Enthalpy as a function of temperature

This equation is coupled to the component balances in Equation (3.9) and with an equation for the pressure e.g., one of Equations (3.14), (3.15), (3.17). There are A +2 equations and some auxiliary algebraic equations to be solved simultaneously. Numerical solution techniques are similar to those used in Section 3.1 for variable-density PFRs. The dependent variables are the component fluxes , the enthalpy H, and the pressure P. A necessary auxiliary equation is the thermodynamic relationship that gives enthalpy as a function of temperature, pressure, and composition. Equation (5.16) with Tref=0 is the simplest example of this relationship and is usually adequate for preliminary calculations. 

Throughout the preceding discussion, simplified expressions of stream-specific enthalpy as a function of temperature are used. They have to be updated during process operation to consider changes in steady-state compositions. 

It frequently is necessary to express the Joule-Thomson coefficient in terms of other partial derivatives. Considering the enthalpy as a function of temperature and pressure H T, P), we can write the total differential... 

The Calorimetrically Obtained van t Hoff Enthalpy In a manner analogous to that used to obtain the van t Hoff enthalpy from the fractional change in the optical absorbance, one can use the temperature dependence of the fractional enthalpy as a function of temperature to determine an effective enthalpy. We will adopt the notation to represent the total enthalpy associated with the denaturation transition. It can be obtained from an integration of the excess heat capacity, corrected for the baselines, as discussed before ... 

Fig. 2.3. Enthalpy as a function of temperature, pressure, and extent of reaction for an ideal gas reaction.

FIG. 20.1 Free enthalpy as a function of temperature for a polymer and for its monomer. The full line refers to the stable state. 

Heatup path calculations are simplified by putting enthalpy-as-a-function-of-temperature equations in cells D8-J8 of matrix Table 11.2. The equations are listed in Appendix G. They change Eqn. (11.7) to ... 

A heatup path matrix is entered as described in Tables 11.2 and 1.1. Cells D28 to J28 contain enthalpy-as-a-function-of-temperature equations, Table 1.1. Cells C22 to C24 contain input kg-mole S02,02 and N2, Section 11.13. 

Hence it is necessary to measure the heat capacity of a substance from near 0 K to the temperature required for equilibrium calculations to derive the enthalpy as a function of temperature according to equation tB 1.27.15). 

For single-phase systems, we consider the enthalpy as a function of temperature, pressure and number of moles, and express its differential as... 

Figure 5. Iridium Set of thermophysical properties obtained from a single pulse-calorimetric experiment on an iridium sample a) basic electrical quantities and the pyrometer signal as a function of experimental duration b) specific enthalpy as a function of temperature c) electrical resistivity, at initial geometry and with volume expansion, as a function of temperature d) thermal conductivity as a function of temperature e) thermal diffusivity as a frmction of temperature f)...

Thermal analysis (measuring changes, e.g., enthalpy, as a function of temperature)... 

Here, the only unknown is the final temperature T. Still, to obtain a numerical solution we must express the excess enthalpy as a function of temperature, while the data are available in graphical form and at a few temperatures only. We accomplish this by linearizing the excess energy around its value at T = 323.15 K, by writing... 

Figure 4.10. Schematic diagram of the free enthalpy as a function of temperature, illustrating the effect of orientation in the amorphous regions and partial disordering of the crystals to a mesophase. Compare to Figures 4.3 and 4.7.

Equation (8.37) is a practical working relationship for the infinite dilution aqueous molecular component partial molar enthalpy as a function of temperature. There is, however, an underlying assumption made. The relationship derived above assumes equilibrium between liquid and vapor, or, in other words a saturated liquid-vapor situation. In a subcooled liquid the enthalpy predicted by equation (8.37) should be corrected for the enthalpy difference between the pure component at the prevailing pressure and the saturation pressure. 

Different functions may be used to represent the ideal gas heat capacity or enthalpy as a function of temperature values in Appendix 1 are based on a fourth-degree polynomial for the enthalpy of formation, where the zero-order coefficient has been adjusted to obtain the enthalpy of formation at 25 C. 

The enthalpy of any substance increases as the temperature increases at constant pressure via Eq. 4.12. The slope of a graph of enthalpy as a function of temperature is the heat capacity of the substance [Cp(A°, where "A" designates the substance and the means standard state]. The heat capacity is the amount of energy that a substance absorbs as a function of temperature. It relates to all the different ways the substance can store internal energy at constant pressure. For example, vibrational and rotational modes can absorb thermal energy, and a compound that has more such modes will be expected to have a larger heat capacity. 

Table K. 1 shows a 2" catalyst bed heatup path matrix with enthalpy-as-a-function-of-temperature equations in cells D15 to K15. The only difference between this matrix and the Table 14.2 matrix is that Eqn. 14.9 in row 15 has been changed to ...

The Appendix K heatup path matrix is entered into cells C21 to K28. Cells D28 to K28 contain enthalpy-as-a-function-of-temperature equations, Appendix K (- in cells D28 to G28). Cells C21 to C24 contain Fig. 14.2 s 2" catalyst bed input kg-mole SO3, SO2, O2 and N2. 

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