Thermodynamic Framework

This framework uses the modern International System of Units (SI). A scalar quantity has a real numerical value with the stated unit; a vector quantity has a value in the stated real coordinate space with the stated component unit. The numerical coordinates used elsewhere on this page are divided by their displayed SI units before they are placed in a Euclidean coordinate space.

The model's energy coordinate $U$ and its process-transfer values $Q$ and $W_{\mathrm{o}\mathrm{n}}$ are scalar energies measured in joules (J). Volume $V$ is a scalar measured in cubic metres (m³); amount of substance $n_j$ for species $j$ is a scalar measured in moles (mol); and a particle count $N_j$, when separately used in a microscopic model, is dimensionless. Time $t$ is measured in seconds (s), mass $M$ in kilograms (kg), and a translational velocity ${\mathbf{v}}_{\mathrm{c}\mathrm{m}}$ is a vector in ${\mathbb{R}}^3$ with components measured in metres per second (m/s). The physical meanings of $U,Q,W_{\mathrm{o}\mathrm{n}}$ are fixed by the state and process definitions, not by their units alone.

The relevant exact SI relations are

$1\mathrm{J}=1\mathrm{k}\mathrm{g}{\mathrm{m}}^2{\mathrm{s}}^{-2},1\mathrm{N}=1\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-2},1\mathrm{P}\mathrm{a}=1\mathrm{J}{\mathrm{m}}^{-3}\mathrm{.}$

The kelvin (K) is the SI unit of thermodynamic temperature. The Boltzmann constant has the exact defining value $k_{\mathrm{B}}=1.380649\times {10}^{-23}\mathrm{J}{\mathrm{K}}^{-1}$. The mole is the SI unit of amount of substance; the Avogadro constant has the exact defining value $N_{\mathrm{A}}=6.02214076\times {10}^{23}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$. When a model uses particle counts, $n_j=N_j\mathrm{/}{N}_{\mathrm{A}}$. The molar gas constant is derived as $R=N_{\mathrm{A}}{k}_{\mathrm{B}}=8.31446261815324\mathrm{J}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$. This framework does not set $k_{\mathrm{B}}=1$. These SI definitions do not themselves introduce a thermodynamic temperature state function or entropy function.

Within the macroscopic framework, $(U,V,n_j)$ are adopted state coordinates. A quantity's status as primitive in this model is separate from whether its SI unit is a base or derived unit.

A thermodynamic system $s$ is a model-selected body or collection of bodies together with a specified outer boundary $\mathrm{\partial }s$. The boundary separates the system from its environment for the purpose of classifying transfers; it need not be a material wall. A compound system has named subsystems inside one outer boundary. Transfers between its subsystems are internal to the compound; only transfers across $\mathrm{\partial }s$ are external.

The model specifies which transfers across $\mathrm{\partial }s$ are permitted. A system is closed when no matter crosses its outer boundary; energy may cross. A system is open when only the named matter species and other named transfer channels may cross its boundary. A system or compound is isolated when no matter or energy crosses its outer boundary. These terms classify the declared model boundary, not the accuracy with which a real apparatus approximates it.

This boundary classification alone asserts neither an energy-balance law nor equilibration.

For this nonrelativistic framework, a Newtonian event-coordinate domain is the Cartesian product $I\times {\mathbb{R}}^3$ (The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$), where $I\subseteq \mathbb{R}$ is an open time interval as in Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, measured in seconds, and ${\mathbb{R}}^3$ has the Euclidean inner product and norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ (the norm properties follow from Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation); lengths are measured in metres as in SI quantities and unit conventions. A Galilean inertial chart labels events in such a domain by coordinates $(t,\mathbf{x})$. If a second chart uses the translated time domain $I^{\mathrm{\prime }}=I+\tau$, the allowed coordinate changes are $t^{\mathrm{\prime }}=t+\tau ,{\mathbf{x}}^{\mathrm{\prime }}=R\mathbf{x}+\mathbf{a}-\mathbf{b}t,$ where $\tau$ is a constant time shift measured in seconds, $\mathbf{a}\in {\mathbb{R}}^3$ is a constant spatial translation measured in metres, $\mathbf{b}\in {\mathbb{R}}^3$ is a constant boost velocity measured in metres per second, and $R\in M_{3\times 3}(\mathbb{R})$ is a dimensionless constant matrix (The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations) satisfying $R^{\mathsf{T}}R=I_3$ using matrix transpose and product (The transpose $A^{\mathsf{T}}$ of a matrix, Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes), with $I_3$ the identity matrix. In the coordinate formula, $(R\mathbf{x}{)}_i:={\sum }_{k<3}{R}_{ik}{x}_k$ for $i<3$. Thus time is absolute in this model and no Lorentzian structure is used. Here “frame” means this coordinate chart, not an observer congruence or a basis/tetrad field.

For a finite point-mass representation with $N\in \mathbb{N}$ (The natural numbers $\mathbb{N}$ (von Neumann)), $N\ge 1$, masses $m_i>0$ in kg at positions ${\mathbf{x}}_i\in {\mathbb{R}}^3$ in metres for $i<N$, assume $M:={\sum }_{i<N}{m}_i>0$ (Finite sums and finite products, by recursion) and define the center-of-mass coordinates by $X_{\mathrm{c}\mathrm{m},j}:=M^{-1}{\sum }_{i<N}{m}_i{x}_{i,j}$ for $j<3$; these three real coordinates form ${\mathbf{X}}_{\mathrm{c}\mathrm{m}}$. The masses are invariant under the allowed chart changes. Other mechanical models must supply their total mass and center-of-mass map; this thermodynamic item does not construct those quantities for a continuum or field model. Every admitted mechanical model must also supply center-of-mass maps and velocities that transform under the displayed Galilean chart changes by ${\mathbf{X}}_{\mathrm{c}\mathrm{m}}^{\mathrm{\prime }}(t^{\mathrm{\prime }})=R{\mathbf{X}}_{\mathrm{c}\mathrm{m}}(t)+\mathbf{a}-\mathbf{b}t$ and ${\mathbf{v}}_{\mathrm{c}\mathrm{m}}^{\mathrm{\prime }}(t^{\mathrm{\prime }})=R{\mathbf{v}}_{\mathrm{c}\mathrm{m}}(t)-\mathbf{b}$, where $t^{\mathrm{\prime }}=t+\tau$; these are kinematic compatibility conditions, not dynamical theorems about arbitrary mechanical models. When a model supplies three coordinate functions $X_{\mathrm{c}\mathrm{m},j}:I\to \mathbb{R}$ for $j<3$ as in The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$, assemble its center-of-mass position as ${\mathbf{X}}_{\mathrm{c}\mathrm{m}}(t):=(X_{\mathrm{c}\mathrm{m},0}(t),X_{\mathrm{c}\mathrm{m},1}(t),X_{\mathrm{c}\mathrm{m},2}(t))\in {\mathbb{R}}^3$. At a time $t\in I$ where each component limit exists, define the velocity by $v_{\mathrm{c}\mathrm{m},j}(t)={\lim }_{h\to 0}(X_{\mathrm{c}\mathrm{m},j}(t+h)-X_{\mathrm{c}\mathrm{m},j}(t))\mathrm{/}h$ through $H_t:=\{h\in \mathbb{R}:t+h\in I,h\mathrm{\ne }0\}$, for each $j<3$, using the scalar limit of The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$; these three components form ${\mathbf{v}}_{\mathrm{c}\mathrm{m}}(t)$. A statewise instantaneous center-of-mass rest chart at a specified time $t$ is the Galilean chart obtained by the pure boost $\mathbf{b}={\mathbf{v}}_{\mathrm{c}\mathrm{m}}(t)$ with $R=I_3$, $\mathbf{a}=0$, and $\tau =0$. Its center-of-mass velocity vanishes at that event, but need not vanish at other times because the boost is constant. For an interval $J\subseteq I$ on which ${\mathbf{v}}_{\mathrm{c}\mathrm{m}}$ exists at every time, a single Galilean chart has zero center-of-mass velocity throughout $J$ if and only if ${\mathbf{v}}_{\mathrm{c}\mathrm{m}}$ is constant on $J$: in such a chart $R{\mathbf{v}}_{\mathrm{c}\mathrm{m}}(t)=\mathbf{b}$ for every $t\in J$, so ${\mathbf{v}}_{\mathrm{c}\mathrm{m}}(t)=R^{\mathsf{T}}\mathbf{b}$ is constant, and conversely a constant velocity ${\mathbf{v}}_0$ is made zero by the pure boost $\mathbf{b}={\mathbf{v}}_0$. For a fixed closed body with total mass $M>0$ in kg, its separately tracked translational kinetic energy in a selected inertial chart is $K_{\mathrm{c}\mathrm{m}}=\frac{1}{2}M\mathrm{\parallel }{\mathbf{v}}_{\mathrm{c}\mathrm{m}}{\mathrm{\parallel }}_2^2$ in joules; under a pure boost it is $K_{\mathrm{c}\mathrm{m}}^{\mathrm{\prime }}=\frac{1}{2}M\mathrm{\parallel }{\mathbf{v}}_{\mathrm{c}\mathrm{m}}-\mathbf{b}{\mathrm{\parallel }}_2^2$.

In this framework $U$ denotes the scalar internal-energy coordinate evaluated at a state in its statewise instantaneous center-of-mass rest chart. Thus an equilibrium preparation's internal energy is evaluated using the boost selected by its center-of-mass velocity at that preparation; a process using one chart with the center of mass at rest throughout its interval must separately satisfy the constant-velocity condition above. Bulk translational kinetic energy and any separately tracked external-field potential energy are excluded from $U$. For a given state, the model treats $U$ as invariant between its center-of-mass rest charts that differ only by spatial translation or orthogonal orientation, and fixes its numerical zero, including any retained or subtracted constituent rest-energy offsets, consistently across the composition changes it admits. This is a nonrelativistic modeling convention, not a derived energy-transformation law.

Fix a system $s$ as in Thermodynamic systems and permitted exchanges. A material constitution $c$ specifies a finite ordered list of named species $({\sigma }_1,\dots ,{\sigma }_r)$, with $r\in \mathbb{N}$ (The natural numbers $\mathbb{N}$ (von Neumann)) and $r\ge 1$, and the model parameters that identify the material. An equilibrium-preparation label in the set $E_c$ selects a single macrostate that the model designates as an equilibrium state for constitution $c$. The equilibrium designation is primitive model input here; it is not a dynamical equilibration theorem or a global-stability claim. For each admitted preparation, its mechanical model supplies a positive total mass and a defined center-of-mass velocity as in Galilean inertial charts and the center-of-mass rest frame, so the allowed boost can select a center-of-mass rest chart in which its internal energy is recorded. A preparation for which these mechanical data are unavailable lies outside this framework. A statistical preparation that instead supplies a normalized probability measure on a declared measurable space is as in Probability measures and probability spaces and belongs to a separate statistical branch; it is an element of $E_c$ only if that model also assigns it a single equilibrium macrostate.

Each $e\in E_c$ is assigned a unique coordinate tuple $(U,V,n_1,\dots ,n_r)$, where $U$ is the adopted internal-energy coordinate in joules as in Galilean inertial charts and the center-of-mass rest frame, $V>0$ is volume in cubic metres, and each $n_j>0$ is an amount in moles; the SI unit assignments are those of SI quantities and unit conventions. These are real-valued mole amounts in the bulk continuous chart, not integer particle counts. Define the dimensionless numerical coordinates $\widehat{U}=U\mathrm{/}(1\mathrm{J})$, $\widehat{V}=V\mathrm{/}(1{\mathrm{m}}^3)$, and ${\widehat{n}}_j=n_j\mathrm{/}(1\mathrm{m}\mathrm{o}\mathrm{l})$. The representation map ${\pi }_c:E_c\to {\mathbb{R}}^{r+2}$ is a function as in A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain, sends $e$ to $(\widehat{U},\widehat{V},{\widehat{n}}_1,\dots ,{\widehat{n}}_r)$, and the equilibrium state domain is its image $D_c:={\pi }_c(E_c)\subseteq {\mathbb{R}}^{r+2}$ using inclusion as in Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$. For $x=(\widehat{U},\widehat{V},{\widehat{n}}_1,\dots ,{\widehat{n}}_r)\in D_c$, write $U_c(x):=\widehat{U}\mathrm{J}$, $V_c(x):=\widehat{V}{\mathrm{m}}^3$, and $n_{c,j}(x):={\widehat{n}}_j\mathrm{m}\mathrm{o}\mathrm{l}$ for its dimensioned coordinates. Preparation labels with the same coordinate tuple represent the same thermodynamic state in this framework. Statistical preparations that specify a probability law belong to a separate branch and are not used in this state-domain definition.

For fixed $n=(n_1,\dots ,n_r)$ in a chart used by the model, write ${\mathrm{\Gamma }}_{c,n}=\{(\widehat{U},\widehat{V}):(\widehat{U},\widehat{V},\widehat{n})\in D_c\}\subseteq {\mathbb{R}}^2$, with inclusion notation as in Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$. This designated slice is a specified nonempty open convex subset. With the dimensionless Euclidean norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_2$ of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ (a norm by Cauchy-Schwarz $\mid ⟨x,y⟩\mid \le \parallel x{\parallel }_2\parallel y{\parallel }_2$ with its equality case, the triangle inequality for $\parallel \cdot {\parallel }_2$, the parallelogram law and polarisation), openness means that every $z\in {\mathrm{\Gamma }}_{c,n}$ has some $ϵ>0$ for which $\{w\in {\mathbb{R}}^2:\mathrm{\parallel }w-z{\mathrm{\parallel }}_2<ϵ\}\subseteq {\mathrm{\Gamma }}_{c,n}$. Convexity means that $(1-t)z_0+t{z}_1\in {\mathrm{\Gamma }}_{c,n}$ whenever $z_0,z_1\in {\mathrm{\Gamma }}_{c,n}$ and $t\in [0,1]$, as in A convex subset of ${\mathbb{R}}^m$ contains every line segment between two of its points. Both properties are assumptions about the particular slice, not conclusions about every $D_c$; no coordinate tuple is presumed physically realizable merely because it lies in an ambient Euclidean space. A zero amount for an absent species belongs only to a separately specified boundary domain and is not in this positive-mole chart.

In a simple-compressible chart, volume $V$ is the only mechanical coordinate in the equilibrium representation; composition changes, fields, polarization, surface terms, and other mechanical coordinates require a separately declared state domain and exchange rules. No pressure or reversible-work formula follows from this definition.

For a system $s$ of constitution $c$, the process model fixes one Galilean inertial chart $g_s$ for kinetic-energy bookkeeping and transfer records and holds that chart fixed throughout each process. A process record is a tuple $P=(I_P,x_P^{-},x_P^{+},z_P,{\kappa }_P,r_P)\mathrm{.}$ For each finite closed time interval $I_P=[t_0,t_1]\subseteq \mathbb{R}$ with $t_0<t_1$ as in Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, measured in seconds, its component fields have these types: $x_P^{-},x_P^{+}\in D_c$ are equilibrium endpoint coordinates from Equilibrium state domains; $Z_s$ is the model-declared configuration space of the system, its auxiliaries and devices; $Z_s^{\mathrm{e}\mathrm{q}}\subseteq Z_s$ is the model-declared set of equilibrium configurations and ${\xi }_s:Z_s^{\mathrm{e}\mathrm{q}}\to D_c$ assigns their state coordinates; $z_P:I_P\to Z_s$ is a function (A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain); ${\mathcal{K}}_s$ is the model-declared set of control/contact records and ${\kappa }_P\in {\mathcal{K}}_s$; and ${\mathcal{R}}_s$ is the model-declared set of signed transfer records and $r_P\in {\mathcal{R}}_s$. These fields form a finite Cartesian product (The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$). The admitted records ${\mathcal{P}}_s$ are the model-specified subset of tuples with these types that additionally satisfy $z_P(t_0),z_P(t_1)\in Z_s^{\mathrm{e}\mathrm{q}}$, ${\xi }_s(z_P(t_0))=x_P^{-}$, and ${\xi }_s(z_P(t_1))=x_P^{+}$. No continuity or differentiability is imposed on $z_P$, and its values at interior times need not represent equilibrium states.

For an endpoint bookkeeping quantity $F$, write $\mathrm{\Delta }F=F_P^{+}-F_P^{-}$. A closed record in the sense of Thermodynamic systems and permitted exchanges has no matter crossing the chosen outer boundary throughout $I_P$; zero net matter transfer alone does not suffice. The core closed-record class ${\mathcal{P}}_{\mathrm{c}\mathrm{l}}(s)\subseteq {\mathcal{P}}_s$ (Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$) consists exactly of the closed records $P\in {\mathcal{P}}_s$ for which the system has a supplied fixed total mass $M>0$ and defined center-of-mass velocities at both endpoints in the fixed chart $g_s$, and $\mathrm{\Delta }{K}_{\mathrm{c}\mathrm{m}}=\mathrm{\Delta }{U}_{\mathrm{e}\mathrm{x}\mathrm{t}}=0$. Here $K_{\mathrm{c}\mathrm{m}}^{\pm }=\frac{1}{2}M\mathrm{\parallel }{\mathbf{v}}_{\mathrm{c}\mathrm{m}}^{\pm }{\mathrm{\parallel }}_2^2$ are defined as in Galilean inertial charts and the center-of-mass rest frame, and $U_{\mathrm{e}\mathrm{x}\mathrm{t}}^{\pm }$ are supplied endpoint values of any model-specified external-field potential energy tracked separately from the internal-energy coordinate in $g_s$; if no such term is tracked, both values are set to zero. Other process classes require an expanded energy bookkeeping rule.

Heat is the protocol-assigned thermal mode of energy transfer across the selected outer boundary, not a quantity stored in the equilibrium state. For each admitted $P$, its transfer record $r_P$ supplies cumulative incoming and outgoing thermal energies $H_P^{\mathrm{i}\mathrm{n}},H_P^{\mathrm{o}\mathrm{u}\mathrm{t}}:I_P\to [0,\mathrm{\infty })$, with numerical values in joules as in SI quantities and unit conventions. Both maps start at zero at $t_0$ and satisfy $H_P^a(t)\le H_P^a(u)$ whenever $t\le u$ in $I_P$, for $a\in \{\mathrm{i}\mathrm{n},\mathrm{o}\mathrm{u}\mathrm{t}\}$. Their increments record the total thermal energy transferred in each direction over the corresponding time interval, across all declared outer-boundary thermal channels, before cancellation between directions or channels. No continuity or differentiability of these maps is required. Transfers between subsystems inside that boundary are internal and are excluded from these outer-boundary totals.

Define $Q:{\mathcal{P}}_s\to \mathbb{R}$ by $Q[P]:=H_P^{\mathrm{i}\mathrm{n}}(t_1)-H_P^{\mathrm{o}\mathrm{u}\mathrm{t}}(t_1)$. The protocol also supplies $W_{\mathrm{o}\mathrm{n}}:{\mathcal{P}}_s\to \mathbb{R}$, with values in joules; both transfers use the fixed bookkeeping chart $g_s$. Thus $Q[P]>0$ means net thermal energy enters the system, and $W_{\mathrm{o}\mathrm{n}}[P]>0$ means net mechanical work is done on it. These are functionals of the process record, not state functions of $x_P^{-}$ and $x_P^{+}$. A record $P\in {\mathcal{P}}_s$ is adiabatic exactly when $H_P^{\mathrm{i}\mathrm{n}}(t)=H_P^{\mathrm{o}\mathrm{u}\mathrm{t}}(t)=0$ for every $t\in I_P$: no thermal energy crosses the selected boundary throughout the record. In particular an adiabatic record has $Q[P]=0$, but zero net heat alone does not specify an adiabatic record. On ${\mathcal{P}}_{\mathrm{c}\mathrm{l}}(s)$, use the restrictions of these same maps; defining them on the larger class supplies no energy-balance law for records outside the core class. A closed record may still exchange energy; it is not thereby isolated.

A preparation is a model-specified rule selecting the particular initial equilibrium state $x_P^{-}$. A statistical extension may instead specify a normalized probability measure on a declared measurable space, as in Probability measures and probability spaces; such a statistical preparation is outside this deterministic process class. A preparation does not assert later equilibration. An admitted record may pass through nonequilibrium configurations even though both endpoints lie in $D_c$.

Let $A$ and $B$ be two named subsystems with disjoint physical regions as in Thermodynamic systems and permitted exchanges, combined as a compound system $A\bigsqcup B$ inside a declared outer boundary. Fix their material constitutions $c_A,c_B$, each with a finite ordered list of distinct species labels as in Equilibrium state domains. Fix an ordering $({\sigma }_1,\dots ,{\sigma }_r)$ of the labels occurring in either list, each occurring once, and use this as the species list of the compound constitution $c_{A\bigsqcup B}$. For $s\in \{A,B,A\bigsqcup B\}$, write $D_s:=D_{c_s}$ and use $U_s,V_s$ for its dimensioned coordinate maps. For every $1\le j\le r$, define the aligned amount map $a_{s,j}:D_s\to [0,\mathrm{\infty })\mathrm{m}\mathrm{o}\mathrm{l}$ by $a_{s,j}(x):=n_{c_s,k}(x)$ if ${\sigma }_j$ is the $k$-th species in $c_s$'s list, and $a_{s,j}(x):=0\mathrm{m}\mathrm{o}\mathrm{l}$ if it is absent. Distinctness makes $k$ unique. The absent-species value is a bookkeeping map on the existing $D_s$; it does not insert a zero coordinate into that constitution's positive-mole chart or assert a boundary-state extension.

The model's admitted separated pairs form a specified subset ${\mathcal{C}}_{AB}\subseteq D_A\times D_B$ using Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$ and The Cartesian product $A\times B:=\{z\in \mathcal{P}(\mathcal{P}(A\cup B)):\mathrm{\exists }a\in A\mathrm{\exists }b\in Bz=(a,b)\}$. Adopt a composition function ${\kappa }_{AB}:{\mathcal{C}}_{AB}\to D_{A\bigsqcup B}$ (A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain) such that, for every $(x_A,x_B)\in {\mathcal{C}}_{AB}$ and every $1\le j\le r$, its output satisfies $U_{A\bigsqcup B}({\kappa }_{AB}(x_A,x_B))=U_A(x_A)+U_B(x_B),V_{A\bigsqcup B}({\kappa }_{AB}(x_A,x_B))=V_A(x_A)+V_B(x_B),a_{A\bigsqcup B,j}({\kappa }_{AB}(x_A,x_B))=a_{A,j}(x_A)+a_{B,j}(x_B)\mathrm{.}$ All quantities have the SI units of SI quantities and unit conventions. Each union species is present in at least one constituent with positive amount, so the summed amount is positive and agrees with the compound's $j$-th coordinate. The existence of an output in the declared $D_{A\bigsqcup B}$ is part of the adopted model, not a consequence of positivity.

Here “extensive coordinates” means precisely the named coordinates declared additive by this postulate. This is an idealized composition rule: cross-interface and long-range interaction energy is omitted from $U_{A\bigsqcup B}$, rather than asserted to vanish in nature. The postulate does not assert that every pair of subsystem states is physically admissible.

The framework's bulk family is a tagged family of material types. For each dimensionless scale $\lambda >0$ the model may designate a tag $c^{(\lambda )}$ with the same ordered species list and SI coordinate conventions as $c$, and a subset $B_{c,\lambda }\subseteq D_c$ of states admitting a scaled copy, with inclusion as in Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$. For $x=(\widehat{U},\widehat{V},{\widehat{n}}_1,\dots ,{\widehat{n}}_r)\in B_{c,\lambda }$, the scaling map is the function ${\delta }_{\lambda }:B_{c,\lambda }\to D_{c^{(\lambda )}}$ (A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain) given by ${\delta }_{\lambda }(x)=(\lambda \widehat{U},\lambda \widehat{V},\lambda {\widehat{n}}_1,\dots ,\lambda {\widehat{n}}_r)$. The tag matters: $c^{(\lambda )}$ is not silently identified with $c$. This postulate specifies the coordinate scaling model and energy additivity only; it does not postulate entropy additivity or exact scaling for every finite physical body.

This is an adopted physical composition rule, not a deduction from SI units or from the mathematics of Cartesian products.

For every admitted $P\in {\mathcal{P}}_{\mathrm{c}\mathrm{l}}(s)$ with equilibrium endpoints $x_P^{-},x_P^{+}\in D_c$, fixed material constitution, zero change in center-of-mass kinetic energy, and zero change in separately tracked external-field potential energy in the fixed chart selected by the process model, the adopted model balance is $U_c(x_P^{+})-U_c(x_P^{-})=Q[P]+W_{\mathrm{o}\mathrm{n}}[P],$ where $U_c$ is the dimensioned energy-coordinate map of Equilibrium state domains.

Here the internal-energy coordinate is evaluated in the center-of-mass rest frame as in Galilean inertial charts and the center-of-mass rest frame, while $Q[P]$ and $W_{\mathrm{o}\mathrm{n}}[P]$ are the protocol transfers defined by Thermodynamic process records and transfer conventions, in joules as in SI quantities and unit conventions. The sign convention is positive $Q$ into the system and positive $W_{\mathrm{o}\mathrm{n}}$ for work done on it. If $P$ is adiabatic, no thermal energy crosses its selected outer boundary throughout the record, so $H_P^{\mathrm{i}\mathrm{n}}(t)=H_P^{\mathrm{o}\mathrm{u}\mathrm{t}}(t)=0$ for every $t\in I_P$ and hence $Q[P]=0$; the same adopted balance gives $W_{\mathrm{o}\mathrm{n}}[P]=U_c(x_P^{+})-U_c(x_P^{-})$.

This equation is an adopted physical postulate of the stated framework. It is not deduced here from a mechanical model or from experimental observations. Records with matter exchange or nonzero changes in omitted kinetic or external-field energy require a separately specified balance and are outside this item.