Physical Chemistry: Multicomponent Systems and Thermodynamics

Composition and Concentration Measures of Multicomponent Solutions

• Definitions and Terminology:

  • A solution is a homogeneous system consisting of two or more mixed components possessing variable concentrations.
  • The component constituting the major proportion of the solution is defined as the solvent (typically designated as component $1$).
  • All other components present in lower proportions are defined as dissolved substances or solutes.

• Molar and Molecular Quantities:

  • In a volume $V$ containing $k$ components, where the quantity of component $i$ is $n_i$ moles, the total molar quantity $n$ is:     $n = \sum_{i=1}^{k} n_i$
  • Expressed in terms of particle counts, if $N_i$ represents the number of molecules of component $i$, the total number of molecules $N$ is:     $N = \sum_{i=1}^{k} N_i$

• Molar Concentration ($c_i$) and Particle Concentration ($C_i$):

  • Molar concentration $c_i$ is defined as the number of moles of component $i$ per unit volume:     $c_i = \frac{n_i}{V}$
  • The standard SI unit for molar concentration is $\text{mol}\,\text{m}^{-3}$ or $\text{mol}\,\text{dm}^{-3}$ (replacing the older convention $\text{mol/l}$).
  • Particle concentration $C_i$ expresses the number of particles per unit volume:     $C_i = \frac{N_i}{V}$
  • The SI unit for particle concentration is $\text{m}^{-3}$ or $\text{dm}^{-3}$.

• Mole Fraction ($x_i$):

  • The mole fraction $x_i$ is the dimensionless ratio of the molar quantity of component $i$ to the total moles in solution:     $x_i = \frac{n_i}{n} = \frac{N_i}{N} = \frac{c_i}{\sum_{k} c_k}$
  • Mole fractions fulfill the normalization condition:     $\sum_{i=1}^{k} x_i = 1$
  • For a solution containing $k$ distinct components, exactly $k - 1$ mole fractions are independent parameters.

• Mole Ratio ($r_i$):

  • The mole ratio $r_i$ is defined as the number of moles of solute $i$ per mole of solvent ($n_1$):     $r_i = \frac{n_i}{n_1}$
  • By definition, for the solvent, $r_1 = 1$, and for solutes $i \neq 1$, r_i > x_i$.\n\n* **Molality (m_i)**:\n * Molality m_i$represents the number of moles of component$i contained within one kilogram of solvent:\n    m_i = \frac{1000\,n_i}{n_1 M_1}\n    where M_1 is the molecular weight of the solvent.\n\n* **Interrelations and Limiting Behavior**:\n * The conversion between mole fraction and mole ratio is given by:\n    x_i = \frac{r_i}{\sum_{k} r_k}\n * The relationship constraint between the solvent fraction and solute ratios obeys:\n    \sum_{k} x_1 r_k = 1\n * In extremely dilute solutions, the solute mole ratio approaches the mole fraction (r_k \approx x_i), and all concentration metrics become directly proportional to one another:\n    x_i \propto c_i \propto m_i \propto r_i\n\n# State Variables and Partial Molar Quantities\n\n* **Degrees of Freedom in Multicomponent Systems**:\n * For a simple one-component system, specifying two thermodynamic parameters (such as temperature T$and pressure$P) uniquely defines the state.\n * For a system containing k components, defining the thermodynamic state requires knowledge of system composition in addition to temperature and pressure.\n * Because only k - 1$mole fractions are independent out of$k$total fractions, the total number of state variables required for a$k$-component system is$k + 1$.

• Thermal Equation of State and Partial Molar Volume:

  • The thermal equation of state for a multicomponent volume is expressed as:     $V = V(P, T, n_1, n_2, n_3, \dots)$
  • The total differential $dV$ is expanded as:     $dV = \left(\frac{\partial V}{\partial P}\right)_{T, \{n_i\}} dP + \left(\frac{\partial V}{\partial T}\right)_{P, \{n_i\}} dT + \sum_{i=1}^{k} \left(\frac{\partial V}{\partial n_i}\right)_{P, T, \{n_{j \neq i}\}} dn_i$     where the notation $\{n_i\}$ indicates that molar quantities of all components, except the variable in the partial derivative, are held constant.
  • The partial molar volume $v_i$ of component $i$ is defined by:     $v_i = \left(\frac{\partial V}{\partial n_i}\right)_{P, T, \{n_{j \neq i}\}}$
  • Since total volume $V$ is an extensive quantity and a homogeneous function of first order with respect to mole numbers, applying Euler's theorem for homogeneous functions yields:     $V = \sum_{i=1}^{k} v_i n_i$
  • Partial molar volumes are generally distinct from the molar volumes of pure components. In certain instances, partial molar volumes can be negative (for example, magnesium sulfate, $\text{MgSO}_4$, in dilute aqueous solutions). A negative partial molar volume indicates that adding the solute causes a net compression or decrease in total solution volume.

• General Definitions of Partial Molar Properties:

  • Chemical Potential (Partial Gibbs Free Energy):     $\mu_i \equiv g_i = u_i - T s_i + P v_i$     where $u_i$ is the partial internal energy and $s_i$ is the partial entropy.
  • Partial Entropy:     $s_i = -\left(\frac{\partial \mu_i}{\partial T}\right)_{P, \{n_k\}}$
  • Partial Volume:     $v_i = \left(\frac{\partial \mu_i}{\partial P}\right)_{T, \{n_k\}}$
  • Partial Enthalpy:     $h_i = u_i + P v_i = \mu_i - T \left(\frac{\partial \mu_i}{\partial T}\right)_{P, \{n_k\}}$
  • Partial Isobaric Heat Capacity:     $C_{P,i} = T \left(\frac{\partial s_i}{\partial T}\right)_{P, \{n_k\}} = \left(\frac{\partial h_i}{\partial T}\right)_{P, \{n_k\}}$
  • Extensive Total Properties as Sums of Partial Quantities:     $S = \sum_{i} n_i s_i$$H = \sum_{i} n_i h_i$

Average Molar Quantities and the Gibbs-Duhem Equation

• Average Molar and Density Characteristics:

  • Average molar quantities represent extensive parameters divided by the total number of moles $n$ in the solution, independent of component type.
  • Average Molar Entropy ($s$):     $s = \frac{S}{n} = \frac{1}{n} \sum_{i} n_i s_i = \sum_{i} x_i s_i$
  • Average Molar Volume ($v$):     $v = \frac{V}{n} = \frac{1}{n} \sum_{i} n_i v_i = \sum_{i} x_i v_i$
  • Property Densities (per unit volume), such as Enthalpy Density ($h$):     $h = \frac{H}{V} = \frac{1}{V} \sum_{i} n_i h_i = \sum_{i} c_i h_i$

• Gibbs-Duhem Relationship in Solutions:

  • Expressed using average molar quantities and mole fractions, the Gibbs-Duhem equation is written:     $-s\,dT + v\,dP - \sum_{i} x_i\,d\mu_i = 0$
  • The chemical potential of component $i$ is a function of temperature, pressure, and composition:     $\mu_i = \mu_i(T, P, x_1, x_2, \dots)$
  • The total differential of chemical potential is:     $d\mu_i = v_i\,dP - s_i\,dT + \hat{D}\mu_i$
  • The differential operator $\hat{D}$ isolates the change in chemical potential due strictly to composition changes:     $\hat{D}\mu_i = \sum_{k} \left(\frac{\partial \mu_i}{\partial x_k}\right) dx_k$
  • Substituting $d\mu_i$ into the Gibbs-Duhem equation and invoking the relations between average and partial quantities yields the fundamental compositional constraint:     $\sum_{i} x_i \hat{D}\mu_i = 0$

Thermodynamics of Ideal Solutions

• Definition of Ideal Solutions:

  • An ideal solution is rigorously defined as a mixture in which the chemical potential of every component obeys:     $\mu_i = \mu_i^0 + RT \ln(x_i)$     where the standard chemical potential $\mu_i^0$ is a function strictly of temperature and pressure ($T, P$), independent of molecular structure or concentration.

• Partial Molar Quantities in Ideal Solutions:

  • Partial Molar Entropy:     $s_i = -\left(\frac{\partial \mu_i}{\partial T}\right) = s_i^0 - R \ln(x_i)$
  • Partial Molar Volume:     $v_i = \left(\frac{\partial \mu_i}{\partial P}\right) = v_i^0$     The partial molar volume in an ideal solution is concentration-independent and identical to its pure/standard value. Consequently, mixing ideal solution components occurs at constant $T$ and $P$ with zero volume change ($\Delta V_{mix} = 0$):     $V = \sum_{i} n_i v_i = \sum_{i} n_i v_i^0$
  • Pressure Dependence of Partial Volume:     $v_i = v_i^* (1 - \beta_i P)$     where $v_i^*$ corresponds to the volume at zero pressure and $\beta_i$ represents the thermodynamic compression factor of component $i$. The parameters $v_i^*$ and $\beta_i$ are independent of pressure and composition, but depend on temperature.
  • Total Enthalpy and Heat of Dissolution:     $H = \sum_{i} n_i h_i = \sum_{i} n_i h_i^0$     The heat of mixing or dissolution for ideal solutions is precisely zero ($\Delta H_{mix} = 0$).

• Free Energy Expressions:

  • Total Gibbs Free Energy ($G$):     $G = \sum_{i} n_i \mu_i^0 + RT \sum_{i} n_i \ln(x_i) = V \sum_{i} c_i \mu_i^0 + RT V \sum_{i} c_i \ln(x_i)$
  • Total Helmholtz Free Energy ($F$):     $F = V \sum_{i} c_i \mu_i^0 + RT V \sum_{i} c_i \ln(x_i) - P V$

• Average Molar Characteristics of Ideal Mixtures:

  • Average Molar Gibbs Free Energy:     $g = \sum_{i} x_i \mu_i^0 + RT \sum_{i} x_i \ln(x_i)$
  • Average Molar Entropy:     $s = -\sum_{i} x_i \left(\frac{\partial \mu_i^0}{\partial T}\right) - R \sum_{i} x_i \ln(x_i) = \sum_{i} x_i s_i^0 - R \sum_{i} x_i \ln(x_i)$
  • Average Molar Enthalpy:     $h = \sum_{i} x_i \left(\mu_i^0 - T \frac{\partial \mu_i^0}{\partial T}\right) = \sum_{i} x_i h_i^0$
  • Average Molar Volume:     $v = \sum_{i} x_i \left(\frac{\partial \mu_i^0}{\partial P}\right) = \sum_{i} x_i v_i^0$

• Entropy of Mixing:

  • Mixing individual components to form an ideal solution generates an entropy increase termed the entropy of mixing. For a binary ideal solution with mole fractions $x$ and $1 - x$:     $\Delta S_{mix} = -R \left[x \ln(x) + (1 - x) \ln(1 - x)\right]$

• Series Expansion in Solvent-Excess Limit ($1 - x_1 \ll 1$):

  • When component $1$ functions as the primary solvent:     $\mu_1 \approx \mu_1^0 - RT \sum_{i \neq 1} x_i$$s_1 \approx s_1^0 - \left(\frac{\partial \mu_1^0}{\partial T}\right) + R \sum_{i \neq 1} x_i$$v_1 \approx v_1^0$

• Approximations for Highly Dilute Solutions:

  • In extreme dilution, alternative concentration units yield the approximate chemical potential formulations:     $\mu_i \approx \mu_i^0 + RT \ln(r_i) \approx \mu_i^0 + RT \ln(m_i) \approx \mu_i^0 + RT \ln(c_i)$
  • Standard state definitions must be maintained strictly when using dimensional quantities ($m_i$, $c_i$) inside logarithmic arguments.

Nonideal Solutions and Osmosis

• Activity and Activity Coefficients:

  • Real solutions exhibit deviations from ideal logarithmic equations. To maintain formal simplicity, non-ideality is incorporated into the activity parameter $a_i$:     $\mu_i = \mu_i^0 + RT \ln(a_i)$
  • The activity $a_i$ relates to molar concentration $c_i$ via the activity coefficient $\gamma_i$:     $a_i = \gamma_i c_i$
  • Under infinite dilution, $\gamma_i \rightarrow 1$ and $a_i \rightarrow c_i$.\n\n* **Osmotic Coefficient (\phi$)**:\n * For solvent component 1$, non-ideality in dilute solution behavior is frequently parametrized using the osmotic coefficient$\phi:\n    \mu_1 = \mu_1^0 - \phi RT \sum_{i eq 1} r_i\n    where r_i$is the mole ratio of solute$i$.

• Comparison of Gibbs and Helmholtz Free Energies in Liquids:

  • In simple fluid systems, $G$ and $F$ differ by the product $P V$. In liquid phases, $P V \ll T S$; therefore, the numerical difference between Gibbs and Helmholtz free energies is negligible in simple liquid systems.

• Osmotic Equilibrium and Osmotic Pressure ($\Pi$):

  • Consider a solution phase $\beta$ separated from a pure solvent phase $\alpha$ by a semi-permeable membrane permeable exclusively to solvent molecules. Solvent passage generates an equilibrium pressure differential called osmotic pressure $\Pi = P^\beta - P^\alpha$.
  • Equilibrium condition:     $\mu_1^\alpha(P^\alpha, T) = \mu_1^\beta(P^\beta, T)$
  • Substituting the osmotic coefficient representation:     $\mu_1^0(P^\alpha, T) = \mu_1^0(P^\beta, T) - \phi RT \sum_{i \neq 1} r_i$
  • Expanding $\mu_1^0(P^\beta, T)$ in a Taylor series around $P^\alpha$:     $\mu_1^0(P^\beta, T) = \mu_1^0(P^\alpha, T) + \left(\frac{\partial \mu_1^0}{\partial P}\right) (P^\beta - P^\alpha) = \mu_1^0(P^\alpha, T) + v_1 \Pi$
  • Solving for osmotic pressure $\Pi$:     $\Pi = P^\beta - P^\alpha = \frac{\phi RT \sum_{i \neq 1} r_i}{v_1} = \frac{\phi RT \sum_{i \neq 1} n_i}{V_1}$     where $V_1$ is the solvent volume.
  • Since solvent volume $V_1$ is approximately equal to total solution volume $V$:     $\Pi = \phi RT \sum_{i \neq 1} c_i$
  • Setting $\phi = 1$ yields van 't Hoff's law for ideal solutions:     $\Pi = RT \sum_{i \neq 1} c_i$

• Macromolecular Solution Exception:

  • When dissolved solutes consist of macromolecular species (such as hemoglobin inside red blood cells), the solute volume occupies a substantial fraction of the total solution volume even when the mole fraction x_i \ll 1$.\n * In such macromolecular systems, van 't Hoff's law deviates significantly, and the exact molar ratio formula using solvent volume V_1 must be applied even for ideal behavior:\n    \Pi = \frac{RT \sum_{i eq 1} n_i}{V_1}\n\n# Chemical Equilibrium and Electrolyte Dissociation\n\n* **Stoichiometric Formulation**:\n * Any generic chemical reaction among solution species B_i is written:\n    \sum_{i} u_i B_i = 0\n    where u_i represents the stoichiometric coefficient (positive for products, negative for reactants).\n * Example: For water synthesis (2\text{H}2 + \text{O}_2 = 2\text{H}_2\text{O}$), rewritten as$2\text{H}_2\text{O} - 2\text{H}_2 - \text{O}_2 = 0:\n     u{\text{H}2\text{O}} = 2, \quad u{\text{H}2} = -2, \quad u{\text{O}2} = -1\n\n* **Extent of Reaction / Chemical Variable (\xi)**:\n * Molar changes dn_i$during reaction progress are related by the chemical variable$d\xi:\n    \frac{dn_1}{ u_1} = \frac{dn_2}{ u_2} = \dots = d\xi\n\n* **Thermodynamic Equilibrium Criterion**:\n * At constant T$and$P$, the Gibbs free energy differential$dG is minimized:\n    dG = \sum{i} \mu_i dn_i = \left(\sum_{i} u_i \mu_i\right) d\xi = 0\n * Consequently, chemical equilibrium requires:\n    \sum_{i} u_i \mu_i = 0\n\n* **Equilibrium Constants and Law of Mass Action**:\n * Substituting dilute solution chemical potentials \mu_i = \mu_i^0(P,T) + RT \ln(x_i) into the equilibrium criterion yields:\n    \prod_{i} x_i^{ u_i} = K(P, T)\n    where the dimensionless equilibrium constant K(P, T) is:\n    K(P, T) = \exp\left(-\frac{1}{RT} \sum_{i} u_i \mu_i^0(P, T)\right)\n * Converting mole fractions to concentrations c_i produces the Law of Mass Action:\n    \prod_{i} c_i^{ u_i} = K(P, T)\n * For nonideal systems, activities a_i replace concentrations in the mass action expression.\n\n* **Dissociation of Electrolytes**:\n * Consider electrolyte dissolution forming cations B$and anions$A$with respective valence charges$z_B$and$z_A:\n    B_{ u_+} A_{ u_-} \rightleftharpoons u_+ B^{z_B} + u_- A^{z_A}\n * According to the Law of Mass Action:\n    \frac{c_B^{ u_+} c_A^{ u_-}}{c_{BA}} = K\n * Defining the degree of dissociation \alpha$(where$c$is total electrolyte concentration and$c_{BA} is undissociated concentration):\n    \alpha = \frac{c - c_{BA}}{c} \implies c_{BA} = c (1 - \alpha)\n * Ionic concentrations are c_A = u_- \alpha c$and$c_B = u_+ \alpha c. Substituting gives:\n    \frac{( u_+ \alpha c)^{ u_+} ( u_- \alpha c)^{ u_-}}{c (1 - \alpha)} = K \implies \frac{\alpha^{ u_+ + u_-} c^{ u_+ + u_- - 1} u_+^{ u_+} u_-^{ u_-}}{1 - \alpha} = K\n * Ostwald's Dilution Law (for a 1:1 electrolyte where u_+ = 1$and$ u_- = 1):\n    \frac{\alpha^2}{1 - \alpha} c = K\n * As concentration c \rightarrow 0$, the degree of dissociation$\alpha \rightarrow 1, demonstrating that electrolytes undergo complete dissociation in infinitely dilute solutions.\n\n# Gibbs Phase Rule\n\n* **Derivation for Multiphase Systems**:\n * Consider a heterogeneous system containing n$phases and$k components.\n * State variables: Each phase requires temperature T$, pressure$P$, and$k - 1$mole fractions. Across$n$phases, there are$n(k - 1)$compositional variables plus$2$state parameters ($T$and$P$), giving a total of$n(k - 1) + 2 potential variables.\n * System Equilibrium Restrictions:\n * Thermal equilibrium (T^{(1)} = T^{(2)} = \dots = T^{(n)}$): Gives$n - 1 restrictive equations.\n * Mechanical equilibrium (P^{(1)} = P^{(2)} = \dots = P^{(n)}$): Gives$n - 1 restrictive equations.\n * Chemical equilibrium for k$components ($\mu_i^{(1)} = \mu_i^{(2)} = \dots = \mu_i^{(n)}$): Gives$k(n - 1) restrictive equations.\n * Total system restrictions = (n - 1) + (n - 1) + k(n - 1) = (k + 2)(n - 1).\n\n* **Maximum Phase Limit and Degrees of Freedom (f)**:\n * The number of independent thermodynamic variables (degrees of freedom f) is:\n    f = \left[n(k - 1) + 2\right] - (k + 2)(n - 1) = k + 2 - n\n * Because degrees of freedom cannot be negative (f \ge 0$), the maximum number of coexisting phases$n is restricted by:\n    n \le k + 2\n * The Gibbs phase rule equation is strictly stated as:\n    f = k + 2 - n\n\n# Thermodynamics in External Fields\n\n* **Total Chemical Potential (\bar{\mu}_i)**:\n * When exposed to an external force field, temperature remains uniform at equilibrium, but pressure and chemical potential become non-uniform across spatial positions.\n * To account for field interactions, the standard internal chemical potential \mu_i$is combined with potential energy$u_{pot}$to yield the total chemical potential$\bar{\mu}i:\n    \bar{\mu}_i = \mu_i + u{pot}\n * General Field Equilibrium Condition:\n    \bar{\mu}i = \text{const}\n\n* **Specific External Fields**:\n * Electrical Field (Electrochemical Potential):\n    \bar{\mu}_i = \mu_i + z_i F \varphi\n    where z_i$is charge number,$F$is the Faraday constant, and$\varphi is electric potential.\n * Gravitational Field:\n    \bar{\mu}_i = \mu_i + M_i g h\n    where M_i$is molar mass of component$i$,$g$is gravitational acceleration, and$h is height coordinate.\n\n* **Thermodynamic Potentials in Non-Uniform Systems**:\n * A system in an external field cannot be treated as a single homogeneous phase; instead, it is modeled as a continuous sequence of infinitesimally differing phases (\alpha, \beta, \gamma, \dots).\n * Differential expressions for phase \alpha$using total chemical potential$\bar{\mu}_i:\n    dU^\alpha = T^\alpha dS^\alpha - P^\alpha dV^\alpha + \sum{i} \bar{\mu}i^\alpha dn_i^\alpha\n    dH^\alpha = T^\alpha dS^\alpha + V^\alpha dP^\alpha + \sum{i} \bar{\mu}i^\alpha dn_i^\alpha\n    dF^\alpha = -S^\alpha dT^\alpha - P^\alpha dV^\alpha + \sum{i} \bar{\mu}i^\alpha dn_i^\alpha\n    dG^\alpha = -S^\alpha dT^\alpha + V^\alpha dP^\alpha + \sum{i} \bar{\mu}i^\alpha dn_i^\alpha\n    d\Omega^\alpha = -S^\alpha dT^\alpha - P^\alpha dV^\alpha - \sum{i} n_i^\alpha d\bar{\mu}i^\alpha\n * Integrated formulations:\n    G^\alpha = \sum{i} n_i^\alpha \bar{\mu}i^\alpha\n    F^\alpha = \sum{i} n_i^\alpha \bar{\mu}i^\alpha - P^\alpha V^\alpha\n\n* **Single-Component Pure System in Gravitational Field**:\n * Condition for uniform total chemical potential:\n    d\bar{\mu} = d\mu + M g\,dh = 0\n * At constant temperature, d\mu = v\,dP$. Substituting density$\rho = \frac{M}{v} produces the hydrostatic equation:\n    dP = -\rho g\,dh\n * Assuming an incompressible fluid (\rho = \text{const}), integration gives:\n    P = -\rho g h + \text{const}\n\n* **Multicomponent Solutions in Gravitational/Centrifugal Fields**:\n * Equilibrium condition per component i:\n    d(\mu_i + M_i g h) = 0 \implies v_i\,dP + \hat{D}\mu_i + M_i g\,dh = 0\n * Multiplying by mole fraction x_i$and summing over all$k components:\n    \sum{i} x_i v_i\,dP + \sum_{i} x_i \hat{D}\mu_i + \sum_{i} x_i M_i g\,dh = 0\n * Applying \sum x_i \hat{D}\mu_i = 0$, average molar volume$v = \sum x_i v_i$, and average molar weight$M = \sum x_i M_i yields the general hydrostatic equation:\n    v\,dP + M g\,dh = 0 \implies dP = -\frac{M}{v} g\,dh\n * Eliminating pressure dP generates the spatial composition variation differential equation:\n    \hat{D}\mu_i + \left(M_i - v_i \frac{\sum_{k} x_k M_k}{\sum_{k} x_k v_k}\right) g\,dh = 0\n * For highly dilute ideal solutions (\hat{D}\mu_i = RT d\ln(x_i)$) where solvent component$1 dominates:\n    RT d\ln(c_i) + \left(M_i - v_i \frac{M_1}{v_1^0}\right) g\,dh = 0\n * Integrating for an incompressible solution across two spatial positions \alpha$and$\beta$separated by height$h = h_\beta - h_\alpha:\n    \frac{c_i^\beta}{c_i^\alpha} \approx \frac{x_i^\beta}{x_i^\alpha} \approx \exp\left[-\frac{g h}{RT} \left(M_i - v_i \frac{M_1}{v_1^0}\right)\right]\n    This formula describes solute equilibrium distribution under gravity or centrifugal force fields.\n\n# Thermodynamics of Nonequilibrium States\n\n* **Internal Parameters and Auxiliary Fields (Leontovich Formalism)**:\n * Describing nonequilibrium states requires additional internal structural parameters \xi (such as spatial concentration gradients or temperature distributions).\n * An auxiliary external field X$is introduced conceptually to stabilize the non-equilibrium parameter value$\xi as an equilibrium state.\n\n* **Thermodynamic Transformation Path (0 \rightarrow 1^* \rightarrow 1)**:\n * State 0$: Initial true equilibrium state without external field$X$.
  • State $1^*$: System reaches a quasistatic equilibrium in the presence of applied field $X$, attaining the target internal parameter \xi$.\n * State 1$: Desired non-equilibrium state, obtained by rapidly removing field$X$from state$1^$such that internal variables$\xi$and entropy$S remain unchanged.\n\n* **Energy and Work Governing Relations**:\n * Work performed during quasistatic variation of auxiliary field X:\n    \delta W_X = \xi\,dX\n * Energy differential in state 1^:\n    dU^* = T dS^* - P dV - \xi\,dX\n * Transition to state 1$maintains entropy ($S = S^), but energy adjusts by field energy removal:\n    U = U^ + \xi X\n * Differentials of non-equilibrium state thermodynamic potentials:\n    dU = T\,dS - P\,dV + X\,d\xi\n    dF = -S\,dT - P\,dV + X\,d\xi\n\n* **Physical Significance and Applications**:\n * The term X\,d\xi$represents the work done on the system by the auxiliary external field$X$.
  • Any change in non-equilibrium internal energy due to shifts in internal parameter $\xi$ equals the work done by the conjugate external field required to hold that state in equilibrium.
  • Application to Multiphase Non-equilibrium: A system with internal phase equilibrium but non-equilibrium across phase boundaries (e.g., pressure or temperature differences) can be modeled by applying border fields (such as auxiliary pressure). Total thermodynamic potentials equal the sum of equilibrium phase potentials plus the work performed by auxiliary boundary fields.