The enthalpy function

the derivative of the energy with respect to the temperature of a closed system at constant volume is equal to the heat capacity of the system at constant volume. 

The volume appears as an independent variable in the differential expression of the energy function in Equation (2.26). However, the experimental use of the volume as an independent variable is rather inconvenient, whereas the use of the pressure is more convenient for most experimental work. It is thus desirable to change the independent variable from the volume to the pressure. In so doing we define a new function in terms of its differential, dH, based on Equation (2.26) as 

Furthermore, as we are defining a new function, we can assign the value zero to the integration constant involved in the integration of Equation (2.30), and thus define 

This function is called the enthalpy function, or more loosely the enthalpy.2 By its definition the enthalpy function is a function of the state of the system. The change in the values of this function in going from one state to another depends only upon the two states, and not at all upon the path. Its differential is exact. Its absolute value for any system in any particular state is not known, because the absolute value of the energy is not known. 

We may illustrate one use of Equation (2.30) by considering a change of state taking place under constant pressure and involving no other work. Such a change would take place in an open or constant-pressure calorimeter. 

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For processes taking place at constant pressure it is convenient to introduce the enthalpy function, H, defined as... 

We introduced the enthalpy function particularly because of its usefulness as a measure of the heat that accompanies chemical reactions at constant pressure. We will find it convenient also to have a function to describe the temperature dependence of the enthalpy at constant pressure and the temperature dependence of the energy at constant volume. Eor this purpose, we will consider a new quantity, the heat capacity. (Historically, heat capacity was defined and measured much earlier than were enthalpy and energy.)... 

Normally, the standard state is the most stable state at one atmosphere pressure and at the given temperature. Most tabular data, as used for the calculation of reaction temperatures, are given at 0 °C or 298 K. The overall calculation for the heat of reaction of black powder at different temperatures is simplified by using tabulated data of the enthalpy function. Hr — for the reaction products, since no enthalpy measurements can be made in the sense of an absolute quantity. 

The first law of thermodynamics leads to a broad array of physical and chemical consequences. In the following Sections 3.6.1-3.6.8, we describe the formal theory of heat capacity and the enthalpy function, the measurements of heating effects that clarified the energy and enthalpy changes in real and ideal gases under isothermal or adiabatic conditions, and the general first-law principles that underlie the theory and practice of thermochemistry, the measurement of heat effects in chemical reactions. 

As with the energy, we are often interested in determining the change in the value of the enthalpy function of a closed system for some change of state without having to measure the heat absorbed and without being confined to constant-pressure processes. We choose the temperature and pressure as the two convenient independent variables to use, and write the differential of the enthalpy as... 

The calculation of the change in the value of the enthalpy function for some change of state would then involve the integration of this equation after the coefficients have been evaluated in terms of experimentally determined quantities. Again, we cannot evaluate (8H/8P)T until Chapter 4, but we can evaluate (8H/8 T )P n in the same way that we did for the energy, obtaining... 

The only two functions actually required in thermodynamics are the energy function, obtained from the first law of thermodynamics, and the entropy function, obtained from the second law of thermodynamics. However, these functions are not necessarily the most convenient functions. The enthalpy function was defined in order to make the pressure the independent variable, rather than the volume. When the first and second laws are combined, as is done in this chapter, the entropy function appears as an independent variable. It then becomes convenient to define two other functions, the Gibbs and Helmholtz energy functions, for which the temperature is the independent variable, rather than the entropy. These two functions are defined and discussed in the first part of this chapter. 

The enthalpy function for the surface is obtained from the defining equation... 

This equation is identical to Equation (13.28). Consequently, we conclude that the energy function and the enthalpy function are identical for the surface. By similar arguments we find that... 

Similarly, the entropy difference of the native and denatured states according to Eq. (7) should also decrease with a decrease in temperature, but nonlinearly, and should reach zero at a somewhat higher temperature, rf", than that at which the enthalpy function inverts its sign... 

Relative Values of the Enthalpy Function, H for Solid, Liquid and Gas... 

The enthalpy function, H, was first introduced in Frame 10, equation (10.8). In Figure 18.1, Frame 18 was shown the general variation of the Gibbs energy G as a function of temperature, T and using the equation ... 

For the third law evaluations of the reaction enthalpies from mass spectrometric equilibrium measurements the Gibbs free energy functions of the reactants are needed. Likewise the enthalpy functions are needed in order to correct the second law reaction enthalpies obtained at the average temperature of measurement to the desired reference temperature. These thermal functions can be calculated according to standard statistical... 

The enthalpy function is important in the analysis of open systems, as we will show in the next section. It can also be shown, however, that if a closed system expands (or contracts) against a constant external pressure, and are negligible, and the only work done by or on the system is the work of the expansion, then the energy balance equation reduces to Q = AT/. A proof of this statement is required in Problem 7.15. 

In general, 2 types of calorimeters are used for measurement of specific heats, the adiabatic heated and the drop calorimeter. We chose the latter type for our study of specific heats because we needed the enthalpy function for our thermochemical calculations. This is the integral of the specific heat up to the experimental temperature, and cannot be measured directly by an adiabatically heated calorimeter. At higher temperatures the integration error will be considerable, so that... 

The first step is to formulate the equation of internal energy in terms of enthalpy. The enthalpy function is defined by (e.g., [89] [87]) ... 

There are two alternative paths available, either to formulate the complete differentiation of the enthalpy function in terms of — 1 species or in terms of N species and the constraint that = 1 (e.g., [103], pp. 66-70). 

Example 2-5 Use the compositions and temperatures found in Examples 2-3 and 2-4 and the enthalpy functions given in Example 1-5 to compute Qc, L2, and V3. 

Formulation of the Vapor-Liquid Equilibrium and the Enthalpy Functions... 

The system shown in Fig. 7-2 may be formed from the one shown in Fig. 7-1 by returning the bottoms B2 from column 2 to column 1. The addition of the recycle mass transfer stream does not change the general form of the equations for the capital 0 method where the reflux ratios and boilup ratios are specified. The heat balance enclosing column 1 must be modified, however, in order to account for the recycle stream B2. The resulting expression for the enthalpy function r10 is... 

The enthalpy function is formulated by use of the enthalpy balance enclosing the top of the column and the rectifying pinch [see the first expression given by Eq. (11-19)]. This expression for Lr contains the condenser duty Qc. However since dj, D, and V2 are known, the top of the column is uniquely determined at the outset (Note v2i = lu 4- d,), and Qc may be computed by use of Eq. (11-18). Use of Eq. (11-23) permits the enthalpy balance enclosing the rectifying pinch to be restated in the following functional form... 

II. Vapor Enthalpies Curve fit constants for the enthalpy function... 

Since dq is not a state function, c will depend on the path. The problem can be simplified by introducing the enthalpy function, defined as... 

The enthalpy function //(T) employed in this work is particularly suitable for the calculation of enthalpy changes in technical processes. More calculation is involved if [//(T)-7/(298.15)] (T) in combination with A7/f(7 ) is employed for the same purpose. The latter enthalpy values are listed alone in many other tabular compilations. 

The BDEs in diatomic species have usually been measured by spectroscopy or mass spectrometry. In the absence of data on the enthalpy function, the values at 0 K, D°(A-B), are converted to by the approximate equation ... 

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