Thermodynamics 2

Formeln

PV = nRT = NkT

Equation of state for an ideal/perfect gas, relating pressure (P), volume (V), temperature (T), number of moles (n), and the ideal gas constant (R) or Boltzmann's constant (k).

U = 3/2nRT = 3/2Nkt

Expression for the internal energy of an ideal gas, indicating that it is directly proportional to the temperature and number of moles or molecules.

dU = dQ + dW, so ΔU = ∫dQ + ∫dW = Q + W

This equation represents the first law of thermodynamics, stating that the change in internal energy (ΔU) of a system is equal to the heat added to the system (dQ) plus the work done on the system (dW).

U is a state function

closed system (n = const.)

dQ|_V = C_VdT, where C_V = (∂U/∂T)_V

dQ|_V: infinitesimal amount of heat added to a system while the volume is held constant; at constant volume, the system does no mechanical work (dW = pdV = 0) —> dQ = dU and since the change in internal energy at constant volume depends only on temperature —> dU = C_VdT

• C_V = (∂U/∂T)_V: The rate at which the system’s internal energy U changes with temperature T at fixed volume.

• C_V: heat capacity at constant volume; the amount of heat required to raise the system’s temperature by one unit without changing its volume.

Heat capacity of a perfect gas is C_V = 3/2nR

dQ|_P = C_PdT, where C_P = (∂H/∂T)_P

• dQ|_P: infinitesimal amount of heat added to a system while the pressure is held constant; at constant pressure, the system generally does mechanical work as it expands or contracts —> dQ = dU + PdV and H = U + PV, its differential is dH = dU + PdV + VdP (at const. pressure dP = 0) —> dH = dQ; since enthalpy at constant pressure depends only on temperature, dH = C_PdT

• C_P = (∂H/∂T)_P: The rate at which is the system’s internal energy U changes with temperature T at fixed pressure.

• C_P: heat capacity at constant pressure; the amount of heat required to raise the system’s temperature by one unit without changing its pressure.

Heat capacity of a perfect gas is C_P = 5/2nR

dS = dQ^rev/T ≥ dQ/T

Entropy is a state function, this change depends only on the initial and final states, not the path.

Definition of entropy/Clausius inequality, expressing the second law of thermodynamics. It relates the change in entropy (S) to heat transfer (Q) and absolute temperature (T), and distinguishes between reversible and irreversible processes. (spontaneous process in closed system)

• dQ^rev: Infinitesimal heat transferred in a reversible process.

• dQ: Infinitesimal heat transferred in a general (possibly irreversible) process

• T: Absolute temperature at which the heat exchange occurs (in Kelvin)

• ≥: Indicates that entropy change is greater for irreversible processes than for reversible ones

Interpretation: For a reversible process: dS = dQ/T

For an irreversible process: dS > dQ/T

Entropy of an isolated system never decreases.

dS_tot = dS + dS_surr ≥ 0

Total entropy change / entropy balance, expressing the second law of thermodynamics. It states that the total entropy change of a system plus its surroundings is never negative. (spontaneous process in closed system)

• dS_tot: Infinitesimal change in total entropy (system + surroundings)

Interpretation: Reversible process: dS_tot = 0 (in equilibrium)

Irreversible (spontaneous) process: dS_tot > 0

A process can occur spontaneously only if it increases the total entropy of the universe.

S = klnW

Boltzmann entropy formula, giving the statistical definition of entropy. It relates the macroscopic entropy of a system to the number of accessible microscopic configurations.

• k: Boltzmann’s constant

• W: Number of micro states of the system consistent with the macroscopic constraints (energy, volume, number of particles)

Interpretation: Entropy increases as the number of possible microscopic arrangements of a system increases. Systems naturally evolve toward macrostates with larger W, corresponding to higher entropy.

dU = -PdV + TdS + ∑_iμ_idn_i

Fundamental thermodynamic relation, expressing the differential change in internal energy of a system in terms of its natural variables.

• -PdV: Mechanical work done by the system

• TdS: Energy change associated with heat transfer in a reversible process

• μ_i: Chemical potential of component i

• dn_i: Infinitesimal change in the amount (moles or particles) of component i.

• ∑_iμ_i​dn_i​: Energy change due to matter exchange or chemical reactions.

Interpretation: This equation combines the first and second laws of thermodynamics and shows how internal energy changes due to; volume work, entropy (heat) changes, and changes in composition.

H = U + PV and dH = VdP + TdS + ∑_iμ_idn_i

Definition and differntial form of enthalpy, a thermodynamic potential useful for processes at constant pressure.

• PV: Pressure-volume energy associated with system expansion

• VdP: Energy change due to a change in pressure

• TdS: Energy change associated with entropy (heat) transfer

• μ_i: Chemical potential of component i

• dn_i: Infinitesimal change in the amount of component i.

• ∑_iμ_i​dn_i​: Energy change due to changes in composition.

Interpretation: Enthalpy is a natural thermodynamic potential with natural variables (S, P, {n_i}). At constant pressure and composition, dH = TdS = dQ_rev. It is a state function that represent the total energy content of a system that is useful for describing processes occuring at constant pressure.

At constant P (dP = 0): dH_P = dQ

A = U -TS and dA = -PdV -SdT + ∑_iμ_idn_i

Definition and differntial form of the Helmholtz free energy, a thermodynamic potential useful for processes at constant temperature and volume.

• A: Maximum (free) work that can be done by the system during an isothermal process.

• TS: Energy unavailable to do useful work at temperature T.

• -PdV: mechanical work term

• -SdT: Change in free energy due to temperature variation

• μ_i: Chemical potential of component i

• dn_i: Infinitesimal change in the amount of component i.

• ∑_iμ_i​dn_i​: Energy change due to changes in composition.

interpretation: The Helmholtz free energy has natural variables (T, V, {n_i}). At constant temperature and volume dA ≤ 0 (spontaneous process), with equality for reversible processes.

G = H - TS and dG = VdP - SdT + ∑_iμ_idn_i

Definition and differential form of the Gibbs free energy, the thermodynamic potential most useful/the energy in a system available to do useful work for/at processes at constant temperature and pressure.

• G: State funcion and also called free enthalpy

• TS: Energy unavailable to do useful work at temperature

• VdP: Change in free energy due to pressure variation

• -SdT: Change in free energy due to temperature variation

• μ_i: Chemical potential of component i

• dn_i: Infinitesimal change in the amount of component i.

• ∑_iμ_i​dn_i​: Energy change due to changes in composition.

Interpretation: The Gibbs free energy has natural variables (T, P, {n_i}). At constant temperature and pressure, dG ≤ 0 (spontaneous process), with equality for reversible processes.

At constant temperature and pressure, a process is spontaneous if the Gibbs free energy decreases. At equilibrium, G is minimized and dG = 0.

Δ_rG = (∂G/∂ξ)_P,T = Δ_rG° + RTlnQ with Q = Π_ia_i^v_i

• Δ_rG: Gibbs free energy change of the reaction at the current conditions

• (∂G/∂ξ)_P,T: Change in Gibbs free energy with respect to the extent of reaction ξ, at constant pressure and temperature

• ξ: Extent of reaction, a measure of reaction progress

• Δ_rG°: Standard Gibbs free energy change of reaction, defined for standard-state conditions

• Q: Reaction quotient, describing the current composition of the system

• a_i: Activity of species i (effective concentration)

• v_i: Soichiometric coefficient of species i (positive for products, negative of reactants)

• Π_i: Product over all species in the reaction

Interpretation: Δ_rG < 0: reaction proceeds spontaneously in the forward direction

Δ_rG = 0: reaction is at equilibrium

Δ_rG > 0: reaction is non-spontaneously in the forward direction

The Gibbs free energy change of a reaction depends on its standard free energy change and on how far the system’s current composition is from equilibrium.

ΔrG° = -RTlnK with K = (Πiaivi )eq

Standard Gibbs free energy change-equilibrium relation, linking thermodynamics with chemical equilibrium.

• Δ_rG°: Standard Gibbs free energy change of reaction, defined for standard-state conditions

• K: Equilibrium constant of the reaction

• a_i: Activity of species i

• v_i: Soichiometric coefficient of species i (positive for products, negative of reactants)

• Π_i: Product over all species in the reaction

• (…)_eq: Evaluated at equilibrium

Interpretation: K > 1 —> Δ_rG° < 0: products favored at equilibrium

K = 1 —> Δ_rG° = 0: neither side favored

K < 1 —> Δ_rG° > 0: reactants favored

The standard Gibbs free energy change of a reaction determines the position of equilibrium: the more negative Δ_rG° is, the larger the equilibrium constant and the more product-favored the reaction.

Δ_rG = -νFE, so E = E° -(RT/νF)lnQ

Thermodynamic-electrochemical relation (Nernst equation), linking the Gibbs free energy change of a redox reaction to the cell potential.

• Δ_rG: Gibbs free energy change of the electrochemical reaction

• ν: Number of moles of electrons transferred in the reaction

• F: Faraday constant, change per mole of electrons

• E: Cell (electrode) potential under non-standard conditions

• E°: Standard cell potential

• Q: Reaction quotient, expressed in terms of activities of reactants and products

Interpretation: Δ_rG < 0 —> E > 0: reaction is spontaneous (galvanic cell)

Δ_rG = 0 —> E = 0: system at equilibrium

Δ_rG > 0 —> E < 0: reaction is non-spontaneous

At equilibrium (Q = K), E° = (RT/vF)lnK.

dW = -P_ext.dV + dW’ and dW’_max = (dG)_P,T and dW’ = Edq

Work decomposition and maximum non-expansion work, relating mechanical work, GIbbs free energy and electrical work.

dW: Total infinitesimal work done by the system

• P_ext: External pressure

• -P_extdV: Expansion (pressure-volume) work

• dW’: Non-expansion work (e.g. electrical, surface, magnetic work)

• dW’_max: Maximum non-expansion work obtainable from the system

• (dG)_P,T: Change in Gibbs free energy at constant pressure and temperature —> at constant P and T: dW’_max: -dG

• E: Electrical potential (cell voltage)

• dq: Infinitesimal electric charge transferred

• dW’ = Edq: Electrical work done by the system

Interpretation:

• Total work is the sum of expansion work and non-expansion work

• At constant temperature and pressure, Gibbs free energy determines the maximum useful (non-PV) work

• In electrochemical systems, this useful work appears as electrical work, directly related to the cell potential

μ_i = (∂G/∂n_i)_P,T,nj≠i = (∂A/∂n_i)_V,T,nj≠i = (∂H/∂n_i)_P,S,nj≠i = (∂U/∂n_i)_V,S,nj≠i

Definition of the chemical potential, expressing how the energy of a system changes when the amount of a component is varied.

• μ_i: Chemical potential of component i; the partial molar Gibbs free energy

• n_i: Amount (moles or particles) of component i

• n_j≠i: Amounts of all other components, held constant

Interpretation of each expression:

• (∂G/∂n_i)_P,T,nj≠i: Change in Gibbs free energy when one adds component i at constant pressure and temperature. (Most commonly used definition, especially in chemistry and phase equilibrium)

• (∂A/∂n_i)_V,T,nj≠i: Change in Helmholtz free energy at constant volume and temperature.

• (∂H/∂n_i)_P,S,nj≠i: Change in enthalpy at constant pressure and entropy.

• (∂U/∂n_i)_V,S,nj≠i: Change in internal energy at constant volume and entropy.

The chemical potential measures the tendency of a species to enter or leave a system.

G_P,T = Π_iμ_in_i

Euler equation for Gibbs free energy, expressing the total Gibbs free energy of a system at constant pressure and temperature in terms of chemical potentials.

• GP,T​: Gibbs free energy of the system at fixed pressure and temperature.

• μ_i: Chemical potential of component i

• n_i: Amount (moles or particles) of component i

• ∑_i: Sum over all components in the system

• Forms the basis for equilibrium conditions: minimizing G at constant P, T determines chemical and phase equilibrium

• Leads directly to the Gibbs-Duhem equation when differentiated

Δ^ideal_mixS = -nR(x_Aln(x_A)+x_Bln(x_B))

Entropy of mixing for an ideal binary mixture, giving the increase in entropy when two ideal substances are mixed.

• ΔS^ideal_mix​: Entropy change upon mixing.

• n: Total number of moles in the mixture.

• x_A,x_B​: Mole fractions of components A and B.

Interpretation: This formula applies to ideal mixtures (ideal gases or ideal solutions), where interactions between unlike and like particles are equivalent. Because 0 < x_i < 1, each lnx_i < 0, making the overall entropy change positive: Δ^ideal_mixS > 0

Mixing increases entropy because particles become more randomly distributed, increasing the number of accessible microstates.

General form (multicomponent system):

ΔS^ideal_mix = −nR∑_ix_iln⁡x_i

(∂V/∂T)_P,W’,ni = -(∂S/∂P)_{T,W’, ni}

Maxwell relation (from the Gibbs free energy), linking how volume changes with temperature to how entropy changes with pressure.

• W′ : Other non-expansion work variables held constant (e.g. electrical work).

• n_i: Amounts of each component, held constant.

Origin: Starting from the differential of the Gibbs free energy,

dG = VdP−SdT+∑_iμ_idn_i + (other work terms), the equality of mixed second derivatives of G leads to this relation.

Physical interpretation:

• (∂V/∂T)_P​ describes thermal expansion at constant pressure.

• −(∂S/∂P)_T​ describes how entropy decreases under compression.

X_i = (∂X/∂n_i)_P,T,nj≠i

Partial molar quantity, describing the contribution of component i to an extensive property of a mixture.

• X_i:​ Partial molar value of the extensive property X for component i

• X: Any extensive thermodynamic property (e.g. V,U,H,S,G)

• n_i: Amount (moles) of component i

• P, T: Pressure and temperature held constant.

• n_j≠i: Amounts of all other components held constant.

The partial molar quantity tells how much the total property X changes when a small amount of component i is added to a large mixture at fixed P and T.

∑_jn_jdμ_j = 0

Gibbs-Duhem equation, expressing a fundamental constraint between the chemical potentials in a multicomponent system.

• n_j: Amount (moles) of component j.

• μ_j: Chemical potential of component j.

• dμ_j: Infinitesimal change in chemical potential.

This form applies at constant temperature and pressure.

Interpretation: Because the Gibbs free energy is an extensive property, G_P,T = Π_iμ_in_i. Differentiating at constant T and P gives ∑_jn_jdμ_j = 0.

Physical meaning: The chemical potentials in a mixture are not independent: if one chemical potential changes, at least one other must adjust to satisfy this constraint.

P_j = x_jP_j^*

Raoult’s law for ideal solutions, relating the partial pressure of component j to its mole fraction in the liquid phase.

• P_j: Partial pressure of component j in the vapor.

• x_j: Mole fraction of component j in the liquid

• P_j^*: Vapor pressure of pure component j at the same temperature

In an ideal solution, each component contributes to the vapor pressure in proportion to its mole fraction.

P_j = y_jP

Dalton’s law, relating the partial pressure of a gas to its mole fraction in the vapor phase.

• P_j: Partial pressure of component j

• y_j: Mole fraction of component j in the vapor

• P: Total pressure of the gas mixture

Each gas in a mixture exerts a fraction of the total pressure proportional to its mole fraction.

P_B = x_BK_B

Henry’s law for dilute solutions, describing the behavior of a solute at low concentration.

• P_B: Partial pressure of solute B in the vapor.

• x_B: Mole fraction of solute B in the liquid.

• K_B: Henry’s law constant for solute B (depends on temperature and solute-solvent pair)

At low concentrations, the vapor pressure of a solute is proportional to its mode fraction, but with a proportionality constant different from the pure-component vapor pressure.

(∂µ_β/∂P)_T − (∂µ_α/∂P)_T = ∆_trsV

Pressure dependence of chemical potential/phase transition relation, expressing how the difference in chemical potentials between two phases changes with pressure.

• µ_α, µ_β: Chemical potentials of a substance in phases α and β (e.g. solid and liquid).

• (∂µ/∂P)_T: Change in chemical potential with pressure at constant temperature

• ∆_trsV: Volume change of transition between two phases: Δ_trs​V = V_β​ − V_α​

Explains why increasing pressure favors the phase with smaller molar volume.

Forms part of the Clapeyron equation.

Used to analyze phase equilibria and pressure-induced phase transitions.

(∂µ_β/∂T)_P − (∂µ_α/∂T)_P = −∆_trsS

Temperature dependence of chemical potential/phase transition relation, expressing how the difference in chemical potentials between two phases changes with temperature.

• µ_α, µ_β: Chemical potentials of a substance in phases α and β (e.g. solid and liquid).

• (∂µ/∂T)_P: Change in chemical potential with temperature at constant pressure

• ∆_trsS: Entropy change of transition between two phases: Δ_trs​S = S_β​ − S_α

Explains why increasing temperature favors the phase with higher entropy.

Together with the pressure relation, leads to the Clapeyron equation.

Central to understanding phase stability and phase diagrams.

P = P*exp(V_m∆P/RT)

Pressure dependence of vapor pressure, describing how the vapor pressure of a condensed phase changes with pressure at constant temperature.

• P: Vapor pressure of the substance at pressure P^* + ΔP.

• P^*: Vapor pressure at a reference pressure (usually the saturation vapor pressure at the same temperature).

• V_m: Molar volume of the condensed phase (liquid or solid)

Interpretation: This equation shows that increasing the pressure on a condensed phase increases its vapor pressure exponentially. The effect is usually small for liquids and solids because their molar volumes are small.

Assumptions: Constant V_m and T, Vapor behaves ideally

dP/dT =∆_trsS/∆_trsV = ∆_trsH/T∆_trsV

Clapeyron equation, describing the slope of a phase boundary in a pressure-temperature phase diagram.

• dP/dT: Slope of the phase equilibrium line between two phases

• Δ_trs​S: Entropy change of the phase transition.

• Δ_trsV: Volume change of the phase transition.

• Δ_trsH: Enthalpy change (latent heat) of the phase transition.

Predicts how phase equilibrium shifts with temperature and pressure.

Explains unusual behavior such as the negative melting curve slope of ice (because Δ_trsV < 0)

dlnP/dT ≈ ∆_trsH/RT²

Clausius-Clapeyron equation (approximate form), describing how the vapor pressure depends on temperature during a phase transition involving a gas (typically vaporization or sublimation).

• P: Vapor pressure of the substance

• Δ_trs​H: Enthalpy change of transition (e.g. enthalpy of vaporization or sublimation)

Assumptions:

• The vapor behaves as an ideal gas

• The molar volume of the condensed phase is negligible compared to that of the vapor.

• Δ_trs​H is approximately constant over the temperature range

Physical interpretation: The vapor pressure of a substance increases rapidly with temperature, with the rate of increase governed by the enthalpy of the phase transition.

Integrated form (commonly used):

lnP = -(Δ_trs​H​/R) (1/T) + C

or, between two temperatures,

ln⁡ ⁣(P₂/P₁) = −(Δ_trsH/R) (1/T₂ − 1/T₁)

∆T = (RT^*2/∆_trsH)x_B

Boiling-point elevation (ideal dilute solution)

• ΔT: Increase in boiling temperature of the solution relative to the pure solvent.

• T^*: Boiling temperature of the pure solvent.

• Δ_trsH: Enthalpy of vaporization of the solvent.

• x_B: Mole fraction of the solute B.

Interpretation: Adding a nonvolatile solute lowers the vapor pressure of the solvent, so a higher temperature is required for boiling

lnx_B = ∆_fusH/R[1/T_fus − 1/T]

Freezing-point depression (ideal solution)

• x_B: Mole fraction of solute B in the liquid phase.

• Δ_fus​H: Enthalpy of fusion of the solvent.

• T_fus: Freezing (melting) temperature of the pure solvent.

• T: Freezing temperature of the solution.

Interpretation: The presence of solute stabilizes the liquid phase, lowering the temperature at which solid-liquid equilibrium occurs.

For very dilute solutions, this leads to the familiar linear freezing-point depression: ΔT_fus ​∝ x_B

µ_i = µ_i° + RTlna_i = µ_i° + RTlnx_i + RTlnγ_i

Chemical potential in terms of activity, describing real (non-ideal) solution behavior.

• μ_i: Chemical potential of component i

• μ_i°: Standard chemical potential of component i (defined by the chosen standard state)

• a_i:​ Activity of component i, a measure of its effective concentration.

• x_i: Mole fraction of component i

• γ_i: Activity coefficient of component i, accounting for deviations from ideality.

Key relations: a_i = x_iγi_i

• For an ideal solution: γ_i = 1, so μ_i = μ_i° + RTln⁡x_i

• For a non-ideal solution: γ_i ≠ 1, and the additional term RTln⁡γ_i​ quantifies intermolecular interaction effects.

Activity replaces mole fraction to account for non-ideal interactions between particles.

µ_i = µ_i° + RTlna_i = µ_i° + RTln(b_i/b°) + RTlnγ_i

Chemical potential in terms of activity and molality, used for real solutions (especially electrolytes).

• μi_i: Chemical potential of component i

• μ_i°: Standard chemical potential of component i

• a_i: Activity of component i

• b_i: Molality of component i (mol*kg⁻¹)

• b°: Standard molality (1 mol*kg⁻¹)

• γ_i: Activity coefficient on the molality scale.

Key relation: a_i = γ_i(​b_i/b°)​​

• For an ideal dilute solution: γ_i ​= 1, so µ_i = µ_i° + RTln(b_i/b°)

• For a non-ideal solution: γ_i ≠ 1, and the additional term RTlnγ_i quantifies intermolecular or ionic interaction effects

Central to solution thermodynamics and phase equlibrium.

Essential for describing electrolytes and real solutions.

Activity corrects molality to reflect the true thermodynamic “escaping tendency” of a component

F = C − P + 2

Gibbs phase rule, giving the number of independent variables required to describe a system at equilibrium.

• F: Degrees of freedom —> Number of independent intensive variables (e.g. temperature, pressure, composition) that can be changed without altering the number of phases in equilibrium

• C: Number of components —> Minimum number of chemically independent species needed to describe the composition of all phases

• P: Number of phases —> Physically and chemically uniform, mechanically separable parts of the system (e.g. solid, liquid, gas)

• 2: The two fundamental intensive variables, temperature and pressure

Fundamental to phase diagrams.

Predicts the maximum number of phases that can coexist.

As the number of phases increases, the number of variables that can be independently adjusted decreases.

Examples:

• Single-phase system (P = 1): F = C + 1

• Two-phase equilibrium (P = 2): F = C

• Triple point of a one-component system (C = 1, P = 3): F = 0 (No freedom: both T and P are fixed)

n_αl_α = n_βl_β

Lever rule, used in phase diagrams to relate the amounts of coexisting phases in a two-phase region.

• n_α: Amount (moles or mass) of phase α

• n_β: Amount (moles or mass) of phase β

• l_α: Length of the tie-line segment opposite phase α in the phase diagram

• l_β: Length of the tie-line segment opposite phase β

Allows calculation of phase amounts in binary phase diagrams.

Widely used in liquid-liquid, solid-liquid, and solid-solid equilibria.

γ_± = (γ^p₊γ^q₋)^1/(p+q)

Mean ionic activity coefficient, used to describe the non-ideal behavior of electrolytes.

• γ_±: Mean ionic activity coefficient of the electrolyte

• γ₊: Activity coefficient of the cation

• γ₋: Activity coefficient of the anion

• p: Stoichiometric coefficient of the cation

• q: Stoichiometric coefficient of the anion

Physical interpretation: For an electrolyte that dissociates as A_pB_q → pA⁺ + qB⁻, individual ionic activity coefficients cannot be measured independently. Instead, their combined effect is represented by the mean ionic activity coefficient.

Central to electrolyte thermodynamics.

Used in equilibrium constants, electrochemistry, and colligative properties.

Appears in models such as Debye-Hückel theory

Special case (1:1 electrolyte): For NaCl (p = q = 1): γ_± = √γ₊γ₋​

logγ_± = −|z₊z₋|A√I

Debye-Hückel limiting law, describing the activity coefficients of electrolytes in very dilute solutions.

• γ_±: Mean ionic activity coefficient

• log: Base-10 logarithm

• z₊: Charge number of the cation

• z₋: Charge number of the anion

• |z₊z₋|: Product of ionic charges (absolute value)

• A: Debye-Hückel constant (depends on temperature and solvent)

• I: Ionic strength of the solution

Physical interpretation: As ionic strength increases, electrostatic interactions between ions become stronger, causing deviations from ideal behavior and reducing the activity coefficient below 1.

Fundamental equation in electrolyte thermodynamics.

Explains why electrolyte solutions are non-ideal, even at low concentrations.

Used in calculating equilibrium constants, cell potentials and activities.

Higher ionic strength —> stronger ion-ion interactions —> smaller γ_±

I = 1/2∑_i z_i²(b_i/b°)

Ionic strength (molality scale), measuring the total electrostatic effect of ions in a solution.

• I: Ionic strength of the solution.

• ∑_i: Sum over all ionic species in solution.

• z_i: Charge number of ion i

• b_i: Molality of ion i (mol*kg⁻¹).

• b°: Standard molality (1 mol*kg⁻¹).

• 1/2: Factor accounting for double counting of ion–ion interactions.

Central parameter in electrolyte thermodynamics.

Appears in Debye-Hückel equation for activity coefficients.

Used in equilibrium, electrochemistry and colligative property calculations

Ionic strength weights each ion by the square of its charge, so highly charged ions contribute much more strongly than singly charged ions.

Ionic strength increases with both ion concentration and ion charge.

A = (F³/(4πN_Aln10)) (ρb°/2e³R³T³)^1/2

Debye-Hückel constant A, appearing in the Debye-Hückel limiting law for electrolyte solutions. It determines how strongly ionic activity coefficients depend on ionic strength.

• A: Debye-Hückel constant (depends on temperature and solvent)

• F: Faraday constant

• N_A: Avogadro constant

• ρ: Density of the solvent

• b°: Standard molality (1 mol*kg^-1)

• ε₀: Permittivity of free space

• ε_r: Relative permittivity (dielectric constant) of the solvent

Physical interpretation: The constant A reflects how effectively the solvent screens electrostatic interactions between ions. A high dielectric constant (large ε_r) weakens ion-ion interactions, reducing deviations from ideality.

Depends only on solvent properties and temperature, not on the electrolyte.

P_in = P_out + 2γ/r

Laplace’s equation/Kelvin equation for a curved interface

P_in: Pressure inside a droplet or bubble

P_out: Pressure outside the droplet or bubble

γ: Surface tension of the liquid (N/m)

r: Radius of curvature of the droplet or bubble.

Physical interpretation: The pressure inside a curved surface (droplet or bubble) is higher than outside due to surface tension. Smaller droplets/bubbles have higher internal pressure because P_in - P_out ∝1/r

Explains phenomena like capillarity, bubble formation and emulsion stability

P = ρgh

Hydrostatic pressure

• P: Pressure at depth h in a fluid

• ρ: Density of the fluid

• g: Acceleration due to gravity

• h: Height (depth) of the fluid column

Pressure increases linearly with depth in a static fluid.

Fundamental to fluid statics and capillary phenomena.

Often combined with Laplace’s equation to study curved interfaces under gravity

w_ad = γ_sg + γ_lg − γ_sl

Work of adhesion/wetting equation, describing the thermodynamic of a liquid spreading on a solid surface.

• w_ad: Work of adhesion (energy required to separate unit area of the solid-liquid interface)

• γ_sg​: Surface tension of the solid–gas interface.

• γ_lg: Surface tension of the liquid–gas interface.

• γ_sl: Surface tension of the solid–liquid interface.

Physical interpretation:

• Determines wettability of a liquid on a solid surface.

• If w_ad > 0, the liquid spreads on the solid (good wetting)

• If w_ad < 0, the liquid beads up (poor wetting)

Predicts whether liquids will spread, form droplets or stick to surfaces.

γ_sg = γ_sl + γ_lgcosΘ_c

Young’s equation, describing the contact angle of a liquid droplet on a solid surface.

• γ_sg​: Surface tension of the solid–gas interface.

• γ_lg: Surface tension of the liquid–gas interface.

• γ_sl: Surface tension of the solid–liquid interface.

• Θ_​c: Contact angle of the liquid droplet on the solid surface

Physical interpretation:

• Relates surface tensions to the shape of a droplet on a solid surface.

• The contact angle Θ_​c indicates wettability: Θ_​c < 90° —> good wetting and Θ_​c > 90° —> poor wetting

• If Θ_​c = 0, the liquid spreads completely

Used to calculate work of adhesion:

w_ad = γ_lg(1 + cos⁡Θ_c)

n_i/N = (exp(-ε_i/kT))/q with q = ∑_iexp(-ε_i/kT) and <X> = N<x> = N∑_ix_i(n_i/N)

n_i/N = (exp(-ε_i/kT))/q: Boltzmann distribution —> Gives the fraction of particles in a particular energy state at temperature T

q = ∑_iexp(-ε_i/kT): Partition function —> Encodes the statistical properties of the system

<X> = N<x> = N∑_ix_i(n_i/N): Average value of an arbitrary quantity X of the entrie system at a certain temperature —> GIves the expected total value of a property over the system

• n_i: Number of particles in state i

• N: Total number of particles.

• ε_i: Energy of state i.

• k: Boltzmann constant.

• T: Absolute temperature.

• q: Partition function; normalizes the probabilities of each state.

• ⟨X⟩: Ensemble average of property X over the system.

• ⟨x⟩: Average value of x per particle.

• x_i: Value of property x for a particle in state i.