Enthalpy is a measure of the energy (or heat) content of a substance [3]. It is a thermodynamic quantity whose absolute value cannot be determined; however, enthalpy of a substance with respect to its value at some reference conditions can be calculated [4, 5]. The reference state, also called the standard state, is specified in terms of pressure and temperature of the system, usually 1 bar and 25°C (298.15 K) [6]. The standard specific enthalpies (enthalpy per mole) of formation of various substances (from its constituent elements) at the reference state are available from various sources, including books on thermodynamics [3], handbooks [7], and Web databases such the one maintained by the National Institute of Standards and Technology (www.nist.gov).4 Thus, the specific enthalpy of any substance at any other condition can be calculated from its functional dependence on system variables and the reference state enthalpy.
4. By convention, the enthalpies of formation of elements in their natural states of occurrence are taken to be zero.
Enthalpy is a function of temperature and pressure of the system, and its dependence on temperature at constant pressure is described as follows:
The left side of this equation represents the partial derivative of enthalpy with respect to temperature at constant pressure; h is the specific enthalpy of the substance—that is, enthalpy per unit mole—and CP is the specific heat capacity of the substance at constant pressure. The SI units of h and CP are joule per mole (J/mol) and joule per mole per kelvin (J/mol K), respectively. The specific enthalpy h does not depend on the quantity of substance present, making it an intensive property. The total enthalpy H, on the other hand, is an extensive property, which depends on the quantity of material present in the system.5 H is obtained simply by multiplying the specific enthalpy by the number of moles present and has the unit of J (joule).
5. The intensive and extensive properties are discussed in detail in the thermodynamics courses.
If the information about the specific heat capacity CP is available, then integration of equation 7.6 enables us to calculate the change in enthalpy (Δh) when the temperature of the substance changes from T1 to T2 at constant pressure:
where h1 and h2 are the specific enthalpies of the substance at the temperatures T1 and T2, respectively.
Note that equation 7.7 is valid only when CP, the specific heat capacity at constant pressure, does not depend on the temperature and hence is constant over the temperature range under consideration. Typically, however, CP is a function of temperature, the dependence often being expressed as polynomial in T, with one such function shown by equation 7.8 [5].
Coefficients A through E are constants characteristic of the substance and are available from the same sources previously stated. The enthalpy change per unit mole of the substance is then calculated by integrating equation 7.6:
Equation 7.7 or 7.9 is used for calculating the change in the specific enthalpy of a substance when its temperature changes from T1 to T2 under constant pressure conditions. When the process is not conducted under constant pressure (isobaric) conditions, enthalpy dependence on pressure also needs to be taken into account while performing the energy balance computations. The pressure dependence of enthalpy is complex and requires an understanding of the volumetric behavior of the substances—that is, an understanding of the relationship between pressure, volume, and temperature for the substance. This is generally covered in the thermodynamics courses and is not considered in this text.
The assumption implicit in the development of equations 7.7 and 7.9 is that the substance does not undergo a phase change; that is, it does not change its state from solid to liquid or liquid to gas, and vice versa. Thus, the substance undergoes only a sensible heat change that is reflected in the temperature of the substance. However, if the substance does experience a phase change at a temperature intermediate between T1 and T2, then the enthalpy change should include a latent heat component. For example, if the boiling point of the substance Tb is greater than T1 but less than T2, then the substance is a liquid at the beginning of the process at T1, but at T2, at the end of the process, it is a vapor. The enthalpy change for this situation is described by equation 7.10.
In this equation, and are the specific heat capacities of the liquid and vapor form of the substance, respectively, and ΔHv is the latent heat of vaporization at the temperature Tb. If the phase change involves melting/fusion or sublimation/condensation, then the corresponding latent heat value must be used.
If T1 is chosen as 298.15 K—that is, the standard state temperature—then the specific enthalpy of a substance can be calculated at any temperature using equation 7.11.
Here, Δh is calculated using equation 7.9 or 7.10, with the lower and upper temperature limits of integration being 298.15 and T K, respectively. The specific enthalpy at 298.15 K, h298.15, is equal to the standard enthalpy of formation,
, as previously discussed. Equation 7.11 allows us to compute the specific enthalpy of any substance at any temperature, provided the information on the standard enthalpy of formation and the dependence of the specific heat capacity on temperature are known.
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