation of state (EoS) is the computational engine that turns the property relations into numbers. Chemical engineers lean on three families: the ideal gas, the cubic EoS of the van der Waals class, and semi-empirical or virial forms.

4.1 The Ideal Gas

\[ PV = nRT \]

is the zero-density limit of every real-gas model. Its internal energy depends on \( T \) only, and \( c_P - c_V = R \). It is accurate for dilute gases, mild conditions, and small molecules, but it cannot describe condensation.

4.2 The Van der Waals Equation

\[ P = \frac{RT}{v - b} - \frac{a}{v^2}. \]

The term \( a/v^2 \) represents attractive forces; \( b \) accounts for molecular co-volume. Critical-point constraints \( (\partial P/\partial v)_{T_c} = 0 \), \( (\partial^2 P/\partial v^2)_{T_c} = 0 \) fix \( a = 27 R^2 T_c^2 / (64 P_c) \), \( b = R T_c / (8 P_c) \). The van der Waals EoS predicts a universal critical compressibility \( Z_c = 3/8 = 0.375 \); real substances show \( 0.23 \le Z_c \le 0.30 \), revealing the model’s quantitative limits.

4.3 Redlich–Kwong and Soave Modifications

Redlich–Kwong (1949) improves the attractive term:

\[ P = \frac{RT}{v - b} - \frac{a}{T^{1/2} v ( v + b )}. \]

Soave (1972) generalized the attractive term with an acentric-factor-dependent \( \alpha(T,\omega) \):

\[ P = \frac{RT}{v - b} - \frac{a\, \alpha(T,\omega)}{v ( v + b )}. \]

4.4 The Peng–Robinson Equation

Peng–Robinson (1976) is the industry workhorse for light hydrocarbons, natural gas, and petrochemicals:

\[ P = \frac{RT}{v - b} - \frac{a\, \alpha(T,\omega)}{v ( v + b ) + b ( v - b )}, \]

with \( \alpha = [ 1 + \kappa ( 1 - \sqrt{T_r} ) ]^2 \) and \( \kappa = 0.37464 + 1.54226 \omega - 0.26992 \omega^2 \). It gives superior liquid densities relative to SRK and matches vapor pressures well along saturation lines.

4.5 Cubic EoS Solution and Root Selection

In \( Z = Pv/RT \) form, Peng–Robinson becomes a cubic in \( Z \). At a given \( (T,P) \) one obtains either one real root (single-phase) or three real roots. The largest root is the vapor-phase compressibility, the smallest the liquid-phase, and the middle root is thermodynamically unstable. Fugacity-based root selection picks the phase with the lower molar Gibbs energy.

4.6 The Virial Equation

A series in density,

\[ Z = 1 + B(T) \frac{P}{RT} + \cdots, \]

where \( B(T) \) is the second virial coefficient, connects directly to intermolecular pair potentials. It is excellent for moderate pressures but fails near and below the critical density.

4.7 Mixing Rules

For mixtures, the cubic-EoS parameters are combined using quadratic mixing:

\[ a_m = \sum_i \sum_j x_i x_j \sqrt{a_i a_j} ( 1 - k_{ij} ), \qquad b_m = \sum_i x_i b_i, \]

where binary interaction parameters \( k_{ij} \) are fitted to VLE data.

Chapter 5: Fugacity and Activity

The chemical potential \( \mu_i = ( \partial G / \partial n_i )_{T,P,n_{j\ne i}} \) is the master driver of phase and reaction equilibrium. Because \( \mu_i \to -\infty \) for ideal gases as \( P \to 0 \), we recast the potential through fugacity.

5.1 Fugacity of a Pure Substance

Fugacity \( f \) is defined so that \( \mu(T,P) = \mu^\circ(T) + RT \ln( f/f^\circ ) \) with \( f/P \to 1 \) as \( P \to 0 \). The fugacity coefficient is

\[ \varphi = \frac{f}{P}, \qquad \ln \varphi = \int_0^P \frac{Z - 1}{P} \, dP \quad (T \text{ const}). \]

Any cubic EoS yields an analytic expression for \( \ln \varphi \).

5.2 Fugacity in Mixtures

For component \( i \) in a mixture,

\[ \hat f_i = \varphi_i^{L} x_i P = \varphi_i^{V} y_i P \quad \text{(φ-φ formulation)}. \]

5.3 Activity and Activity Coefficients

For a liquid, fugacity is often referenced to a pure-liquid standard:

\[ \hat f_i^{L} = \gamma_i x_i f_i^{L}, \]

where \( \gamma_i \) is the activity coefficient. The dimensionless activity \( a_i = \gamma_i x_i \) equals \( x_i \) in an ideal solution (Lewis–Randall). Classical excess-Gibbs-energy models — Wilson, NRTL, UNIQUAC, UNIFAC — all produce \( \gamma_i \) formulas.

5.4 Consistency: Gibbs–Duhem

At fixed \( T,P \),

\[ \sum_i x_i \, d \ln \gamma_i = 0, \]

a constraint used to test experimental VLE data sets and to derive one activity coefficient from another in a binary.

Chapter 6: Phase Equilibria — VLE, LLE, SLE

Phase equilibrium is the condition \( \mu_i^{\alpha} = \mu_i^{\beta} \) for every component \( i \) across every pair of coexisting phases \( \alpha,\beta \). Translating into fugacity, \( \hat f_i^{\alpha} = \hat f_i^{\beta} \). This section presents the three most common specializations.

6.1 Vapor–Liquid Equilibrium (VLE)

6.1.1 Raoult’s Law

The simplest VLE model, valid for ideal-gas vapors and ideal-solution liquids,

\[ y_i P = x_i P_i^{sat}(T), \]

yields linear bubble- and dew-point calculations. It fails when molecules are chemically dissimilar or at elevated pressure.

6.1.2 Modified Raoult (γ-φ)

\[ y_i \varphi_i^{V} P = x_i \gamma_i P_i^{sat} \varphi_i^{sat} \exp \left[ \frac{v_i^{L} ( P - P_i^{sat} )}{RT} \right]. \]

The exponential Poynting term corrects the liquid fugacity for pressures far from saturation. This is the dominant low-to-moderate pressure framework.

6.1.3 The φ-φ Method

At higher pressures a single cubic EoS describes both phases, and the equal-fugacity condition is solved directly:

\[ \varphi_i^{V}(T,P,\{y\}) \, y_i = \varphi_i^{L}(T,P,\{x\}) \, x_i. \]

6.1.4 Azeotropy

An azeotrope is a composition at which \( x_i = y_i \) for all \( i \), so distillation cannot proceed past it. Minimum-boiling azeotropes (e.g., ethanol–water) arise from positive deviations \( ( \gamma > 1 ) \); maximum-boiling azeotropes from negative deviations \( ( \gamma < 1 ) \).

6.2 Liquid–Liquid Equilibrium (LLE)

If \( G^E \) is large and positive, the liquid phase splits. For a binary, the spinodal condition \( (\partial^2 G/\partial x_1^2)_{T,P} = 0 \) delimits thermodynamic instability; the binodal, lying outside the spinodal, is determined by equal-activity conditions \( \gamma_1 x_1 |_{\alpha} = \gamma_1 x_1 |_{\beta} \) and similarly for component 2. LLE underpins liquid–liquid extraction design.

6.3 Solid–Liquid Equilibrium (SLE)

For a solid that does not form solid solutions, the solubility of a pure solid in a liquid solvent satisfies

\[ \ln ( \gamma_1 x_1 ) = -\frac{\Delta h_{fus}}{R} \left( \frac{1}{T} - \frac{1}{T_m} \right) + \frac{\Delta c_P}{R} \left[ \ln \frac{T}{T_m} - 1 + \frac{T_m}{T} \right]. \]

The first term is usually dominant; the \( \Delta c_P \) correction is small away from the melting point.

Chapter 7: Reaction Equilibrium

For the reaction \( \sum_i \nu_i A_i = 0 \), equilibrium requires

\[ \Delta G_{rxn} = \sum_i \nu_i \mu_i = 0, \]

leading to the equilibrium constant

\[ K(T) = \prod_i \left( \frac{\hat f_i}{f_i^\circ} \right)^{\nu_i} = \exp \left[ -\frac{\Delta G^\circ(T)}{RT} \right]. \]

7.1 Temperature Dependence — Van ’t Hoff

Assuming constant \( \Delta H^\circ \),

\[ \ln \frac{K(T_2)}{K(T_1)} = -\frac{\Delta H^\circ}{R} \left( \frac{1}{T_2} - \frac{1}{T_1} \right). \]

Endothermic reactions favour products as \( T \) rises; exothermic reactions favour reactants. The more exact form retains \( \Delta c_P(T) \).

7.2 Gas-Phase Equilibrium and Non-Ideality

For an ideal-gas reaction,

\[ K = \left( \frac{P}{P^\circ} \right)^{\sum \nu_i} \prod_i y_i^{\nu_i}. \]

At high pressure the fugacity coefficients \( \varphi_i \) are inserted, and \( K_y \) (mole-fraction product) becomes pressure-dependent even though \( K \) itself depends only on \( T \).

7.3 Extent of Reaction

Mole balances are cleanest in terms of the extent \( \xi \): \( n_i = n_{i,0} + \nu_i \xi \). For single reactions the one unknown \( \xi \) is found by inserting \( y_i(\xi) \) into the equilibrium expression.

7.4 Multiple Simultaneous Reactions

For \( R \) independent reactions among \( N \) species, the equilibrium state minimizes the total Gibbs energy

\[ G = \sum_i n_i \mu_i \]

subject to \( N - R \) element balance constraints. Lagrange-multiplier or direct-minimization algorithms handle the coupled non-linear system.

Chapter 8: Power Cycles

The engineering payoff of thermodynamics is power. Two cycles dominate modern practice: Rankine for steam, Brayton for gas turbines.

8.1 The Rankine Cycle

Four steady-flow devices arranged in a loop:

1. Boiler (heat added at high pressure, \( q_{in} = h_3 - h_2 \)).
2. Turbine (shaft work, \( w_T = h_3 - h_4 \)).
3. Condenser (heat rejected, \( q_{out} = h_4 - h_1 \)).
4. Pump (work input, \( w_P = h_2 - h_1 \approx v_1 ( P_2 - P_1 ) \)).

Thermal efficiency is

\[ \eta = \frac{w_T - w_P}{q_{in}}. \]

Efficiency gains come from increasing \( T \) of heat addition (superheat), reducing \( T \) of heat rejection (condenser vacuum), reheat (expand, re-superheat, expand again), and regeneration (extract steam to feed-heat the boiler water).

8.2 The Brayton Cycle

Air-breathing gas turbines use compression, combustion, and expansion. For cold-air-standard analysis with \( \gamma \) constant,

\[ \eta = 1 - r_p^{-(\gamma - 1)/\gamma}, \]

where \( r_p = P_2/P_1 \). Combined-cycle power plants couple a gas-turbine topping cycle with a Rankine bottoming cycle and reach integrated efficiencies above 60 %.

8.3 Second-Law Critique

Carnot efficiency \( \eta_C = 1 - T_L/T_H \) is the ceiling. Real-cycle irreversibilities — finite-\( \Delta T \) heat exchange, turbine/compressor non-ideality, pressure drops — are made visible by an exergy balance around each component.

Chapter 9: Refrigeration and Air Conditioning

Refrigeration moves heat from a cold reservoir to a hot one at a cost in work; its figure of merit is the coefficient of performance (COP).

9.1 Vapor-Compression Refrigeration

Four components: compressor, condenser, expansion valve, evaporator. The key states circle the saturation dome on a \( P\)-\(h \) diagram.

\[ \text{COP}_R = \frac{q_L}{w_c} = \frac{h_1 - h_4}{h_2 - h_1}, \qquad \text{COP}_{HP} = \frac{q_H}{w_c} = \frac{h_2 - h_3}{h_2 - h_1}. \]

Refrigerant choice is driven by saturation curves, toxicity, flammability, and GWP. HFOs (e.g., R-1234yf) have displaced HFCs (e.g., R-134a) in new systems.

9.2 Absorption Refrigeration

Replaces mechanical compression with a generator–absorber pair driven by low-grade heat. Ammonia–water and lithium-bromide–water are the two industrial systems. The COP is modest but the work input is nearly zero, so waste heat and solar thermal inputs are attractive energy sources.

9.3 Air Conditioning and Psychrometrics

Moist air is a mixture of dry air and water vapor. Key properties:

• Specific humidity \( \omega = m_w / m_a \).
• Relative humidity \( \phi = P_w / P_{w,sat}(T) \).
• Dew-point temperature, wet-bulb temperature, enthalpy of moist air per unit dry air.

Cooling and dehumidifying coils, evaporative coolers, and cooling towers are analyzed with mass, dry-air, water, and energy balances on a psychrometric chart.

9.4 Cooling Towers

A counter-current tower evaporates a small fraction of circulating water to reject heat at near the wet-bulb temperature. The integrated balance (Merkel analysis) relates tower height, air flow, water flow, and the approach \( T_{water,out} - T_{wb} \).

Chapter 10: Liquefaction of Gases

Turning a gas into a liquid for transport, storage, or separation requires refrigeration below the critical temperature.

10.1 Joule–Thomson Effect

A valve reduces pressure at constant \( h \). Cooling occurs only if

\[ \mu_{JT} = \left( \frac{\partial T}{\partial P} \right)_h > 0, \]

which for many gases requires being below the inversion temperature \( T_{inv}(P) \). Hydrogen and helium must first be precooled before throttling can liquefy them.

10.2 The Linde Process

A single compression, aftercool, counter-current heat exchange against the returning cold low-pressure stream, throttle, and phase separation. The fraction liquefied \( y \) follows from an energy balance around the exchanger–throttle subsystem:

\[ y = \frac{h_1 - h_2}{h_1 - h_f}, \]

where 1 is the high-pressure feed entering the exchanger, 2 is the warm gas leaving, and \( f \) is the saturated liquid product.

10.3 The Claude Process

Replaces part of the Linde throttle with an expansion engine (turboexpander), recovering work and producing deeper cooling. Used in air separation, natural-gas liquefaction (APCI, Cascade), and cryogenic hydrogen plants.

Chapter 11: Surface Thermodynamics

At a liquid–vapor or liquid–liquid interface, molecules experience unbalanced forces. The resulting surface tension \( \sigma \) is an energy per unit area and a force per unit length — identical in SI units (\( \text{J/m}^2 = \text{N/m} \)).

11.1 Gibbs Dividing Surface

Excess quantities per unit area \( \Gamma_i \) (surface concentration) satisfy the Gibbs adsorption isotherm,

\[ d\sigma = -\sum_i \Gamma_i \, d\mu_i, \]

which explains why surfactants (species with strong positive \( \Gamma \)) lower surface tension dramatically at sub-millimolar bulk concentrations.

11.2 Young–Laplace and Kelvin Equations

Across a curved interface with principal radii \( R_1, R_2 \),

\[ \Delta P = \sigma \left( \frac{1}{R_1} + \frac{1}{R_2} \right). \]

A droplet of radius \( r \) has elevated saturation pressure given by the Kelvin equation

\[ \ln \frac{P(r)}{P^{sat}(\infty)} = \frac{2 \sigma v^{L}}{R T r}. \]

This underlies nucleation theory, capillary condensation in porous catalysts, and atmospheric aerosol growth.

Chapter 12: Electrochemical Thermodynamics

Electrochemistry couples material transfer with charge transfer. Equilibrium of a half-cell is set by

\[ E = E^\circ - \frac{RT}{nF} \ln Q, \]

the Nernst equation, where \( Q \) is the reaction quotient, \( n \) is electrons transferred, and \( F = 96485 \) C/mol is Faraday’s constant.

12.1 Cell Potential and Gibbs Energy

For a cell reaction proceeding spontaneously,

\[ \Delta G = -nFE, \qquad \Delta G^\circ = -nFE^\circ. \]

Standard electrode potentials are tabulated relative to the standard hydrogen electrode.

12.2 Fuel Cells

A hydrogen–oxygen fuel cell at 25 C has \( E^\circ = 1.229 \) V. The ideal (thermodynamic) efficiency is

\[ \eta_{FC,id} = \frac{\Delta G}{\Delta H} = 1 - \frac{T \Delta S}{\Delta H}, \]

which for \( \text{H}_2 + \tfrac{1}{2} \text{O}_2 \to \text{H}_2\text{O}(l) \) is about 0.83 — not bounded by the Carnot efficiency, a key advantage. Real cells lose potential to activation, ohmic, and concentration overpotentials.

12.3 Electrolysis

An external potential \( E_{app} > |E^\circ| \) drives non-spontaneous reactions such as water splitting. The minimum (reversible) energy per mole of \( \text{H}_2 \) is \( \Delta G / nF \); actual industrial alkaline electrolyzers run at 1.7–2.0 V cell voltage.

Chapter 13: Biological Thermodynamics

Living cells obey the same second law as chemical plants; they simply exploit coupled reactions and compartmentalization to keep local entropy generation small.

13.1 Coupled Reactions and ATP

Many biosynthetic reactions have \( \Delta G > 0 \) in isolation and would not proceed. Coupling to ATP hydrolysis,

\[ \text{ATP} + \text{H}_2\text{O} \to \text{ADP} + P_i, \qquad \Delta G^{\circ\prime} \approx -30.5 \text{ kJ/mol}, \]

provides the downhill step. Under cellular concentrations, \( \Delta G \) for ATP hydrolysis is closer to \( -50 \) kJ/mol because \( Q \ll K \).

13.2 Biochemical Standard State

Biochemists redefine standards at \( \text{pH} = 7 \), \( [H^+] = 10^{-7} \) M, denoted \( \Delta G^{\circ\prime} \). The transformation is \( \Delta G^{\circ\prime} = \Delta G^\circ + \nu_{H^+} R T \ln 10^{-7} \) summed over proton stoichiometry.

13.3 Membrane Transport and the Chemiosmotic Gradient

A species crossing a membrane against an electrochemical potential difference requires work

\[ \Delta \tilde \mu_i = RT \ln \frac{c_i^{in}}{c_i^{out}} + z_i F \Delta \phi, \]

where \( \Delta \phi \) is the membrane potential. Mitochondrial ATP synthesis is driven by the proton motive force, an example of chemiosmotic coupling that Peter Mitchell resolved into strict thermodynamic equivalence.

13.4 Enzyme-Catalyzed Equilibria

Enzymes do not shift equilibrium — \( K \) is a state-function ratio — but they lower kinetic barriers, making the approach to equilibrium practically achievable. Thermodynamic analysis fixes the feasibility window; kinetics determines the rate.

Chapter 14: Putting It Together

14.1 Solution Strategy

A recurring four-step pattern solves nearly every CHE 330 problem.

1. Draw the control volume and label every stream and energy transfer.
2. Write mass, energy, and entropy (or exergy) balances.
3. Choose an EoS (ideal gas, tables, cubic) and an activity/fugacity model as warranted.
4. Solve the closed set — often a small non-linear system — and check limits (ideal-gas, dilute-solution, low-pressure) for sanity.

14.2 Common Pitfalls

• Using \( h = c_P ( T - T_{ref} ) \) across a phase change, ignoring latent heat.
• Applying Raoult’s law at high pressure without the Poynting and \( \varphi \) corrections.
• Computing a single equilibrium \( K \) when two or more reactions are independent.
• Confusing the sign convention for work in the closed-system and open-system first laws.
• Treating an isentropic process as isothermal (or vice versa) in a turbine or compressor.

14.3 Unit Vigilance

Always carry units through an equation. \( R = 8.314 \) J/(mol K) \( = 0.08314 \) L·bar/(mol K). Pressure in a cubic EoS must match volume units. Gibbs-energy tables are molar, kJ/mol; entropy tables are J/(mol K). A wrong prefactor of \( 1000 \) is the most common error in equilibrium calculations.