Thermodynamic systems, variables, and equilibrium

Two still readings, one unanswered question

Two rigid bodies sit inside an insulated box. Body A is uniformly at 320 K; body B is uniformly at 280 K. An ideal internal insulating barrier prevents energy transfer between them, and their readings remain unchanged. Have they reached thermal equilibrium with each other? What changes if the internal barrier is replaced by a conducting contact while the outer box stays isolated?

The readings alone do not settle the question. The barrier prevents one kind of change, so an unequal-temperature arrangement can persist under that constraint. Once thermal contact is permitted, that arrangement can change without anything crossing the outer boundary. Each body can already be internally equilibrated even though the pair is not in mutual thermal equilibrium.

Thermodynamics describes systems through macroscopic properties, their allowed exchanges and their equilibrium states. “Macroscopic” means that we describe aggregates of many microscopic constituents through quantities such as mass, volume, pressure and temperature. Equilibrium does not mean that molecules stop moving. It concerns the absence of a macroscopic tendency to change under the stated conditions.

On a first reading, draw the outer boundary, classify exchanges, compare the replicated sample and follow the thermal-contact calculation. Then compare a stationary temperature gradient with an equilibrium state. Sections marked “Further” develop state-variable independence, boundary contributions, local equilibrium and metastability. Published preparation is “Physical systems, states, boundaries, and environments”, “Dimensions, scaling, and similarity”, and “Conservation of mass, momentum, and energy”. Essential thermodynamic terms are introduced here; later thermodynamics lessons supply their detailed laws.

Draw the boundary before classifying the system

The system is the matter or region selected for study. Its surroundings are everything outside the chosen boundary. A boundary can be a physical wall or an imaginary surface, and it can be fixed or moving. The same apparatus can contain several legitimate systems: one body, both bodies, or the bodies together with a surrounding thermal bath. State which choice a claim describes.

A closed system has no matter crossing its boundary during the process under consideration. It may exchange energy. An open system permits matter transfer; a fixed region around a flowing fluid is a common example. Equal inflow and outflow can keep its stored mass constant while matter continually crosses, so zero net accumulation does not make it closed. Following a specified parcel instead selects a different boundary.

An isolated system exchanges neither matter nor energy with its surroundings. It therefore excludes heat transfer and work transfer as well as mass transfer. An adiabatic boundary blocks heat transfer; it need not block work. A capped insulated cylinder can remain closed and adiabatic while a moving piston does work on its contents. A rigid container can exchange energy through a conducting wall or an electrical connection.

The ideal outer box in the opening is closed, thermally insulated, fixed and free of other work connections, so the composite is isolated in this model. Each body is a closed subsystem. When the internal contact conducts, energy crosses a body boundary but not the composite boundary. Changing the boundary changes the energy ledger, not the underlying event. Real isolation is an approximation over a stated observation time and accuracy.

Name what belongs to the state

A thermodynamic state is a specification of the relevant macroscopic condition. A state variable, or property, has a value associated with that state. Mass m measures matter in kg; volume V measures occupied space in m³; pressure p measures isotropic mechanical force per area in Pa; temperature T characterizes thermal condition, here in K. Amount of substance n, in mol, counts specified microscopic entities through the mole. Composition specifies which substances and amounts are present.

Internal energy U, in J, is the system’s energy associated with its internal degrees of freedom in the chosen macroscopic description, with a consistently chosen reference. It is distinct from the kinetic energy of overall motion and the potential energy of the system as a whole in an external field. When those terms matter, the complete energy accounting includes them. Changing a reference zero does not alter a consistent energy difference.

Heat and work describe energy transferred during an interaction; they are not extra amounts of substance stored inside a body. “The body contains 10 kJ of heat” does not specify a thermodynamic property. State its internal energy relative to a reference, or state that 10 kJ was transferred by heating during a specified process. The later lesson “Heat, work, internal energy, and the first law” develops the accounting and sign conventions.

A state function has a change that depends only on its initial and final states: ΔX = Xfinal − Xinitial. Returning to the same complete state returns every state function to its previous value. Returning one thermometer or pressure gauge to an old reading does not establish that the complete state has returned: composition, amount, another independent property or a slowly relaxing internal variable may differ.

Replicate a sample: which values scale?

An extensive quantity is additive over the specified subsystems. For ordinary bulk samples with negligible interaction and boundary contributions, mass, volume and internal energy add. An intensive quantity does not increase simply because the same homogeneous state is replicated. Temperature and pressure are intensive. Two identical noninteracting copies at 300 K do not make a 600 K sample.

Consider a supplied homogeneous bulk sample with m₀ = 0.25 kg, V₀ = 0.00025 m³ and U₀ = 10 kJ at T = 300 K and p = 100 kPa. These are teaching inputs with a fixed energy reference, not a measured property table for a named material. Replicating the same state by a factor λ gives m = λm₀, V = λV₀ and U = λU₀ while T and p stay the same. A homogeneous fraction of the sample obeys the same bulk scaling under the stated assumptions.

Density ρ = m/V is 1000 kg/m³ in every row. Specific volume v = V/m is 0.001 m³/kg, and specific internal energy u = U/m is 40 kJ/kg. “Specific” here means per unit mass. A molar property instead divides by amount n, for example molar volume Vm = V/n in m³/mol. A ratio of two extensive quantities can be intensive when their scale factors cancel; it need not be dimensionless.

Intensive does not mean numerically small, constant in every process or uniform in every nonequilibrium system. Temperature can change with heating, and a rod can have a temperature field T(x). Extensivity describes an addition or scaling operation under specified conditions; it is not a rule based on the size of a number or its unit.

Further: adding subsystems differs from merging their boundaries

Additivity needs a precise partition. If two subsystems interact significantly, the total energy also includes their interaction contribution unless it has already been assigned consistently. Gravitational interactions over large distances, fields and interface contributions can matter. Ordinary bulk thermodynamics often works where such effects are negligible compared with the bulk energy, or where they are included through additional state variables.

A geometric comparison makes the boundary issue visible. Eight separate cubes of edge 1 cm have total volume 8 cm³. Each has six exterior faces of 1 cm², so their total exterior area is 48 cm². One merged cube with the same total volume has edge 2 cm and exterior area 24 cm². The amount is the same, but the available surface has changed. This is a comparison of two specified arrangements, not a calculated joining process.

Surface area is additive across separate objects when each exterior surface is counted. It is not proportional to volume under geometrically similar enlargement: area scales as edge squared and volume as edge cubed. Thus additive and proportional-to-amount bulk scaling must not be treated as interchangeable definitions without specifying what is replicated. Merging removes internal exposed surfaces and can change surface-related contributions even at fixed total amount.

If those contributions affect the question, include the required area, curvature, interface state or field information. Do not conclude that every total energy must equal eight times one small object’s energy after its boundary arrangement changes. “Surface tension and capillarity” provides the mechanical and energetic interface development; no surface energy or heat release is computed by the cube geometry alone.

Equilibrium depends on what can relax

Thermal equilibrium between bodies means that permitting thermal contact creates no net thermal driving tendency between them; in the ordinary setting used here, their equilibrium temperatures match. The next lesson “Temperature and the zeroth law” develops the operational meaning of temperature and thermal comparison. The opening pair at 320 K and 280 K fails this mutual thermal test when the insulating constraint is removed.

Mechanical equilibrium requires a balance of forces and stresses for the allowed mechanical motions. In a simple fluid without relevant body forces or interfacial stresses, pressure equality across a movable frictionless partition is the familiar condition. A fixed wall can support a pressure difference. A fluid in gravity can be at equilibrium with pressure varying with height, and a curved interface can sustain a pressure jump. The published statics and capillarity lessons explain these cases; equilibrium does not require every pressure reading everywhere to match.

Material-transfer equilibrium concerns whether an allowed species transfer between regions has a net thermodynamic driving tendency. Reaction equilibrium concerns allowed chemical reactions. Equal concentrations are not a universal transfer criterion: different phases can have different equilibrium concentrations. The relevant chemical potentials, which quantify the thermodynamic tendency for species exchange under stated conditions, enter that test. “Chemical potential and phase equilibrium” develops the equations and the treatment of fields and reactions.

A constraint blocks a specified change: an insulating wall blocks thermal transfer, a rigid partition blocks volume exchange, and a selective membrane blocks some species while passing others. A system can be equilibrated with respect to its allowed changes while retaining differences that a blocked change would remove. Call this constrained equilibrium and name the constraint. Removing one constraint enlarges the accessible set; it does not remove every other restriction.

Complete equilibrium for a chosen system requires all relevant allowed thermal, mechanical, transfer and reaction tendencies to be equilibrated. A subsystem may also be in equilibrium with a bath while exchanging energy microscopically in both directions: equilibrium requires no sustained net macroscopic transfer, not that every microscopic exchange ceases. Sustained driven heat or mass currents describe a different situation.

Allow thermal contact inside the same isolated box

For a small calculation, give the two bodies the supplied fixed-volume caloric relations UA = CA(TA − Tref) and UB = CB(TB − Tref), with CA = 2000 J/K, CB = 1000 J/K and Tref = 250 K. These constant capacities are assigned over 260–340 K. Both bodies are stationary and rigid; no phase change or reaction occurs. The contact switch has negligible energy and heat capacity. We use total-energy conservation from the published preparation, without supplying a transfer-rate law.

Initially UA = 140 kJ and UB = 30 kJ, so the composite has Utotal = 170 kJ in the given reference. With the ideal internal barrier present, the two separate energies are constrained. With thermal contact allowed and all outer exchanges still excluded, only their sum is fixed; their individual values can change.

At the final common temperature Tf, conservation gives CA(TAi − Tref) + CB(TBi − Tref) = (CA + CB)(Tf − Tref). Cancel the common reference contribution and solve: Tf = (CATAi + CBTBi)/(CA + CB) = 306.6667 K. The warmer body has twice the capacity, so the common temperature lies closer to 320 K than to 280 K. A simple arithmetic mean of the two initial temperatures would give 300 K and would not conserve the assigned energy.

At that final temperature, UA ≈ 113.3333 kJ and UB ≈ 56.6667 kJ. About 26.6667 kJ has moved from A to B, while the composite total remains 170 kJ. The final temperature lies within the supplied model interval. This calculation selects a possible final equilibrium state under the stated constraints; it does not predict the contact conductance, the transient internal temperature fields or the time required to approach it.

The same final-state data do not determine a unique history. Different contacts can approach the same final state at different rates. Other allowed work or material transfers can change the conserved quantities and therefore the final state. “Heat capacities and calorimetry” and “Heat, work, internal energy, and the first law” supply the broader measurement and process treatment.

Further: a state postulate needs conditions and independent coordinates

For a simple compressible, homogeneous equilibrium bulk phase of fixed composition, two genuinely independent intensive properties can locally specify its intensive state. “Simple compressible” means that ordinary pressure–volume deformation is the relevant mechanical work mode; significant interface, electrical, magnetic or elastic-deformation coordinates have been excluded or fixed. The amount of material is also needed to determine total extensive properties. Material identity, phase and constraints are part of the specification.

This is a state-space postulate, not a claim that any two instrument readings always determine any system. Density and specific volume obey ρv = 1, so giving both supplies only one independent quantity. In the replicated-sample table, every row has the same ρ and v but different mass and total U. Knowing those two readings alone determines neither a second independent thermal coordinate nor the amount.

The selected coordinates must also identify the intended branch without ambiguity. Near a phase transition, or where a property reverses its trend, a pair can fail to distinguish states. In liquid–vapor coexistence of a pure substance, saturation temperature and pressure are linked; that pair does not independently specify how much of each phase is present. Mixture composition, phase amounts and additional constraints require appropriate information. “Equations of state and gases” develops constitutive relations; “Chemical potential and phase equilibrium” develops coexistence.

The two-body composite before thermal contact is not one homogeneous simple phase described by one temperature and one pressure. Its internal energy distribution is relevant. Specifying only the total 170 kJ would not distinguish many allowed allocations between A and B. Internal constraints or slowly relaxing structural variables may therefore belong in the state description even when a compact equilibrium description suffices after their removal.

For an ordinary extensive equilibrium bulk representation, the same idea can be written as a fundamental relation with independent energy, volume and component amounts, or an equivalent set of coordinates. Its derivatives and equilibrium restrictions connect other properties. The later entropy and thermodynamic-potential lessons develop those relations. Here the essential step is to state the model class and count independent physical information rather than letters in a list.

A stationary gradient can carry a continuous current

A steady state has no time dependence in its macroscopic fields during the interval considered. It can still be maintained by continuing exchanges. Consider a separate rod of length L = 0.10 m and cross-sectional area A = 10⁻⁴ m², with its left end held at 320 K and right end at 280 K. Assume constant thermal conductivity k = 2 W/(m·K), no lateral heat loss and no internal energy source.

The supplied one-dimensional steady conduction model gives T(x) = 320 K + (280 K − 320 K)x/L. Its gradient is −400 K/m. Fourier heat flux qx = −k dT/dx is +800 W/m², so the total rate Q̇x = Aqx is +0.080 W toward increasing x. Heat enters one end and leaves the other at equal rates. The rod’s stored energy is steady, but an ongoing heat current passes through it.

This is a closed system with energy exchange, not an isolated equilibrium system. Its mean temperature is 300 K, the same numerical mean as a separate uniformly equilibrated insulated body at 300 K. Those equal mean readings do not make the states equivalent. One has a temperature field and a sustained current; the other has no such gradient or sustained thermal exchange.

If the rod is isolated from the unequal-temperature reservoirs, the former imposed profile is no longer maintained. The ensuing relaxation needs a time-dependent transport model and material storage information; no relaxation time is supplied here. This example tests the meaning of equilibrium and stationarity. “Irreversible thermodynamics and heat and mass transfer” develops the transport theory.

Further: local equilibrium is an assumption about scale

An inhomogeneous system may sometimes be divided into small cells that are still large enough to contain many microscopic constituents. If each cell internally relaxes much faster than its macroscopic condition changes, equilibrium property relations may be used at the local values of T, density and composition. Different cells can have different values. This is local thermodynamic equilibrium; it does not assert equilibrium between the cells.

The cell must be small compared with the length over which the macroscopic fields vary substantially, yet large compared with the relevant microscopic correlation or transport lengths. Its internal relaxation time must also be short relative to the imposed variation time. These are model-dependent separations, not one universal cell size or numerical threshold. A cell can exchange energy and matter with neighbors while its local thermodynamic description remains useful.

A local-equilibrium equation of state does not by itself determine a heat flux, viscosity, diffusion rate or reaction rate. Conservation balances and appropriate transport or kinetic relations are additional information. The stationary rod illustrates this division: local temperatures describe thermal condition, while the separately assigned conductivity gives the continuing heat current.

Rapid shocks, rarefied gases, strong driving, slowly relaxing internal modes, glasses and very small interfacial regions can require extra variables or a different kinetic description. A single global temperature or a local equilibrium fit is not automatically justified there. “Microfluidic, reacting, biological, and geophysical flows” introduces several of these regime choices; this lesson keeps the local assumption explicit rather than certifying all nonequilibrium states.

Further: constrained equilibrium, metastability and global stability differ

A state can resist small disturbances yet have a more stable alternative elsewhere in its accessible state space. A metastable state is associated with a local thermodynamic minimum separated from a lower accessible state by a barrier. Its observed persistence depends on kinetics, disturbances and observation time. Long persistence alone does not prove global equilibrium, and the minimum appropriate to a given experiment depends on what is held fixed.

For this return path, introduce Helmholtz energy F = U − TS, where S is entropy, a thermodynamic state function measured in J/K. For a closed system at fixed T, V and material amounts with no nonexpansion work, stable equilibrium minimizes F over the allowed states. At fixed T and pressure a different potential is appropriate. An isolated system instead uses an entropy-maximum criterion at fixed total energy, volume and material constraints. “Entropy and the second law” and “Enthalpy, Helmholtz energy, and Gibbs energy” develop these conditional criteria; they are not interchangeable instructions to minimize every energy.

To see local versus global without claiming a material model, assign a one-coordinate effective landscape F(q) − Fref = E₀[q⁴/4 − q³/3 − q²], with dimensionless internal coordinate q, E₀ = 1 J and −2 ≤ q ≤ 3. Temperature, volume and material amounts are fixed; other variables have already relaxed conditionally. This supplied polynomial is an illustrative landscape, not a measured phase diagram or a trajectory in time.

Its stationary condition is dF/dq = E₀q(q + 1)(q − 2) = 0. At q = −1 and q = 2 the curvatures are positive; at q = 0 the curvature is negative. Comparing these candidates with the endpoints shows the full-domain minimum at q = 2. The left minimum q = −1 is higher by 2.25 J and is metastable within this model. Its barrier to q = 0 is 5/12 J, about 0.416667 J. The curve supplies no escape rate or lifetime.

Now impose an external restriction q ≤ 0. The same left minimum q = −1 is the global minimum within that restricted domain. Calling it a constrained equilibrium states a different claim from calling it metastable after the full domain becomes accessible. Removing the restriction makes the lower right basin available but does not specify how a barrier is crossed. A deterministic transition, thermal activation law or external perturbation would require additional dynamics.

In a many-variable system, curvature must be tested in every allowed direction; positive curvature along one plotted coordinate cannot certify full stability. The later lesson “Maxwell relations and thermodynamic stability” develops those derivative tests. The lesson here is to distinguish a stationary point, a local minimum and a global minimum over a specified accessible set.

A process needs more than its endpoint labels

A process changes the state; a path describes the intervening sequence. In a quasistatic approximation, internal relaxation is fast enough that successive states can be treated as close to equilibrium for the chosen description. “Slow” has meaning relative to the relevant relaxation times. It is not a guarantee that every process variable is spatially uniform to arbitrary accuracy.

Quasistatic and reversible are different requirements. Friction or other dissipative effects can persist even when mechanical motion is slow, and a finite temperature difference can drive irreversible heat exchange. A reversible idealization additionally requires that system and surroundings can be restored without residual change. “Isothermal, adiabatic, isobaric, and isochoric processes” and the entropy lesson develop process conditions and reversibility.

For a new problem, specify the boundary and observation scale, identify allowed matter and energy exchanges, list the needed state variables and constraints, and distinguish equilibrium, constrained persistence and steady driving. Check whether the variables are independent and whether bulk additivity or local equilibrium is justified. Then choose the later equation-of-state, energy, entropy or transport model needed for the actual question.

The opening box shows why these choices matter: its outer isolation does not stop internal relaxation, its two body temperatures cannot be replaced by one arithmetic mean, and a fixed total energy does not by itself specify the initial internal allocation. The rod and landscape supply two further warnings: constant readings can conceal a current, and a stable-looking basin can fail to be the lowest accessible state.

Worked examples

Try it yourself

1. A fixed fluid region has equal mass inflow and outflow and constant stored mass. Is it necessarily a closed thermodynamic system?

   Show the answer and reasoning

   No. It is open when matter crosses the chosen boundary.

   Equal crossing rates make net accumulation zero; they do not remove the crossings. Following a material parcel would choose a different boundary.

2. A sealed insulated cylinder is compressed by a piston. Does insulation alone establish isolation?

   Show the answer and reasoning

   No. Work can cross the moving boundary even when heat and matter do not.

   Adiabatic excludes heat transfer. Isolation additionally excludes work and matter transfer. The piston’s motion must be included in the exchange description.

3. The supplied base sample has U = 10 kJ and m = 0.25 kg. Find u and explain why it remains unchanged under the stated bulk replication.

   Show the answer and reasoning

   u = 40 kJ/kg; both U and m multiply by the same replication factor.

   Their factor cancels in U/m. An intensive quantity can have units and can change in other processes; unchanged replication does not mean universally constant.

4. You know ρ = 1000 kg/m³ and v = 0.001 m³/kg. Have you supplied two independent intensive coordinates for the simple-phase state postulate?

   Show the answer and reasoning

   No. Those readings supply one independent quantity.

   They obey ρv = 1 identically. A second genuinely independent property and the needed amount/composition information must still be specified.

5. For the supplied pair, try Tf = 300 K. What total U would that give with Tref = 250 K, and does it match the initial 170 kJ?

   Show the answer and reasoning

   It gives 150 kJ, which is 20 kJ below the conserved total.

   (CA + CB)(300 − 250) = 3000 × 50 J = 150 000 J. Equal weighting of temperatures is inappropriate when the supplied capacities differ.

6. Two fluids on opposite sides of a fixed rigid wall have different pressures and no wall motion. Must the pressure difference vanish before the constrained arrangement can persist?

   Show the answer and reasoning

   No. The fixed wall and its supports can sustain a pressure difference.

   Mechanical conditions concern allowed displacements and complete force balance. Releasing a movable partition would pose a different problem. Gravity and curved interfaces provide other reasons that equilibrium pressure can be nonuniform.

7. Compare eight separate 1 cm cubes with one cube of the same total volume. What are the two exterior areas, and does this geometry calculate joining heat?

   Show the answer and reasoning

   48 cm² versus 24 cm²; no joining heat has been calculated.

   The merged cube has edge 2 cm and six faces of 4 cm². Surface inventory changes, but an energetic calculation would need material/interface information and process conditions.

8. At the center of the steady rod, T = 300 K. Does that make the whole rod equivalent to a uniformly equilibrated insulated body at 300 K?

   Show the answer and reasoning

   No. A single local reading does not describe the whole temperature field or the current.

   The rod spans 320 to 280 K and carries the supplied 0.080 W heat rate. The separate uniform equilibrium body has no corresponding gradient or sustained thermal current.

9. A proposed cell is smaller than the length over which a field changes, but an internal mode relaxes as slowly as the imposed driving. Is local equilibrium established by the length comparison alone?

   Show the answer and reasoning

   No. The required time-scale separation also needs justification.

   Local equilibrium assumes relevant internal modes relax rapidly enough to use equilibrium property relations at the local state. Slowly relaxing modes can require extra variables even in a spatially small cell.

10. The assigned left-well barrier is 5/12 J. Can that number alone give the lifetime of the metastable state or guarantee an immediate jump when q ≤ 0 is removed?

    Show the answer and reasoning

    No. No transition dynamics or rate information has been supplied.

    The landscape identifies energy ordering and stationary curvature for the fixed constraints. Barrier crossing also depends on kinetic laws, fluctuations or forcing; a static energy plot is not a clock.

11. The landscape has zero slope and positive curvature at q = −1. Does that establish the lowest state over the full domain −2 ≤ q ≤ 3?

    Show the answer and reasoning

    No. It establishes a local minimum along q, while q = 2 has lower energy.

    Global comparison requires the other stationary points and boundaries. With additional internal coordinates, stability must also be checked along every allowed direction.

What to carry forward

• Draw the system boundary before classifying matter and energy exchanges. Adiabatic and closed do not by themselves mean isolated.
• State properties describe a macroscopic condition; heat and work describe transfers during a process.
• Bulk replication scales extensive totals while intensive ratios remain fixed only under the stated replication assumptions. Merging interfaces changes the state information.
• Equilibrium must be tested against allowed changes, remaining constraints and external fields; unchanged readings alone are insufficient.
• The state postulate needs a specified model class and genuinely independent coordinates, with enough information for amount, composition and phase structure.
• A steady field can carry continuing heat or matter currents. Local equilibrium is a conditional local property description, not global equilibrium or a transport law.
• A local minimum, a constrained global minimum and an unconstrained global minimum are different claims; an energy landscape does not supply a lifetime.