Thermodynamic databases for pure substances

Thermodynamic data

Thermodynamic data is usually presented as a table or chart of function values for one mole of a substance (or in the case of the steam tables, one kg). A thermodynamic datafile is a set of equation parameters from which the numerical data values can be calculated. Tables and datafiles are usually presented at a standard pressure of 1 bar or 1 atm, but in the case of steam and other industrially important gases, pressure may be included as a variable. Function values depend on the state of aggregation of the substance, which must be defined for the value to have any meaning. The state of aggregation for thermodynamic purposes is the standard state, sometimes called the reference state, and defined by specifying certain conditions. The normal standard state is commonly defined as the most stable physical form of the substance at the specified temperature and a pressure of 1 bar or 1 atm. However, since any non-normal condition could be chosen as a standard state, it must be defined in the context of use. A physical standard state is one that exists for a time sufficient to allow measurements of its properties. The most common physical standard state is one that is stable thermodynamically (i.e., the normal one). It has no tendency to transform into any other physical state. If a substance can exist but is not thermodynamically stable (for example, a supercooled liquid), it is called a metastable state. A non-physical standard state is one whose properties are obtained by extrapolation from a physical state (for example, a solid superheated above the normal melting point, or an ideal gas at a condition where the real gas is non-ideal). Metastable liquids and solids are important because some substances can persist and be used in that state indefinitely. Thermodynamic functions that refer to conditions in the normal standard state are designated with a small superscript °. The relationship between certain physical and thermodynamic properties may be described by an equation of state.

Enthalpy, heat content and heat capacity

It is very difficult to measure the absolute amount of any thermodynamic quantity involving the internal energy (e.g. enthalpy), since the internal energy of a substance can take many forms, each of which has its own typical temperature at which it begins to become important in thermodynamic reactions. It is therefore the change in these functions that is of most interest. The isobaric change in enthalpy H above the common reference temperature of 298.15 K (25 °C) is called the high temperature heat content, the sensible heat, or the relative high-temperature enthalpy, and called henceforth the heat content. Different databases designate this term in different ways; for example H_T-H₂₉₈, H°-H°₂₉₈, H°_T-H°₂₉₈ or H°-H°(T_r), where T_r means the reference temperature (usually 298.15 K, but abbreviated in heat content symbols as 298). All of these terms mean the molar heat content for a substance in its normal standard state above a reference temperature of 298.15 K. Data for gases is for the hypothetical ideal gas at the designated standard pressure. The SI unit for enthalpy is J/mol, and is a positive number above the reference temperature. The heat content has been measured and tabulated for virtually all known substances, and is commonly expressed as a polynomial function of temperature. The heat content of an ideal gas is independent of pressure (or volume), but the heat content of real gases varies with pressure, hence the need to define the state for the gas (real or ideal) and the pressure. Note that for some thermodynamic databases such as for steam, the reference temperature is 273.15 K (0 °C).

The heat capacity C is the ratio of heat added to the temperature increase. For an incremental isobaric addition of heat:

${\displaystyle C_P(T)=\left\{\underset{\mathrm{\Delta }T\to 0}{lim}\frac{\mathrm{\Delta }H}{\mathrm{\Delta }T}\right\}={\left(\frac{\mathrm{\partial }H}{\mathrm{\partial }T}\right)}_p}$

C_p is therefore the slope of a plot of temperature vs. isobaric heat content (or the derivative of a temperature/heat content equation). The SI units for heat capacity are J/(mol·K).

Enthalpy change of phase transitions

When heat is added to a condensed-phase substance, its temperature increases until a phase change temperature is reached. With further addition of heat, the temperature remains constant while the phase transition takes place. The amount of substance that transforms is a function of the amount of heat added. After the transition is complete, adding more heat increases the temperature. In other words, the enthalpy of a substance changes isothermally as it undergoes a physical change. The enthalpy change resulting from a phase transition is designated ΔH. There are four types of enthalpy changes resulting from a phase transition. To wit:

• Enthalpy of transformation. This applies to the transformations from one solid phase to another, such as the transformation from α-Fe (bcc ferrite) to ${\displaystyle \gamma }$-Fe (fcc austenite). The transformation is designated ΔH_tr.
• Enthalpy of fusion or melting. This applies to the transition of a solid to a liquid and is designated ΔH_m.
• Enthalpy of vaporization. This applies to the transition of a liquid to a vapor and is designated ΔH_v.
• Enthalpy of sublimation. This applies to the transition of a solid to a vapor and is designated ΔH_s.

C_p is infinite at phase transition temperatures because the enthalpy changes isothermally. At the Curie temperature, C_p shows a sharp discontinuity while the enthalpy has a change in slope.

Values of ΔH are usually given for the transition at the normal standard state temperature for the two states, and if so, are designated with a superscript °. ΔH for a phase transition is a weak function of temperature. In some texts, the heats of phase transitions are called latent heats (for example, latent heat of fusion).

Enthalpy change for a chemical reaction

An enthalpy change occurs during a chemical reaction. For the special case of the formation of a compound from the elements, the change is designated ΔH_form and is a weak function of temperature. Values of ΔH_form are usually given where the elements and compound are in their normal standard states, and as such are designated standard heats of formation, as designated by a superscript °. The ΔH°_form undergoes discontinuities at a phase transition temperatures of the constituent element(s) and the compound. The enthalpy change for any standard reaction is designated ΔH°_rx.

Entropy and Gibbs energy

The entropy of a system is another thermodynamic quantity that is not easily measured. However, using a combination of theoretical and experimental techniques, entropy can in fact be accurately estimated. At low temperatures, the Debye model leads to the result that the atomic heat capacity C_v for solids should be proportional to T³, and that for perfect crystalline solids it should become zero at absolute zero. Experimentally, the heat capacity is measured at temperature intervals to as low a temperature as possible. Values of C_p/T are plotted against T for the whole range of temperatures where the substance exists in the same physical state. The data are extrapolated from the lowest experimental temperature to 0 K using the Debye model. The third law of thermodynamics states that the entropy of a perfect crystalline substance becomes zero at 0 K. When S₀ is zero, the area under the curve from 0 K to any temperature gives the entropy at that temperature. Even though the Debye model contains C_v instead of C_p, the difference between the two at temperatures near 0 K is so small as to be negligible.

The absolute value of entropy for a substance in its standard state at the reference temperature of 298.15 K is designated S°₂₉₈. Entropy increases with temperature, and is discontinuous at phase transition temperatures. The change in entropy (ΔS°) at the normal phase transition temperature is equal to the heat of transition divided by the transition temperature. The SI units for entropy are J/(mol·K).

The standard entropy change for the formation of a compound from the elements, or for any standard reaction is designated ΔS°_form or ΔS°_rx. The entropy change is obtained by summing the absolute entropies of the products minus the sum of the absolute entropies of the reactants. Like enthalpy, the Gibbs energy G has no intrinsic value, so it is the change in G that is of interest. Furthermore, there is no change in G at phase transitions between substances in their standard states. Hence, the main functional application of Gibbs energy from a thermodynamic database is its change in value during the formation of a compound from the standard-state elements, or for any standard chemical reaction (ΔG°_form or ΔG°_rx). The SI units of Gibbs energy are the same as for enthalpy (J/mol).

Additional functions

Compilers of thermochemical databases may contain some additional thermodynamic functions. For example, the absolute enthalpy of a substance H(T) is defined in terms of its formation enthalpy and its heat content as follows:

${\displaystyle H(T)=\mathrm{\Delta }{H}_{form,298}^{\circ }+[H_T-H_{298}]}$

For an element, H(T) and [H_T - H₂₉₈] are identical at all temperatures because ΔH°_form is zero, and of course at 298.15 K, H(T) = 0. For a compound:

${\displaystyle \mathrm{\Delta }{H}_{form}^{\circ }=H(T)compound-\sum \left\{H(T)elements\right\}}$

Similarly, the absolute Gibbs energy G(T) is defined by the absolute enthalpy and entropy of a substance:

${\displaystyle G(T)=H(T)-T\times S(T)}$

For a compound:

${\displaystyle \mathrm{\Delta }{G}_{form}^{\circ }=G(T)compound-\sum \left\{G(T)elements\right\}}$

Some tables may also contain the Gibbs energy function (H°_298.15 – G°_T)/T which is defined in terms of the entropy and heat content.

${\displaystyle (H_{298}^{\circ }-G_T^{\circ })/T=S_T^{\circ }-(H_T-H_{298})/T}$

The Gibbs energy function has the same units as entropy, but unlike entropy, exhibits no discontinuity at normal phase transition temperatures.

The log₁₀ of the equilibrium constant K_eq is often listed, which is calculated from the defining thermodynamic equation.

${\displaystyle {\log }_{10}\left(K_{eq}\right)=-\mathrm{\Delta }{G}_{form}^{\circ }/(19.1448T)}$