The two laws, and what the Gibbs energy measures

The pages of this chapter use the enthalpy, the entropy and the Gibbs energy as familiar objects, although the relations between them are rarely obvious on a first reading: a calorimeter measures a change of enthalpy, whereas the solver minimizes a Gibbs energy. This page rebuilds both from the two laws of thermodynamics, mostly in the order in which (Anderson and Crerar, 1993) introduce them, and writes the laws as balances on a chosen boundary, which is where the conditions behind   and behind the minimum of become explicit. The chemical potential and reactions then lets the composition change, and Formation quantities and the database explains the numbers a database stores.

Questions this page answers

• Why does a calorimeter measure an enthalpy, and when does it not?

• Why can a closed, insulated reaction heat up, and what fixes its final temperature?

• What does the Gibbs energy measure, and are and two flows of heat?

• If is a state function, why does the path a reaction takes matter?

1. The system and its boundary

A system is the part of the world under consideration, and its state is fixed by a few variables, the temperature , the pressure and the amounts of its constituents. A state function is a quantity whose value depends on the state only, so that its change between two states does not depend on the way followed from one to the other. Its boundary determines which transfers belong in its balance.

description	matter crosses the boundary	heat crosses the boundary	work crosses the boundary
closed	no	allowed	allowed
open	allowed	allowed	allowed
adiabatic	depends on the boundary	no	allowed
isolated	no	no	no

These descriptions can be combined: a sealed, insulated piston-cylinder is closed and adiabatic, but can do expansion work. A sealed, insulated, rigid vessel with no other work exchange is isolated. A sealed sample can also be isothermal if a thermostat removes the heat of reaction.

Arrows mark transfers that the boundary allows, in either direction. Closing the system blocks matter transfer; adding insulation blocks heat transfer. Work can still cross a closed adiabatic boundary, for example through a moving piston.

In a closed reacting system, the species amounts change even though no matter crosses the boundary. For example,

consumes lime and water and produces portlandite while conserving calcium, hydrogen, oxygen and charge. In the package's notation the fixed quantity is  , not each or the sum of the species amounts.

This is classical chemical thermodynamics: nuclear transformations, and the contribution of exchanged energy to the mass of a sample, are outside the model.

2. Heat, work, and why a calorimeter measures an enthalpy

The internal energy is a state function, whereas the heat and the work received by the system are not: the same change of state can be brought about with more work and less heat, or the reverse. The first law states that their sum is nevertheless fixed by the two end states (Anderson and Crerar, 1993) (§4.6),

In a chemical system held at constant pressure, the only work exchanged is usually that of the pressure against the change of volume,    , and the heat received is then    . It is therefore natural to introduce the enthalpy

another state function, whose change at constant pressure is exactly the heat received,  . A reaction that gives off heat at constant pressure is one whose enthalpy decreases, and the heat recorded by an isothermal calorimeter, counted positive when released, is   . The difference between and is the work  , small for condensed phases and considerable as soon as a gas is produced or consumed.

The same balance written for an infinitesimal change makes its conditions explicit. Neglecting changes in bulk kinetic and gravitational potential energy, the first law for a closed system reads

Heat and work describe transfers along a process: neither is a stored property of the sample. This is why their infinitesimal transfers use whereas a state function uses .

The pressure in expansion work is the external pressure. Replacing it by the system's pressure requires mechanical equilibrium. Constant pressure alone does not rule out electrical work, stirring or other forms of work. With   , a mechanically equilibrated closed system obeys

At constant pressure, with only pressure-volume work, this gives  . With other work, the balance is instead   . A rigid closed calorimeter with no other work measures  ; conversion to requires the change in and any corrections needed to refer the measurement to the stated temperature and standard states.

An enthalpy change is an energy difference, in joules. A heat rate is an energy transfer per unit time, in watts. Enthalpy changes can include reaction, heating, phase change and mixing; they are not restricted to bond energies.

3. Heat capacity, and why an adiabatic reaction can raise the temperature

The heat capacity at constant pressure measures how the enthalpy grows with temperature at fixed composition,

and it governs the other kind of calorimeter, the one that exchanges no heat.

For a closed adiabatic system at constant pressure, with only pressure-volume work, remains constant. For a rigid closed adiabatic system with no work, it is that remains constant. Neither condition requires constant temperature.

Consider the constant-pressure case. Let the initial state have composition at and the final state composition at . Evaluating    through an imaginary path, first changing the composition at and then heating the final composition, gives the balance

The first term is a difference between two compositions, not the enthalpy of either state alone. Both terms are total energies: here is the heat capacity of the final sample, in J/K. If a molar reaction enthalpy is used, it must be integrated over the reaction extent, in moles. Writing     without a common amount basis would mix J/mol and J.

An exothermic composition change makes the first term negative, and heating can compensate it. The integral does not presume that is known: this equation determines it. If the final composition itself depends on , composition and temperature must be solved together. Phase transitions along the imaginary heating path require their enthalpy jumps as well. A vessel's heat capacity belongs in this balance if the vessel is part of the system.

The above is taken at fixed composition. A system in which some reactions stay at equilibrium as it is heated has the larger heat capacity

the second term being the configurational heat capacity of (Richet, 2001; Sec. 6.1d, Eqs. (6.10)–(6.11), p. 129), positive at a stable equilibrium. Reactions faster than the measurement, such as the aqueous speciation, follow the temperature; slower ones, such as the dissolution of the clinker, do not (Kinetics under partial equilibrium).

A semi-adiabatic vessel adds the heat it loses to this balance, and it is the subject of A CEM I 52.5 N mortar in a semi-adiabatic calorimeter, inside the kinetics.

4. Entropy: the inequality of Clausius, and entropy production

The first law says nothing about the direction of a change. The second law supplies it through a further state function, the entropy , which obeys, for any transformation of a closed system at temperature , the inequality of Clausius (Anderson and Crerar, 1993) (§5.2, §5.8)

the equality holding for a reversible transformation only. Unlike and , the entropy can be given a zero common to all substances, the third law fixing it for perfect crystals at 0 K (Richet, 2001; Sec. 4.4b, p. 75); Absolute entropy and entropy of formation defines it, together with the entropy of formation it must not be confused with.

The inequality becomes a balance once the entropy created inside the system is named. For a closed system exchanging heat at a boundary temperature ,

With several heat contacts, sum . The denominator belongs to the heat contact in the entropy balance; it cannot in general be replaced by a single temperature of a sample containing thermal gradients. An adiabatic closed system has    even while its temperature changes. A closed system that exports enough entropy as heat can have  ; the nonnegative quantity is the entropy production.

To evaluate the difference between two thermodynamic states, one may use any reversible reference path connecting them:

This is a path integral of heat transfer divided by the temperature at which each transfer occurs. For reversible heating at fixed pressure and composition,   , so it becomes  . For an isothermal reversible path it becomes . These are different parameterizations of the same definition, not a general replacement of heat by temperature.

Entropy describes the thermodynamic availability of energy and the possible microscopic states. Reading it as disorder is a useful picture, but the picture alone does not establish a balance or the sign of a reaction entropy. The distinction between conservation and direction is developed in (Anderson and Crerar, 1993), Chapter 5.

5. The Gibbs energy, and what it measures

At constant pressure, with no work other than that of the pressure, the heat received is the change of enthalpy,   (§2), and the inequality becomes    . At constant temperature the left-hand side is the differential of one function,          . It is thus the function

the Gibbs energy, that can only decrease in a spontaneous transformation at fixed and ,  , and reaches its minimum at equilibrium. This is the principle the solver of this package implements.

The derivation holds under stated constraints, and the entropy balance of §4 shows which.    is defined at each state, and its existence does not require the temperature to remain constant along a process; the minimum criterion does. For a closed system whose initial and final states share the temperature of one heat reservoir and the same imposed pressure, with only pressure-volume work, the entropy balance applied to the system and the reservoir reads

Equilibrium minimizes over the allowed compositions at fixed , and conservation budget. Other boundaries give other criteria: an isolated rigid system maximizes at fixed , and budget, and an adiabatic constant-pressure system with only pressure-volume work maximizes at fixed , and budget.

What vanishes at equilibrium is a slope, not a difference. The Gibbs energy of every reaction that can proceed both ways,  , is zero there; at a boundary, such as an absent pure phase, only the feasible direction is tested and an inequality replaces that equality. The difference between an initial state and the equilibrium state is negative, and itself has no reason to take any particular value. The distinction is drawn, with the extent of reaction it requires, in The Gibbs energy of a reaction is a slope, not a difference.

What measures is a capacity for work. In a reversible change at fixed and , a system can deliver work other than that of its expansion, the electrical work of a battery for instance, and the most it can deliver is ; a change that delivers none, as a reaction in a beaker does, dissipates that capacity as heat instead. The two terms of weigh two tendencies against each other: a change releasing heat lowers , and a change raising the entropy of the system lowers . A reaction that absorbs heat,  , can therefore still proceed at fixed and when the entropy of the system grows by more than , the entropy that comes in with the heat drawn from the surroundings, as in the dissolution of many salts; the excess,    , is the entropy the reaction produces (Richet, 2001; Sec. 2.3d, Eqs. (2.35)–(2.37), pp. 35–36).

Why and are not two flows of heat

The two terms are easily read as two heats, one chemical and one thermal. They are not.

At fixed and , a reversible process can deliver useful work in addition to expansion work. Then   and the first law gives

With the received-work convention, the maximum useful work delivered is . In a process with no useful work extraction,   instead, and irreversibility accounts for the difference from . These statements concern different processes between the same states. They do not split the actual heat into a "chemical" contribution and a separate "thermal" contribution .

The natural variables of , and the Gibbs-Helmholtz relation

For a closed system of fixed composition with only the work of the pressure, the first law combined with the second,   , gives along a reversible change      (Richet, 2001; Sec. 2.1, Eq. (2.4), p. 27). Differentiating     and inserting it,

The terms in and cancel: subtracting and adding exchanges the variables and for and : no apparatus holds the entropy fixed, and the volume of a condensed phase only with difficulty, whereas a thermostat and a vessel open to the atmosphere set the temperature and the pressure (Richet, 2001; Sec. 2.3d, p. 36). This exchange is called a Legendre transform (Anderson and Crerar, 1993) (§5.4), and it is why is the natural potential of a system held at fixed temperature and pressure. Since , and are state functions, the result holds for any change between two states, reversible or not.

The temperature derivative of follows by the rule for a quotient, and eliminates the entropy:

This is the Gibbs-Helmholtz relation, by which the enthalpy, the quantity calorimetry measures, is read on the temperature dependence of the Gibbs energy, the quantity equilibrium depends on. Applied to a reaction it becomes the van 't Hoff relation of The heat of reaction: measuring how an equilibrium constant changes with temperature gives the heat of the reaction without a calorimeter.

6. State functions and kinetic paths

A state includes temperature, pressure, composition and phase information. Two histories reaching exactly the same initial and final states have the same and . If one history's heating triggers a different reaction and a different final composition, its final state is different and so can be its state-function differences.

At an observation time, a hydrating paste retains slowly reacting clinker and may contain phases whose formation or conversion is inhibited. The coupling of Kinetics under partial equilibrium integrates the slow amounts and re-equilibrates the remaining budget. It computes successive partial equilibria, not thermal jumps over molecular activation barriers. Rate laws supply the time dependence, and a thermal balance supplies a changing temperature when that is modeled.

A state held back by a slow transformation is often pictured as a local minimum of ; Metastable does not mean a local minimum of G explains why it is better described as the minimum of on a smaller set of compositions.

What the package counts as heat

For a closed isothermal constant-pressure sample with only pressure-volume work, the heat released up to time , counted positive as in §2, is

The rate depends on the trajectory; the cumulative value depends on the two states actually reached. Changing temperature, matter transfer or other work requires the corresponding balance terms. In particular, an adiabatic temperature rise is not an outward heat transfer.

The package evaluates    with enthalpy: the standard enthalpies of the species, without the excess enthalpies of the aqueous or solid-solution activity models, as Activity, and the enthalpy of a mixture explains. Using a non-ideal activity model for equilibrium therefore does not by itself add its heat of mixing to the reported calorimetry. For a concentration-based activity, the temperature derivative behind an excess enthalpy also includes the thermal expansion of the solution, which changes molarity but not molality.

heat_release differences this enthalpy between two certified speciations on the same conserved budget, so it counts the precipitation of hydrates as well as the dissolution of the clinker. In a kinetic run, the heat of cumulative_heat depends on the formulation. When the kinetic reactions themselves produce the hydrates, it is heat_rate, summed over those reactions. Under partial equilibrium the kinetic reactions only release ions, so the run instead balances over the whole composition, the equilibrium part included, its state carrying the change of the enthalpy of the cell (Kinetics under partial equilibrium), and refuses a system in which a species has no enthalpy of formation; missing_enthalpy lists such species. Its heat is then the one heat_release computes from the certified replay.

Waiting longer does not by itself guarantee a unique observed assemblage. Accessibility of transformations, boundary conditions, the species list and possible degeneracy of equilibrium all matter. The distinction between physical metastability, a constrained equilibrium and numerical stationarity is developed in Proving that an answer is the answer.

Where to go next

The chemical potential and reactions is the next page: it lets the composition of the system change, which brings in the chemical potential, and arrives at the equilibrium constant of a reaction.

• CEM I 52.5 N and slag in an isothermal calorimeter, read off the states and A CEM I 52.5 N mortar in a semi-adiabatic calorimeter, inside the kinetics, the two calorimeters of §2 and §3 on a cement.