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.
7.1.4 Enthalpy Changes in Processes
The previous discussion should make it clear that it is possible to obtain the values of specific enthalpy of any substance at any temperature. It follows that if a process is carried out at a certain temperature—that is, both the feed and product streams are at that specified temperature—then a certain enthalpy change is associated with that process. The following generic reaction is an example:
A + (b/a) B → (c/a) C + (d/a) D
The enthalpy of reaction (or heat of reaction) is simply the difference between the enthalpies of products and enthalpies of reactants. The following shows this mathematically:
Here, v represents the stoichiometric coefficients of the species involved in the reaction. It should be noted that the equation for the reaction is written such that the stoichiometric coefficients of all the other species are normalized with respect to the stoichiometric coefficient of A; that is, the equation involves 1 mole of A and proportional moles of other species. Thus the enthalpy or heat of reaction, ΔHrxn is based on 1 mole of reactant A. Of course, the equation can be normalized on the basis of the stoichiometric coefficient of any other species involved in the reaction, with the enthalpy of the reaction changing proportionately.
If the process is conducted at standard conditions, then the enthalpy change is termed as the standard enthalpy change. For the reaction shown previously, the standard enthalpy of reaction follows:
If the standard enthalpies of the reactants are higher than those of the products, then the enthalpy of the reaction will be negative. The process involves starting with a material having higher chemical energy and ending up with a material with a lower energy. The difference between the two energies (or enthalpies) appears as the heat evolves during the transformation, making the process exothermic. Conversely, if standard enthalpies of the products are higher than those of the reactants, then the process involves starting with a material of lower energy and ending up with a material having higher energy. Such processes are termed endothermic. Figure 7.2 shows a conceptual schematic of the enthalpy changes in these two types of processes.
Figure 7.2 Conceptual schematic of enthalpy changes in endothermic and exothermic processes.
It is obvious that for an exothermic process, a mechanism for removing heat is necessary if it is desired to maintain a constant temperature. However, if the process is conducted adiabatically—that is, the system does not exchange heat with the surroundings—then the products will be at a higher temperature than the reactants. Conversely, if the process is endothermic, it will require heat input to maintain a constant temperature, and the adiabatic endothermic process will experience a decrease in temperature. Figure 7.3 shows the changes in temperature for an adiabatic system for both endothermic and exothermic processes.
Figure 7.3 Heat effects in transformations: temperature of adiabatic systems.
When the transformation involves a chemical reaction, the enthalpy effect is termed the enthalpy of reaction or the heat of reaction. The enthalpy (or heat) of reaction is termed enthalpy (or heat) of combustion when the reaction is of combustion of a substance. Transformations that are physical in nature—that is, transformations that do not involve chemical reactions—are also frequently (usually) accompanied by an enthalpy change. For example, heat effects accompany dissolution of a solute in a solution, and the change in enthalpy is termed enthalpy of solution or heat of solution. Similarly, enthalpy of mixing refers to the enthalpy change when the process involves mixing of different streams. These transformations can be endothermic or exothermic as well. In all these cases, the discussion presented above for reactive systems can be extended, mutatis mutandis, to other processes and transformations.
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