chapter3 - Chapter 3 Calculation of Changes in Internal...

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Chapter 3 Calculation of Changes in Internal Energy, Enthalpy, and Entropy In the previous chapter, general expressions for calculating changes in internal energy, enthalpy, and entropy are developed. These expressions contain partial derivatives involving temperature, pressure, and molar volume. The purpose of this chapter is to show how to evaluate these derivatives in a systematic manner. 3.1 EQUATIONS OF STATE Any mathematical relationship between temperature, pressure, and molar volume is called an equation of state ,i .e . , f ( T,P, e V )=0 (3.1-1) Equations of state can be expressed either in pressure-explicit form P = P ( T, e V ) (3.1-2) or, in volume-explicit form e V = e V ( T,P ) (3.1-3) Besides, an equation of state can also be expressed in terms of the dimensionless compressibility factor , Z ,as Z = P e V RT (3.1-4) 3.1.1 Ideal Gas Equation of State Theequat iono fstateforanidea lgasisg ivenby P e V = RT (3.1-5) Since Z =1 for an ideal gas, the compressibility factor shows the deviation from ideal behavior. The ideal gas model is dependent on the following assumptions Molecules occupy no volume, Collisions of the molecules are totally elastic, i.e., energy of the molecules before a collision is equal to the energy of the molecules after a collision. In other words, there are no interactions between the molecules. 27
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3.1.2 The Virial Equation of State The virial equation of state is useful for calculating thermodynamic properties in the gas phase. It can be derived from statistical mechanics and is given by an in f nite series in molar volume, e V ,as Z = P e V RT =1+ B ( T ) e V + C ( T ) e V 2 + ... (3.1-6) The coe cient, B ( T ) ,i sca l l edth e second virial coe cient , C ( T ) is called the third virial coe cient ,andsoon .A l lv ir ia lcoe cients are dependent on temperature. In practice, terms above the third virial coe cient are rarely used in chemical thermodynamics. An equivalent form of the virial expansion is an in f nite series in pressure expressed as Z B 0 ( T ) P + C 0 ( T ) P 2 + ... (3.1-7) The coe cients B 0 and C 0 canbeexpressedintermso f B and C as B 0 = B RT and C 0 = C B 2 ( RT ) 2 (3.1-8) In practice, it is recommended to consider only the second virial coe cient for pressures up to 15bar . Under these circumstances, Eq. (3.1-7) takes the form Z BP RT (3.1-9) Van Ness and Abbott (1982) proposed the following equation to estimate the second virial coe cient for nonpolar F uids B = RT c P c h B (0) + ωB (1) i (3.1-10) where B (0) =0 . 083 0 . 422 T 1 . 6 r (3.1-11) B (1) . 139 0 . 172 T 4 . 2 r (3.1-12) The terms T r , reduced temperature , P r , reduced pressure ,and ω , acentric factor ,arede f ned by T r = T T c and P r = P P c (3.1-13) ω = 1 . 0 log P vap ( T r . 7) P c ¸ (3.1-14) The de f nition of the acentric factor is based on the observation that log P vap versus 1 /T is approximately a straight line. The critical constants as well as the acentric factors for selected F uids are given in Appendix A.
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chapter3 - Chapter 3 Calculation of Changes in Internal...

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