The Ideal-Gas State as a Reasonable Approximation
3.8 GENERALIZED CORRELATIONS FOR LIQUIDS
( 83.14 ) ( 65 + 273.15 )
________________ 1021.2 = 27.53 bar
(b) Because the reduced pressure is low (Pr ≈ 27.53/112.8 = 0.244), the general-ized virial-coefficient correlation should suffice. Values of B0 and B1 are given by Eqs. (3.61) and (3.62). With Tr = 338.15/405.7 = 0.834,
B 0 = −0.482 B 1 = −0.232 Substitution into Eq. (3.59) with ω = 0.253 yields:
B ˆ = − 0.482 + ( 0.253 ) ( −0.232 ) = −0.541 B = _____ B ˆ R T c
P c = __________________− ( 0.541 ) ( 83.14 ) ( 405.7 ) 112.8 = −161.8 cm 3 ·mol −1 By the second equality of Eq. (3.36):
P = ____RT
V − B = _____________ ( 83.14 ) ( 338.15 ) 1021.2 + 161.8 = 23.76 bar
An iterative solution is not necessary because B is independent of pressure. The c al-culated P corresponds to a reduced pressure of Pr = 23.76/112.8 = 0.211. Reference to Fig. 3.13 confirms the suitability of the generalized virial-coefficient correlation.
Experimental data indicate that the pressure is 23.82 bar at the given condi-tions. Thus the ideal-gas state yields an answer high by about 15%, whereas the virial-coefficient correlation gives an answer in substantial agreement with exper-iment, even though ammonia is a polar molecule.
3.8 GENERALIZED CORRELATIONS FOR LIQUIDS
Although the molar volumes of liquids can be calculated by means of generalized cubic equa-tions of state, the results are often not of high accuracy. However, the Lee/Kesler correlation includes data for subcooled liquids, and Fig. 3.11 illustrates curves for both liquids and gases.
Values for both phases are provided in Tables D.1 through D.4 of App. D. Recall, however, that this correlation is most suitable for nonpolar and slightly polar fluids.
In addition, generalized equations are available for the estimation of molar volumes of saturated liquids. The simplest equation, proposed by Rackett,22 is an example:
V sat = V c Z c ( 1−T r ) 2/7 (3.68) An alternative form of this equation is sometimes useful:
Z sat = P __r
T r Zc[ 1+ ( 1− T r ) 2/7] (3.69)
22H. G. Rackett, J. Chem. Eng. Data, vol. 15, pp. 514–517, 1970; see also C. F. Spencer and S. B. Adler, ibid.,
3.8. Generalized Correlations for Liquids 113 The only data required are the critical constants, given in Table B.1 of App. B. Results are usually accurate to 1 or 2%.
Lydersen, Greenkorn, and Hougen23 developed a two-parameter corresponding-states correlation for estimation of liquid volumes. It provides a correlation of reduced density ρr as a function of reduced temperature and pressure. By definition,
ρ r ≡ __ρ
ρ c = V ___c
V (3.70)
where ρc is the density at the critical point. The generalized correlation is shown by Fig. 3.15.
This figure may be used directly with Eq. (3.70) for determination of liquid volumes if the value of the critical volume is known. A better procedure is to make use of a single known liquid volume (state 1) by the identity,
reduced densities read from Fig. 3.15
23A. L. Lydersen, R. A. Greenkorn, and O. A. Hougen, “Generalized Thermodynamic Properties of Pure Fluids,”
Univ. Wisconsin, Eng. Expt. Sta. Rept. 4, 1955.
Figure 3.15: Generalized density correlation for liquids.
This method gives good results and requires only experimental data that are usually available.
Figure 3.15 makes clear the increasing effects of both temperature and pressure on liquid den-sity as the critical point is approached.
Correlations for the molar densities as functions of temperature are given for many pure liquids by Daubert and coworkers.24
Example 3.13
For ammonia at 310 K, estimate the density of:
(a) The saturated liquid;
(b) The liquid at 100 bar.
Solution 3.13
(a) Apply the Rackett equation at the reduced temperature, T r = 310 / 405.7 = 0.7641 . With V c = 72.47 and Z c = 0.242 (from Table B.1),
V sat = V c Z c (1− T r ) 2/7 = ( 72.47 ) ( 0.242 ) ( 0.2359 ) 2/7 = 28.33 cm 3 ·mol −1 For comparison, the experimental value is 29.14 cm 3 ·mol −1 , a 2.7% difference.
(b) The reduced conditions are:
T r = 0.764 P r = _____100
112.8 = 0.887
Substituting the value, ρr = 2.38 (from Fig. 3.15), and Vc into Eq. (3.70) gives:
V = V ___c ρ r = 72.47_____
2.38 = 30.45 cm 3 ·mol −1
In comparison with the experimental value of 28.6 cm3 ·mol–1, this result is higher by 6.5%.
If we start with the experimental value of 29.14 cm3·mol–1 for saturated liquid at 310 K, Eq. (3.71) may be used. For the saturated liquid at T r = 0.764, ρ r 1 = 2.34 (from Fig. 3.15). Substitution of known values into Eq. (3.71) gives:
V 2 = V 1 ___ ρ r 1 ρ r 2 = ( 29.14 ) ( _2.34
2.38 ) = 28.65 cm 3 ·mol −1 This result is in essential agreement with the experimental value.
Direct application of the Lee/Kesler correlation with values of Z0 and Z1 interpolated from Tables D.1 and D.2 leads to a value of 33.87 cm3·mol–1, which is significantly in error, no doubt owing to the highly polar nature of ammonia.
24T. E. Daubert, R. P. Danner, H. M. Sibul, and C. C. Stebbins, Physical and Thermodynamic Properties of Pure
3.9. Synopsis 115
3.9 SYNOPSIS
After studying this chapter, including the end-of-chapter problems, one should be able to:
∙ State and apply the phase rule for nonreacting systems
∙ Interpret PT and PV diagrams for a pure substance, identifying the solid, liquid, gas, and fluid regions; the fusion (melting), sublimation, and vaporization curves; and the critical and triple points
∙ Draw isotherms on a PV diagram for temperatures above and below the critical temperature
∙ Define isothermal compressibility and volume expansivity and use them in calculations for liquids and solids
∙ Make use of the facts that for the ideal-gas state Uig and Hig depend only on T (not on P and Vig), and that C Pig = C Vig + R
∙ Compute heat and work requirements and property changes for mechanically reversible isothermal, isobaric, isochoric, and adiabatic processes in the ideal-gas state
∙ Define and use the compressibility factor Z
∙ Intelligently select an appropriate equation of state or generalized correlation for appli-cation in a given situation, as indicated by the following chart:
(a) Ideal-gas state (b) 2-term virial equation (c) Cubic equation of state (d) Lee/Kesler tables, Appendix D
(e) Incompressible liquid (f) Rackett equation, Eq. (3.68) (g) Constant β and κ
(h) Lydersen et al. chart, Fig. 3.15 Gas or
liquid?
Gas
Liquid
∙ Apply the two-term virial equation of state, written in terms of pressure or molar density ∙ Relate the second and third virial coefficients to the slope and curvature of a plot of the
compressibility factor versus molar density
∙ Write the van der Waals and generic cubic equations of state, and explain how the equation-of-state parameters are related to critical properties
∙ Define and use Tr, Pr, and ω
∙ Explain the basis for the two- and three-parameter corresponding-states correlations ∙ Compute parameters for the Redlich/Kwong, Soave/Redlich/Kwong, and Peng/Robinson
equations of state from critical properties
∙ Solve any of the cubic equations of state, where appropriate, for the vapor or vapor-like and/or liquid or liquid-like molar volumes at given T and P
∙ Apply the Lee/Kesler correlation with data from Appendix D
∙ Determine whether the Pitzer correlation for the second virial coefficient is applicable for given T and P, and use it if appropriate
∙ Estimate liquid-phase molar volumes by generalized correlations
3.10 PROBLEMS
3.1. How many phase rule variables must be specified to fix the thermodynamic state of each of the following systems?
(a) A sealed flask containing a liquid ethanol-water mixture in equilibrium with its vapor.
(b) A sealed flask containing a liquid ethanol-water mixture in equilibrium with its vapor and nitrogen.
(c) A sealed flask containing ethanol, toluene, and water as two liquid phases plus vapor.
3.2. A renowned laboratory reports quadruple-point coordinates of 10.2 Mbar and 24.1°C for four-phase equilibrium of allotropic solid forms of the exotic chemical β-miasmone.
Evaluate the claim.
3.3. A closed, nonreactive system contains species 1 and 2 in vapor/liquid equilibrium.
Species 2 is a very light gas, essentially insoluble in the liquid phase. The vapor phase contains both species 1 and 2. Some additional moles of species 2 are added to the system, which is then restored to its initial T and P. As a result of the process, does the total number of moles of liquid increase, decrease, or remain unchanged?
3.4. A system comprised of chloroform, 1,4-dioxane, and ethanol exists as a two-phase vapor/liquid system at 50°C and 55 kPa. After the addition of some pure ethanol, the system can be returned to two-phase equilibrium at the initial T and P. In what respect has the system changed, and in what respect has it not changed?
3.5. For the system described in Prob. 3.4:
(a) How many phase-rule variables in addition to T and P must be chosen so as to fix the compositions of both phases?
(b) If the temperature and pressure are to remain the same, can the overall composi-tion of the system be changed (by adding or removing material) without affecting the compositions of the liquid and vapor phases?
3.10. Problems 117
3.10 117
3.6. Express the volume expansivity and the isothermal compressibility as functions of density ρ and its partial derivatives. For water at 50°C and 1 bar, κ = 44.18 × 10 −6 bar −1 . To what pressure must water be compressed at 50°C to change its density by 1%? Assume that κ is independent of P.
3.7. Generally, volume expansivity β and isothermal compressibility κ depend on T and P.
Prove that:
( ___∂ β ∂ P ) T = − ( ___∂ T ) ∂ κ P
3.8. The Tait equation for liquids is written for an isotherm as:
V = V 0 ( 1 − _AP
B + P )
where V is molar or specific volume, V0 is the hypothetical molar or specific volume at zero pressure, and A and B are positive constants. Find an expression for the isother-mal compressibility consistent with this equation.
3.9. For liquid water the isothermal compressibility is given by:
κ = ______c
V ( P + b )
where c and b are functions of temperature only. If 1 kg of water is compressed iso-thermally and reversibly from 1 to 500 bar at 60°C, how much work is required? At 60°C, b = 2700 bar and c = 0.125 cm3·g–1.
3.10. Calculate the reversible work done in compressing 1(ft)3 of mercury at a constant temperature of 32(°F) from 1(atm) to 3000(atm). The isothermal compressibility of mercury at 32(°F) is:
κ / ( atm ) −1 = 3.9 × 10 −6 − 0.1 × 10 −9 P / ( atm )
3.11. Five kilograms of liquid carbon tetrachloride undergo a mechanically reversible,