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:

${\displaystyle C_{P}(T)=\left\{\lim _{\Delta T\to 0}{\frac {\Delta H}{\Delta T}}\right\}=\left({\frac {\partial H}{\partial T}}\right)_{p}}$

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).

Molar heat content of four substances in their designated states above 298.15 K and at 1 atm pressure. CaO(c) and Rh(c) are in their normal standard state of crystalline solid at all temperatures. S₂(g) is a non-physical state below about 882 K and NiO(g) is a non-physical state at all temperatures.

Molar heat capacity of four substances in their designated states at 1 atm pressure. CaO(c) and Rh(c) are in their normal standard state of crystalline solid at all temperatures. S₂(g) is a non-physical state below about 882 K and NiO(g) is a non-physical state at all temperatures.

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).

Molar enthalpy of zinc above 298.15 K and at 1 atm pressure, showing discontinuities at the melting and boiling points. The ΔH°m of zinc is 7323 J/mol, and the ΔH°v is 115 330 J/mol.

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.

Standard molar heat of formation of ZnBr₂(c,l) from the elements, showing discontinuities at transition temperatures of the elements and the compound.

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).

Absolute entropy of strontium. The solid line refers to the entropy of strontium in its normal standard state at 1 atm pressure. The dashed line refers to the entropy of strontium vapor in a non-physical state.

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).

Standard heat and Gibbs energy change for the reaction:

${\displaystyle Pb(c,l)+2HCl(g)\Rightarrow PbCl_{2}+H_{2}(g)}$

The ΔH°_rx shows discontinuities at the melting points of Pb (600.65 K) and PbCl₂ (771 K). ΔG°_rx is not discontinuous at these phase transition temperatures, but does undergo a change in slope, which is almost imperceptible on the chart.

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:

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

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:

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

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:

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

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.

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

Thermodynamic datafiles

The equation parameters and all other information required to calculate values of the important thermodynamic functions are stored in a thermodynamic datafile. The values are organized in a format that makes them readable by a thermodynamic calculation program or for use in a spreadsheet. For example, the Excel-based thermodynamic database FREED creates the following type of datafile, here for a standard pressure of 1 atm.

Thermodynamic datafile for MgCl₂(c,l,g) from FREED. Some values have truncated significant figures for display purposes. The explanation for the values is shown below.

• Row 1. Molar mass of species, density at 298.15 K, ΔH°_{form 298.15}, S°_298.15. and the upper temperature limit for the file.
• Row 2. Number of C_p equations required. Here, three because of three species phases.
• Row 3. Values of the five parameters for the first C_p equation; temperature limit for the equation.
• Row 4. Values of the five parameters for the second C_p equation; temperature limit for the equation.
• Row 5. Values of the five parameters for the third C_p equation; temperature limit for the equation.
• Row 6. Number of H_T - H₂₉₈ equations required.
• Row 7. Values of the six parameters for the first H_T - H₂₉₈ equation; temperature limit for the equation, and ΔH°_trans for the first phase change.
• Row 8. Values of the six parameters for the second H_T - H₂₉₈ equation; temperature limit for the equation, and ΔH°_trans for the second phase change.
• Row 9. Values of the six parameters for the third H_T - H₂₉₈ equation; temperature limit for the equation, and ΔH°_trans for the third phase change.
• Row 10. Number of ΔH°_form equations required. Here five; three for species phases and two because one of the elements has a phase change.
• Row 11. Values of the six parameters for the first ΔH°_form equation; temperature limit for the equation.
• Row 12. Values of the six parameters for the second ΔH°_form equation; temperature limit for the equation.
• Row 13. Values of the six parameters for the third ΔH°_form equation; temperature limit for the equation.
• Row 14. Values of the six parameters for the fourth ΔH°_form equation; temperature limit for the equation.
• Row 15. Values of the six parameters for the fifth ΔH°_form equation; temperature limit for the equation.
• Row 16. Number of ΔG°_form equations required.
• Row 17. Values of the seven parameters for the first ΔG°_form equation; temperature limit for the equation.
• Row 18. Values of the seven parameters for the second ΔG°_form equation; temperature limit for the equation.
• Row 19. Values of the seven parameters for the third ΔG°_form equation; temperature limit for the equation.
• Row 20. Values of the seven parameters for the fourth ΔG°_form equation; temperature limit for the equation.
• Row 21. Values of the seven parameters for the fifth ΔG°_form equation; temperature limit for the equation.

Most computerized databases will create a table of thermodynamic values using the values from the datafile. For MgCl₂(c,l,g) at 1 atm pressure:

Thermodynamic properties table for MgCl₂(c,l,g), from the FREED datafile. Some values have truncated significant figures for display purposes.

The table format is a common way to display thermodynamic data. The FREED table gives additional information in the top rows, such as the mass and amount composition and transition temperatures of the constituent elements. Transition temperatures for the constituent elements have dashes ------- in the first column in a blank row, such as at 922 K, the melting point of Mg. Transition temperatures for the substance have two blank rows with dashes, and a center row with the defined transition and the enthalpy change, such as the melting point of MgCl₂ at 980 K. The datafile equations are at the bottom of the table, and the entire table is in an Excel worksheet. This is particularly useful when the data is intended for making specific calculations.