I
J. 0. HIRSCHFELDER, R. J. BUEHLERl, H. A. McGEE, JrV2,and J. R. SUTTONa Naval Research Laboratory, The University of Wisconsin, Madison, Wis.
Generalized Thermodynamic Excess Functions for Gases and Liquids Concise formulas for the thermodynamic excess functions of gases and liquids are presented in a form suitable for high speed digital machine computations T H E EQUATIONS OF STATE
of reference
(2) form a thermodynamically consistent
namic properties a t this standard pressure are usually denoted by a subscript zero. T o express these excess functions in reduced units, it is convenient to define the quantities,
description of the entire P-V-T surface. These equations can be used in any thermodynamic calculation requiring a n equation ofstate, whether or not the probSo’ So R In ( P o l P C ) (5) lem is contained within a single region of definition (2). There will be no nonAo’= A. - R T In (Po/P,) (6) physical discontinuities in any thermoGO’ GO - R T In (Po/P,) (7) dynamic calculation, when these equations are used. I n this article, concise where So: Ao, and Go are, respectively, formulas for the thermodynamic excess the standard state entropy, Helmholtz functions of gases and liquids are prefree energy, and Gibbs free energy. sented in a form suitable for high speed There is no need to introduce a n addidigital computations. Detailed exprestive constant to either the standard sions are given for p ( p , t ) , ( + / b ~ ) ~ , state internal energy, UO,or the en(WW,, In(f/Pc), ( H - HO)/RTI (Uthalpy, H,. Small letters indicate the U o ) / R T ,and (C, - C,,)/R. The other reduced properties-e.g., the reduced important relations are simply related to pressure is p = P/Po as in (2). those given in detail,
+
( A - A o ’ ) / R T = In ( f / P c ) ( P V - R T ) / R T , (1) ( G - Go‘)/RT = In ( f / P o )
( S - So’)/R = ( H
(2)
- HO)/RT -
- Cm)/R =
(Cw - C*o)/R
(zc t / P 9 K @ / w , l 2 / ( @ / a P ) t
The internal energy excess function may be derived from the thermodynamic relationship,
+
-
1 (4)
Expressions based on the pseudocritical hypothesis are also given for the partial molal volume and for the fugacity of the first component of an s-component mixture.
Definitions From a n equation of state there may be derived a set of thermodynamic excess functions which are defined as the difference between the values of the thermodynamic properties a t a particular temperature and !pressure (or density) and the values of the same properties a t the same temperature but at a standard pressure, Po. The thermodyPresent address, Statistical Laboratory, Iowa State College, Ames, Iowa. a Present address, Research Laboratories, Arsenal, Huntsville, Ala. Present address, Mechanical Engineering Research Laboratory, East Kilbride, Glasgow, Scotland.
386
> 0.
a t densities such that
For a gas a t density greater than the critical, the range of integration is divided into two zones, such that In ( f / P c ) = In ( f 1 l P A
+
( H - Ho)/RT = ( H i
- Ho)/RT +
Derivations
In ( f / P c ) ( 3 ) (Cp
T o obtain either ( H - H o ) / R T or ln(f/P,) for a gas a t a density less than the critical, Equation 10 or 31 can be used directly with the equation of state of Region I. Equations 10 and 11 may be extrapolated past the saturated vapor density to predict the properties of these metastable vapor states. These extrapolations have physical significance
This differential equation may be expressed in reduced variables and integrated to yield
(13)
where the integrals must be evaluated by using the equation of state of Region IT. The quantities In(fl/P,) and (HI H o ) / R T are evaluated by using the equation of state of Region I from the expressions
(y)[ T- ($),I$ = zc
Or, since H
ES
U
(9)
+ PV,
(zcfillt)
“.p Pt
-1
(10)
-
Similarlv. , , starting from the Maxwell
(g)T
relation = demonstrated that In ( f / P J =
INDUSTRIAL AND ENGINEERING CHEMISTRY
it may be
-1
(15)
In these equations fl, H I , and f l l are the fugacity, enthalpy, and pressure a t the critical density-that is, at the boundary between Regions I and 11. The Region I equation of state permits calculation of the saturated vapor density, p,,, if the vapor pressure, p,,, is known. This vapor pressure might have been computed from any vapor pressure formulation, or obtained from experiment. If p u and p v are known, Equations 10 and 11 define the fugacity,
and the enthalpy, H,, of the saturated vapor:
fY,
Table I. Constants k i i for Dense Gas and Liquid Regions j = O
ln(p,t)
+" d' - 1 - lnzc PVtS L ' v- 1
(16)
0
88.5
i = l
0
-313.3
i = 2
0
i - 3
(17)
5.5
-
p
Ptit
i =4
The enthalpy, H I , of the saturated liquid in equilibrium with the vapor can be determined from the Clausius-Clapeyron equation,
which, rewritten in the reduced variables, becomes ( H i - Ho)/RT = ( H v - HO)/RT -
This relation permits calculation of ( H - Ho)/RT in the liquid region,
+z-"
p -e] -
. - t P .te(pv-l
PI
- p1-l) &??' + (Hv - Ho)/RT dt (20)
where pl is the reduced density of the saturated liquid which must be obtained from an independent relation or from experiment. The integral must be evaluated using the equation of state of Region 111. The condition of equilibrium requires equality of the fugacities of the coexisting phases. From the above relations the fugacity a t any point in the liquid region is given by
-2.25
i = 5
- 3.128
408.9 -237.4
+8
-
47.8
0
j = 3
j = 2
j = l
i = O
+
-124.46 3.848 0.3638'
+
+ 13.428
44.4
405.3 - 6.588 - 1.815@' -457.7 - 7 . 8 p 3.638' 191.9 25.488 - 3.638'
- 21.548
+
+ 15.38 - 4.068
-
-
44.4
19.448
+ 15.668 -
15.668
+ 5.228 0
+ 1.815p2
-- 8 . 4 4 + 4.508
0
133.2
+
cy - 3.35 2
-133.2
- 5.228
-
0
0.363 8'
lation (discussed above) of those properties not given in detail. T o express the formulas as concisely as possible, the following constants are used : 1. Constants used in gas region (Region I)
2. Isothermal change of pressure with density, ( b @ p / b ~ ) ~ Region I. Gas
+
3kz ( - - t ( t / z c ) ( l- b'p2)(1
ko = 5.5, k i = P - ko, kz = ( 1 - ko - a 2p)/2 b = ( 1 / P ) ( 3 B 2 - 6B - 1 ) / ( 3 P - 1 ) b' = ( P - 3 ) / ( 3 P - 1 ) k3 b(4b' - b')"' kq = 2b'(4b' - b2)-"'
+
[A somewhat better fit of the Lydersen, Greenkorn, and Hougen tables is obtained by setting ko = 4.71 instead of 5.5. This is explained in (2).] 2. The constants, ki,, used in the dense gas (Region 11) and liquid regions (Region 111) are given in Table I as functions of a, 8, and ko. They arise from evaluations of ho, hl, hp, h3, and s in terms of a, @, and ko (= 5.5) [see ( 2 ) ] . In the dense gas and liquid regions it is convenient to use the three sums,
Q ~ ( P=) ( h i
+ + b'p')-'
t-')p2
- bp
(29)
Region 11. Dense gas
Region 111. Liquid
3. Constant volume chan e of pressure with temperature, (h@?bt)p Region I. Gas ( d f i / d t ) p = kit-2p2
- kz(1
(PIG)( 1 -
+ t-')p3 +
bp
+ b'p')-'
(32)
Region 11. Dense gas
+ + k2ipa + k31~' + kliP
+ k6iP6)/P ( 2 2 ) + kzlp' + 2ka,p3 f3k4jp4 hip4
( -koj
R,( P )
,
Region 111. Liquid
4k6d)/pa ( 2 3 )
W,(P)= [ - ( 1 / 2 ) k 0 i k3ip3 (1/2)k4,p4
+
where the integral is evaluated using the equation of state of Region 111. Equations 20 and 21 may be extrapolated past the saturated liquid density to predict the properties of these metastable liquid states. These extrapolqtions have physical significance at densities such that Explicit Relations for Thermodynamic Properties
The expressions for the fugacity and the enthalpy excess functions derived above have been evaluated using the equations of state of ( 2 ) . These and several other important relations are presented below. All of these expressions are, of course, thermodynamically interrelated, a fact which permits the calcu-
- kllP + k z d l n ( ~4)-
+ ( 1 / 3 ) k 6 1 ~ ~ 1 / ~( 2' 4 )
Here R, = dQ,/dpand Q, = p2 dW,/dp. The tabulation of the important function may then be written :
1. Pressure, @(p,t) Region I.
P
= -(ko
Gas
+ k i t-') p' + kz( --t + t-') (P
t/zo)/(l
At the critical density PI = ' / ' ( I ko)(t t-')
+
+
'/z(t
+ b$')
- bp (p
p8
f
equation. Therefore, evaluation of
(26)
and is independent of the scope of this article. 4. T h e fugacity, In(f/Pc)
= l),
- - t - * ) c ~ - ko
(g)
(25)
(%)
+
Region 11. Dense Gas
In Equation 34 one should recall that the system of equations of state of the preceding article (2) permits the use of any vapor pressure equation and/or any saturated liquid density
Region 111. Liquid
VOL. 50, NO. 3
MARCH 1958
387
4
4
i_l
J
I
I 0
1.5
1.0
0.5
2.0
REDUCED DENSITY
1-3
I
2.5 r=Vc/V
3.0
Figure 2. Excess Helmholtz free energy as a function of reduced density calculated using parameters characteristic of nitrogen
Figure 1. Excess enthalpy as a function of reduced density calculated using the parameters characteristic of nitrogen
In this discussion the arc tangent is always taken to lie between -n/2 and 7/2. At the critical density ( p = l), In ( f l / P c ) =
- z,[2(kot-l
+ k ~ t P +)
+ + +
At the critical density,
kz(1
+
+
+
l),
+ 3k,t-' + - 2t-91 + ( b - b ' ) / ( l - b + 6') (40)
(Hi- Ho)/RT
( 3 / 2 ) k z ( l - tF')] In t ( b - b')/(l - b b') (1/2) ln(1 - b 6') k3 [tan-' k3 tan-' ( k l - k 3 ) ] - In zc ( 3 6 )
(p =
=
--tc[2kpt-'
Region 11. Dense Gas
6. Internal energy, (U - Uo)/RT Region I. Gas
Region 11. Dense Gas ( U - Ur)/RT =
Region 11. Dense gas
3 g=O
+ 2 k,t-2
- z,[kot-'
Region 111. Liquid Region 111. Liquid
388
INDUSTRIAL AND ENGINEERING CHEMISTRY
-
k2t-21
(44)
T H E R M O D Y N A M I C EXCESS F U N C T I O N S I
\
-? 1.
I
0
,
1.0 REDUCED
0.5
I
1.5 DENSITY
,
I
2.0 2.5 p=Vc/V
I
3.0
0
Figure 3. Excess entropy as a function of reduced density calculated using parameters characteristic of nitrogen
I
I
0.5
1.0
REDUCED
I
I
I
1.5
3.C
25 DENSITY t = V c / V
2.0
Figure 4. Excess Gibbs free energy as a function of reduced density calculated using parameters characteristic of nitrogen
0
I
\
I
I
A
CARBON
DIOXIDE
T x 150'C t i I391
I-
-1000
O
BRIDGEMAN
+ E \
3 -I I
-ZOO(
u-uo cal. / mole
I
i
2-41 a z w
*
I:_ ,
0
0.5
,
,
130.8
,
-3000
; j
1.0 15 20 25 REDUCED DENSITY f=Vc/V
-4000
J 50
Figure 5. Excess internal energy as a function of reduced density calculated using the parameters characteristic of nitrogen
s:'cT'\
der WAALS
\
