Development of an Equation of State for Gases Joseph J

Home , Compressibility, Thermodynamic diagrams

DEVELOPMENT OF AN EQUATION OF STATE FOR GASES JOSEPH J. MARTIN and YU-CHUN HOU University of Michigan, Ann Arbor, Michigan

I. Pressure-Volume-Temperature Behavior of Pure Gases and Liquids 1

More than a hundred equations with any great degree of precision With any isotherm t,aken of state relating the pressure, vol- and not one of them is considered at ume, and temperature of gases suitable for the calculation of ac- Po= 0, 2, = 1.0, have been proposed according to curate thermodynamic diagrams. Dodge (7), but only a very few of This does not imply, however, that (dZ/dP,) = lim (2- 1)/ (P,- 0) them have attained any practical these two-constant equations have importance as the majority do not not been extremely useful. Van der as P, -+ 0 represent the data with sufficient Waals' equation was of the greatest accuracy. In this work the signifi- value in leading to the principle of = lim RT (Z - I)/RTP, cant pressure-volume-temperature corresponding states. as P, 0 (hereafter referred to as PVT) -+ In one form the corresponding- characteristics of pure gases have = (P,/RT)Iim(ZRT/P-RT/P) (5) been examined in detail, and an state principle suggests that the equation has been developed to fit compressibility factor, x = PVIRT, precisely the characteristics com- depends only on the reduced tem- mon to different gases. perature and pressure, which are PVT data may be plotted on dif- defined respectively as T, = TIT, ferent types of graphs, of which and P,= PIP,. On a generalized probably the oldest is that of pres- compressibility chart for many dif- sure vs. volume with temperature ferent compounds single average as a parameter, as shown in Fig- lines are drawn for each isotherm; ure 1. From this graph van der however, to demonstrate that the Waals deduced two properties of principle is approximate, Figure 2 the critical isotherm namely, that has been constructed to emphasize at the critical point the slope is the differences which actually exist zero and an inflection occurs.' Van among compounds. der Waals expressed these two From the compressibility chart properties algebraically in the fol- it is noted that all gases follow the lowing well-known manner : ideal-gas law as the pressure approaches zero, regardless of the temperature. This may be expressed (dP/dVl~= 0 at the critical point as

Z = PV/RT= 1at P = 0 for all (d2P/dV2),= 0 at the critical point FIG. 1. PRESSURE-VOLUMEDIAGRAM. temperatures (3) (2)

~ A curious corollary of where P is pressure, V is volume, this is the seemingly and T is temperature, as indicated contradictory fact that above. Van der Waals employed in general V does not these two conditions to evaluate equal RTIP at P=O. ------the two arbitrary constants in the By definition of a de- equation of state he proposed. rivative at any point A number of other two-constant (Po,2,) on the com- (exclusive of the gas constant) pressibility chart, equations of state have been proposed, the best known being those of Berthelot and Dieterici. None of them, however, actually represent (dZ/dPr)Tr the PVT data over a wide range -co*coyYD I - - COY.OW0 2 *Some investinations (13) indicate that the = lim (2 - Z,>/(p,- P,) critical isotherm- is not the smoothly inflected curve shown in Figure 1. This result seems to 0 Pr.10 be attributed to the indefiniteness of the -PI critical state and possibly to the lack of as P,-b P. attainment of true equilibrium. (4) FIG. 2. COMPRESSIBILITYCHART.

Page 142 A.1.Ch.E. Journal June, 1955 Since by definition Z = PV/RT, or sumed indiscriminately either, or the essential inflection at the criti- ZRTIP = V,(dz/dP,)T, = cal point may not be preserved. In Figure 4 the necessary impli- (P,/RT) lim (V - R TIP)as P, --+ 0 cations of the assumption that the third derivative is zero are con- = - (Pc/BT)(cx)~,~-+O (6) sidered. In Figure 4d four possible curves have been drawn, all of where a is defined as the residual which go through zero according to volume, (RTIP-V). the assumption. If the true situa- If Equation (6) is rewritten, tion is similar to curve I, the fourth derivative must be positive, as at 0 pressure (Y shown in Figure 4e; the second derivative will pass through a = - (RT/PJ (dZ/’d’r)zr (7) minimum, as shown in Figure 4e; the first derivative will have an 0 aomula.Imwaswm which relates a to the slope of an To .I inflection, as shown in Figure 4b; isotherm on the compressibility FIG. 3. T, AS FUNCTIONOF T,. and the primary pressure vs. vol- chart. Since in general this slope ume curve will exhibit a minimum, is not zero at zero pressure, a is a as shown in Figure 4a. As the finite quantity. The one isotherm primary curve must show an in- for which the slope is zero is known flection, it is obvious that curve I as the Boyle point. cannot be correct. A similar analy- Most generalized compressibilty sis of curves I1 and 111, shows charts show the Boyle point occur- that they cannot be used; however, ring at a T, of about 2.5. This is curve 111, does give the required probably true only for such com- character of the pressure vs. vol- pounds as nitrogen, carbon mon- ume curve and therefore one con- oxide, and methane, which were cludes that if the third derivative considered in making the plots. is assumed to vanish, then the ‘These compounds all have critical fourth must also vanish, as shown temperatures of approximately the in Figure 4e, and the fifth must be same magnitude. For hydrogen negative or zero, as shown in Fig- with a much lower critical tempera- ure 4f. ture the data indicate that the Another assumption which can Boyle point is around a T, of 3.3. be made is that the fourth deriva- For compounds with higher critical tive vanishes while the third re- temperatures there are no experi- mains. An analysis similar to that mental data at the required high above shows that for this case the temperatures. However, by extra- third derivative must be negative polation of the experimental data in order to preserve the correct on isometric plots (see Figure 6), inflection at the critical point. The it appears that the Boyle-point re- hypothesis which is advanced here duced temperature goes down as is that in addition to the vanishing the critical temperature goes up FIG. 4. PRESSURE-VOLUMEDERIVATIVES of the first two derivatives, as sug- and that many compounds have gested by van der Waals, the third Boyle points much lower than a T= o AT CRITICAL derivative is either zero or a small T, of 2.5. Figure 3 is a plot of the a V3 negative number and the fourth Boyle point vs. critical tempera- POINT. [IT IS TAKENFOR GRANTED derivative is zero. Assumptions re- ture, prepared by consideration of garding the derivatives higher than ..AT(%), = 0, AND( a2p rI, o the measured points wherever pos- ) the fourth might be made, but sible and utilization of extrapola- a v2 these derivatives would have much tians to higher temperatures where AT CRITICAL POINT.] smaller effects. In current studies necessary, as just mentioned. the fifth and sixth derivatives are A characteristic’ of gas behavior being given some attention. noted on both the compressibility Another characteristic of gas be- chart and the pressure-volume plot havior noted on the compressibility is the straightness of the critical chart is that for compounds with .isotherm for a considerable range different values of 2, (PV/RT at 0n either side of the critical point. the critical point) lines which con- The length of the straight portion nect the critical point to the point indicates the possibility that deriva- Z= 1.0, P,=O are tangent to tives of pressure with respect to isotherms whose reduced tempera- volume higher than the two of van ture is about 0.8. Algebraically der Waals may be zero or at least this condition is very small. Examination of experimental data to prove whether any (dZ/dP,)T, = - (1 - 2,) derivatives higher than the second are zero is difficult, since the data at T’ Ei 0.8TC (8) are often not too accurate. The The meager data available indi- higher derivatives cannot be as- FIG.5. FIT, AS FUNCTIONOF 2,. cate that T’ varies slightly frorr!

Vol. 1, No. 2 A.1.Ch.E. Journal Page 143 x - FIG.6. PRESSURE-TEMPERATURE DIAGRAM. would result in an error in its (d2P/dT2)v = 0 for high T (11) slope. From a thermodynamic point 0.8 according to the value of 2,. of view, the equality of the slope Figure 5 is a plot of the relation Also the isometrics below the of the vapor-pressure curve and between T and Z,, and it is seen critical density curve down, and the critical isometric has been that were it not for the low 2, of those above the critical density justified by the following analysis. water, it would be difficult to draw curve up.* These conditions are From the Clapeyron equation the any curve. Of course, there are given by the equations slope of the vapor-pressure curve other compounds with low Zc)s, but is dP/dT = ASIAV at any tempera- their data at a T,. of 0.8 are insuf- ture. At the critical point ASlAV ficient to warrant their use. becomes (dSldV)., but from the There are other characteristics Maxwell relation this is (dP/dT)Y,. of gas behavior indicated on the and Therefore, the vapor pressure slope compressibility chart that seem to dPldT = (dPldT)~,,which is the 5e of lesser importance than those (d2P/dT2)v> 0 at V < V, (13) desired equality. nreviously mentioned. Some iso- Another important graph is the herms not only go through a mini- A final observation on the iso- reduced-vapor-pressure plot of log mum point but also exhibit inflec- P,. vs. llT7, as shown in Figure 7. tions, notably those between a T, metric chart is that the slope of the vapor-pressure curve at the The different curves are for a num- of 1.0 and about 1.3. Isotherms ber of different compounds. Im- above the Eoyle point are almost critical point is identical with the slope of the critical-density iso- plication of the plot is that if two straight. The compressibility fac- compounds fiall on the same curve tor is approximately 1.1 to 1.2 for metric. The experimental verifica- tion of this fact is difficult. because at any point, they tend to lie to- all isotherms at a reduced pressure gether over the whole range of of about 10. Although these con- the slope of the vapor-pressure curve is changing very rapidly. temperature. The plot is extremely ditions are not utilized in this pa- useful in filling in unexplored per for the obtaining of an equa- Usually the slope obtained by differentiating a vapor-pressure equa- ranges of vapor pressure, provided tion of state, they have been given that the critical temperature and some consideration. tion at the critical point is slightly less than the slope of the critical pressure and one other point on Another important diagram of khe vapor-pressure curve are PVT data is the pressure-tempera- isometric. This may be attributed to the fact that in many cases the known. The plot is also useful for 'we (isometric) plot shown in predicting the slope of the vapor- Figure 6. The most important fact vapor-pressure equation may not reflect the slight upward curvature pressure curve at the critical point. noted here is that the isometrics Each curve approaches the critical are almost straight. In fact there of the vapor-pressure curve on a log P vs. 1JT plot in the region point (T,= 1, P, = 1) with a are three places where the lines unique slope, d log Prld(lJTr). can be considered straight: one is near the critical, as emphasized by Thodos Also the critical vol- This is called M and is designated for large volumes at low pressures, (23). on each curve. If a compound is another is for the critical volume, ume is difficult to determine accurately, and any error in its value known to follow one of the curves, and the third is for all volumes at then M is known. From this (dPl high temperatures. The) three con- *Data on hydrogen indicate that the isometrics above the critical density may curve up for a dT)v, may be obtained as follows: ditions can be expressed as while, but at the highest reduced tem~eratures they start tn curve down. Since these-high reduced temperatures are not encountered with dPr/Pv = - MdT,/T: (14) (d2P/dT2)V= 0 as P -+ 0 (9) nther compounds the negative curvature of all isometrics at sdh high temperatures is not im- from the definition of M. Since P, portant. Also at very high densities indications are that all isometrics curve down for all re- and T, are both 1, duced temperatures. This means that at some vri'y high density the isometrico for moderate temperatures are straight. dPr/dTr = - M (15)

Page 144 A.1.Ch.E. Journal June, 1955 or from the definitions of P, and T,, sure, critical volume, and one point critical temperature, pressure, and on the vapor-pressure curve. The volume determine the compressi- dP/dT = - MP,/T, = one point on the vapor-pressure bility; however, accurate data show (16) curve determines the slope m of that neither of these latter hypoth- the critical isometric, as given eses is correct. This fact is of great value because above. Mathematically this hypoth- It is the primary objective of it means that vapor pressure de- esis is written as this work to develop an equation termines the PVT behavior along which will represent the function the critical isometric. f(P,V, T, P,, T,, Vc,m) = 0 (17) implied by Equations (17) and A study of the interdependency (18). One might also develop of the characteristic properties of or in terms of Z,, graphical representations of those gases, as here presented, has led equations, which could be in the to a consideration of the minimum form of compressibility plots (2 number of conditions or facts vs. P, with T, as a parameter) for necessary to characterize the PVT If this hypothesis is compared with various values of Z,, T,, and M. behavior of any given compound. previous work, it i’s seen that it is Probably a number of plots would A new hypothesis is advanced that not so simple as van der Waals’, be required to cover the widest four properties are necessary to which states that the compressi- ranges, but such plots might prove give the complete characterization bility is dependent only upon the sufficiently useful to justify their for any compound. These are the critical temperature and pressure, preparation. Part I1 considers only critical temperature, critical pres- or as that which says that the the equation and not the graphs.

11. Derivation of the Equation

Although there have been a num- chosen over the others : this variation, a different form of ber of attempts to develop an equa- equation was selected : tion of state from kinetic theory -kT/T, and statistical mechanics, only P = A BT+ Ce token results have been obtained + (21) to date. The important practical Equation (21) was fonnd to repre- equations which are in use tcday, sent the curvature of isometrics of such as the Benedict, Webb, and many different compounds with Rubin(5) and the Beattie and where f’s are functions of tempera- only one value of k, this value Bridgeman (I), are empirical. The ture and b is a constant. being 5.475. (This could be rounded equation about to be developed is In choosing the temperature off, but as with other constants will also empirical; howev&, it is be- functions for Equation (19), one be carried to preserve internal cm- lieved that this equation fits more first considers the character of the sistency.) If Equation (21) is com- of the known behavior characteris- isometrics of Figure 6. Since these pared with Equation (19), it is tics of gases than does any previous are straight at the high-tempera- seen that the temperature func- equation. ture ends and curve only at the tions, f, to f5, must be of the form The form of equation represent- low ends near the saturation curve, any equation to represent the iso- - .5.475T/Tc ing empirically the PVT behavior fl = A1 + BIT + Cle is chosen so that the data are rep- metrics must become linear at high resented as nearly as possible with- temperatures. Beattie and Bridge- -5.475T/TC fz = Az + B2T + Cze in the precision of the experiment. man(1) suggested the form The equation should also be reason- etc. (22) ably simple in order to be useful in where A,, B1,C1, AP,etc., are con- thermodynamic calculations. Obvi- where A, B, and C are functions stants which may be finite or zero. ously, more terms may be put into of volume. Benedict, Webb, and Thus, with the selection of the an equation to get better agree- Rubin employed the same form. complete form of the equation of ment with the data, but complexity The last term, which is the curva- state through Equations (19) and is the penalty. In this study various ture term, was considered care- (22), the problem now is to find polynomial and exponential equa- fully in the more general form a method of evaluating the con- tions were considered. In view of ClTfi. It was found that to repre- stants. the fact that it is the derivatives sent the experimental data for By virtue of the hypothesis given of pressure which are given by the many different compounds n varied in Part I that the PVT behavior van der Waals analysis and that from about 2 to 5, depending large- of a given compound depends only pressure is almost a linear func- ly upon the critical temperature of upon P,, T,, V,, and m, one must tion of temperature at constant the substance in question. For most have recourse to the general proper- volume, equations explicit in pres- compounds with critical tempera- ties of gas behavior to see just sure were preferred. tures near room temperature n was how many arbitrary constants can Because of its symmetry and the about 3, and for compounds with be determined. The general proper- ease of differentiation and integra- high critical temperatures, such as ties to be used are restated in the tion, the following equation was water, n was about 5. In view of following summary :

Vol. 1, No. 2 A.1.Ch.E. Journal Page 145 If (d3P/dV3) in Equation (26) is assumed to be zero, Equations (35) through (38) may be substi- tuted into Equation (26) to determine b as"

b = Vc - 3RTc/15Pc (39)

or in terms of Z,,

b = V,- 3Vc/15Zc (40) The equation of state is now determined along the critical isotherm; however, a comparison of the equation with the data for a number of different gases showed that the equation predicted pressures too high for volumes greater than the critical volume. Previous comparisons using only the two van der Waals derivatives and a correspondingly shorter equation of state (i.e., one terminating at f4 and with b =0) showed that pressures in this region were predicted too low. In another trial the PV = RTas P-+O (23j PV = fi as P+ 0 (32) third derivative condition was completely neglected when an equatioln and in the light of Equation (23), (dp/dV)~= 0 at critical (24) terminating with f4, but otherwise the same as Equation (19), was fi = RT (33) utilized. This implies that f5 is (d2P/dV2)T= 0 at critical (25) arbitrarily set equal to zero; there- The equation of state now becomes fore, from Equation (38),

P = RT/(V-b) +fi/(V-b)'+

(d4P/dv4). = 0 at critical (27) f3/ ( - br + f4 / ( v - b) +fb/ ( v - b) (34) This assumption predicted pres- Next are employed the four de- sures too low; therefore, some in- rivative conditions, (24) to (27), - and the condition that the equa- *Sometime after working out the i(dZ/dP,) ] = 0 at Boyle- tion of state must be satisfied at analysis of the derivatives at the T, P,=O the critical point. There are five critical point, it was found that Plank and Joffe (20,ll) had studied equa- point temperature TB (29) equations containing five unknowns, tions similar to Equation (19). They fZ(T,) through f5(Tc), and b, assumed five equal roots at the criti- where f(T,) means the tempera- cal point, which is the same as assum- (dzPpldTz)v= 0 at V = V, (30) ture function evaluated at the criti- ing the first four derivatives to be zero. This equality may be shown by cal temperature. However, Equa- considering the meaning of five equal i(dP/dT)V = m = - MP,/T, tion (26) cannot be used directly roots at the critical point. Let P be a until the inequality is removed. function of V, with five equal roots at The solution of the remaining four the critical point. Then P = f(V) at at v = vc (31) T = T, and P-Po = f (V)-P,. At equations in terms of b is P = P, the f(V)-P, must be zero, and because of the equality of five By use of the foregoing nine fi (TJ = 9 P,( Vc- b)'- 3.8RTc(Vc- b) roots this must be equivalent to an- Conditions plus the PVT relatian other function F (V)(V-V,) 5 set equal to zero or P-P, = F (V)(V- .at the critical point (since P,, T,, (35) .and are to be given) determina- V,} 5, where F (V,)PO. Differentiating V, with respect to V gives tion of an equation of state with f3(Te)= 5.4RTc(Vc- b)'- ten arbitrary constants might be expected. Equation (26) however 17P, (V,-ZI)~ (36) dP/dV=5(V-VJ4F(V)+ is not determinate as it stands, and so a total of only nine con- (V- VJ5 F' (V) f4(Te)= 12Pc (Vc-b)4- stants can be obtained from the conditions listed. Since V = V,, this is zero. Carrying 3.4RTC(VC-bl3 (37) on three more diffel-entiations will Equation (23) is utilized by continue to give factors oT V-V,, multiplying through Equation (19) which will came the derivatives to by V-b and letting P approach zero f5(Tc)= 0.8RT, (Vc-b)4- vanish, until me comes to the fifth while V, and therefore V-b, ap- derivative, where at least one term 3Pc (Vc-b)5 (38) win not have tr,t",, i.e, €2W(V). proaches infinity.: Thsp the deTivative is not zero. Page 146 A.1.Ch.E. Journal Jurre, f!K& termediate value between Equa- - 5.475T7 fz (T’) = Az B2T’ tions (40) and (41) is required. j3 = A3 + B3T + C3e (45) + + For this purpose let

= b Vc - PVC/15Zc (42) f4 = A4 (46) [(RT’)2(Zc- l)/PC] - bRT’ (52) where fi is a constant for a given compound. It can be shown that p From Equation (29) f5 = B5T (47) cannot be less than 3.0 or more c (dZ/dP,) TI T= TB, p,=o = 0, than 4.0. The problem of evaluating these and inserting this into Equation By trial with the data of a num- eight constants is now mathemati- (51) gives ber of different compounds along cal manipulation of the eight con- the critical temperature line, it was ditions. From Equation (46) f, is fz(TB)=Az + BzTB+ found that 8 depended upon 2,. Use a constant independent of tempera- -5.475Tg/TC = - bRTB of the varying values of p permit- ture and directly equal to A,. Cze (531 ted very accurate representation of Therefore, from Equation (37), Equations (52) and (53) with the data up to densities about 1.5 -5.475 times the critical density. A4=f4 (TJ = 12Pc(Vc-b)4- fz(T,) = A2 B2Tc+C2e (54) Figure S is a plot which gives + as a function of Z,*. 3.4RTc (Vc-b)3 (48) which is known by Equation (35) Since one of the original ten in terms of P,, T,, V,, and b, form conditions was used to evaluate fl Since Equation (47) sets fs = a set of simultaneous equations with three unknowns, AB,B2, and as KT and since one condition was B,T, f5 (T,) = B5T,and from Equa- indeterminate but can be satisfied tion (38), C,. The solution of this set of by setting the magnitude of the simultaneous equations is third pressure-volume derivative through the term p, there are eight conditions left to evaluate the temperature functions f, to f5. Each temperature function may contain a maximum of three constants, A, (55) B, and C, and so a total of twelve constants must be fixed. In view of B5 =f5 (Tc)/Tc = 0.8R (?‘c-b)4 - Bz=[-f2 (T,)- bRTB - C2 the eight conditions, four constants -5.475T3fTC- -5.475 must be set equal to zero. As men- 3Pc (Vc- b)5/T, (49) (e e 11 / ( TB - TC) tioned earlier, more conditions The derivation of the remaining (56) might be utilized, in which case all six constants is a little more in- -5.475 twelve constants might be finite. volved. Starting with Equation A2=f~(Tc)-B2Tc-Cze However, it has been found that (19), one multiplies through by (57) some constants are much less im- V- b, substitutes PV =ZRT to Substituting Equations (43) to portant than others and that drop- eliminate variable V, and then dif- ping four of them still leaves an (47) into Equation (19), rearrang- ferentiates partially with respect to ing in the form of Equation (21), equation which represents the data pressure at constant temperature, with a high order of accuracy. and differentiating twice with re- to obtain spect to volume give, by use of When the eight constants to re- Equation (30), main in the temperature functions (d2ldP)TRT-b = (ZRT-bP) fi- were chosen, it was felt that the { C3 = - Cz ( Vc- b) (58) slope mnstants, B, and the inter- f ZP[ (dZ/dP)TR T - b] / (ZRT - bP)’ cept constants, A, were most im- } Applying Equations (31) and (58) portant, the curvature constants, C, + { (ZRT-bP) f3P (2) - 2f3P2 gives being of less significance. In order B3=m (Vc- b)3-R (Vc-b)2 - to apply Equation (30), however, [(dZ/dP) TRT- b] } / (ZRT- bP) it is necessary that there be at Bz (Vc-b) - Bs/(Ve-b)’ (59) least two C terms in the equation. + { (ZRT-bP) f4P2(3) - 3f4P3 The six constants left were divided From Equation (45), with T = T,, equally between A’s and B’s. The [ (dz/dP)TR T- b] } /(ZR T - bP)4 one obtains followi,ng was decided upon as be- - 5.475 ing at least as good as any othw + { (ZRT-bP) f5P3(4) - 4f5P4 A3 = f3 (T,) - B3 T, - C3e arrangement : (60) [(dZ/dP)TRT -€4 } /ZRT- bP)6 This now completes the solution fi = RT (43) for all the constants in the equa- (50) tion of state. If the critical temperature, critical pressure, critical - 5.475T7 Since at P, = 0, Z = 1.0, Equation f2 = Az + B2T + Cze (44) (50) simplifies to volume, and one point on the vapor- pressure curve are known for a *It will be noted on Figure 8 that f3 never given compound, appropriate exceeds a valire of 3.36 for the gases which have been studied so far. One might wonder graphs will give slope m of the about the nature of the five roots of the equation at the critical point for i3 lying between critical isometric; the Boyle point, 3.0 and 3.35, since for 6~3.0,it has been -bRT (5 1) TB; the point (about 0.8Tc); shown that the five roots are equal. Further T’ analysis shows that there are three real equal Combining Equations (28) and and p, the determinant of the value roots and two complex roots for f3 greater than 3.0 and less than 3.6. (51) at T, = TIT, gives of the third derivative of pressure

VOl. 1, No. 2 A.1.Ch.E. Journal Page 147 with respect to volume at the fairly complete and accurate, and, sufficiently broad to test the versa- critical point. second, the gases had to be as tility of the equation. The applica- When gases were chosen to test different as possible. Seven gases tion of this new equation of state the new equation, two conqldera- have been studied so far, and it is will be discussed and the calculated tions were paramount: first, the felt that the selection of C02, HzO, results for the seven gases studied PVT data for any gas had to be CsH6, N2, C3H6, H2S, and C3Hs is will be presented in Part 111. 111. Application and Discussion of the Equation In the two foregoing parts of The points which are compared The critical volume is the most un- this paper PVT behavior of gases in these tables have been selected reliable measurement, and it is not is considered in some detail and at places where the equation would uncommon to find differences of 5% appropriate algebraic equations are represent the data least well. It is in this value. Naturally slope m in- given to represent that behavior. probable that the average deviation herits any error in the determina- A new hypothesis is stated concern- for the equation if compared at tion of critical volume when fitting ing the minimum amount of in- regularly spaced intervals over all the equation to specific data is at- formation necessary to characterize ranges of temperature and pressure tempted. any given gas and an equation of would be less than the average of The procedures used in develop- state with nine constants is de- the deviations listed in the tables. ing the equation of state might veloped which requires only the which is about 0.4%. well be of value in helping to de- minimum information according to The precision of experimental termine a better value of critical the hypothesis for evaluation of data was important in the equation volume. Usually this value is not the constants. This equation has of state at almost every turn in its measured experimentally but is de- been applied to seven different development. It was mecessary in termined by a rectilinear diameter

TABLEl.-CONSTANTS EVALUATEDFOR THE EQUATION coz HzO CaHs N2 C3Ha HS C3Hs T, 547.5%. 1165.1"R. 562.66"K. 126.1'K. 364.92"K. 672.4"R. 666"R. Pi 1069.4 lb. 32062 1,306.0 lb. 618 1b. lb. 48.7 atm. 33.5 atm. 45.61 atm. - sq. in, sq. in. sq. in. sq.- m. V, 0.03454 cu. tt 0.0503 cu. ft. cc./ liter/ 1.565 cu. ft. 3.22% cu. ft. - - 3.36 g. 90.1 sc*f lh. 1b. g. mole O.lgl g. mole Ib. mole ib.e 2, 0.27671 0.23246 0.27683 0.29171 0.290932 0.28329 0.278465 m 14.0 22.6 0.625 1.647 0.81 12.8 6.47 /3 3.25 3.05 3.25 3.30* 3.25 3.25 3.25 TB/T, 2.3 2.1 2.15 2.5 2.3 2.3 2.3 T'/T, 0.80 0.83 0.80 0.79 0.791 0.794 0.799 b 0.007495 0.00631 01 0.730231 22.1466* 0.0487575 0.368095 0.714598 R 0.24381 0.59545 1.05052 82.055 0.082055 10.73 10.73 At -8.9273631 -85.7394396 -4104.138 - 1592238.2 - 10.2233898 -21206.0776 -44105.1544 B2 0,005262476 0.0312961744 2.62510 3.221.616 0.0081804454 9.7631723 21.127023 C* - 150.97587 -2590.5815 -59333.234 -22,350,930 - 153.061055 -318,608.803 -736,929.32 1 Aa 0.18907819 3.09248249 8553.304 89,845,367 1.219453756 21133.013- 88,025.6002 Ba -0.oooO704617 -0.00082418321 -3.89165685 -134.024.05 0.00074168283 -7.41557245 -28.1649342 Cs +0.0831424 113.95968 156032.7 1,518,821,653 2 1.7717909 381,344.629 1,846,304.2 Ac -0.002112459 -0 0567185967 -8599.2791 -2,467,297,480 -0.068940713 -9897.9772 - 89,907.6635 Bg 1.9565593 X 4.2388378 X lo-' 7.53651 244,915,183 5.0385878X 3.3036675 63.416365 All the constants for the equation of state are given in units of T,, P,, and I/' ,for that compound. * /3 for NZcould be 3.25 depending upon the manner by which the checked points on the critical isotherm are interpolated from exper- mental data reported. gases within a maximum error of many cases to decide which data plot, whereh the mean saturated 176, and usually much less, for points of different investigators liquid and vapor density is extra- densities up to about 1.5 times the and which data points in different polated to the critical temperature, critical density." Table 1 gives the ranges were th.e most reliable, an which has been determined by dis- constants that were calculated from especially necessary decision be- appearance of the meniscus. An- the required P,, T, Vc, and m in- cause of the desire to hold the other procedure which has been formation for each of the gases. maximum deviation under 1%. used is to determine the critical Tables have been made of compari- Data taken in the region of the temperature as the lowest tempera- sons of the pressures predicted critical point are more inconsistent ture at which a measured pressure- from the equation of state with the than those from anywhere else. volume line undergoes a smooth in- experimentally measured pressures This reflects the inherent experi- flection, the point of the inflection reported in the literature. mental difficulties in this region. being the critical volume. These

*Calculations performed after the writing 3 of this paper indicate that at about one and -C2(Vc-b) -c3(vc-b)z one half times the critical density the equa- c5= tion predicts too much curvature of the isometrics. It has been found that this may be improved by adding a CS term to the equation and utilizing the condition stated in the first paper, that at some very high density (approximately twice the critical density) the isometric is straight. With this condition the only TTere 17 is aplrroxiinately 2.

Page 148 A.1.Ch.E. Journal June, 1955. procedures do not agree too well Possibly new ones will be devel- misinterpretation of the term aver- in many cases, and two alternatives oped; for example, even now there age deviation. Although it may be are suggested here. First, one can is a tendency for the reduced slope generally accepted that the average plot the vapor-pressure data on M of the critical isometric to be a deviation of the Beattie-Bridgeman Figure 7 to determine slope m. function of 2,. Several compounds and Benedict-Webb-Rubin equa- Then referring to the experimental seem to fall out of line, but if tions is 0.18 and 0.34% respectively data, he can establish the critical future experimental work were to (7,lo), experience in applying the volume as that volume whose slope change some of the critical values equations within the density limits agrees with the m so selected. This, for these compounds the correla- claimed showed these values to be of course, means that he is relying tion might be good. misleading, because deviations over on the inherent accuracy of the In general it is claimed that the 1% were found in certain places. slopes of the generalized vapor- equation reproduces the experi- Therefore, this work did not try pressure curves. An alternative to mental data within 1%up to densi- to compare different equations but this is to differentiate the vapor- ties of about 1.5 times the critical tried to determine maximum errors pressure curve directly for the density and up to temperatures of between the equation proposed and compound and use that slope m as about 1.5 times the critical tem- the data. the determining factor in picking perature. At about 1.5 times the This new equation is valuable the critical volume. However, as critical density the isotherms be- when applied to a compound whose mentioned in the first paper, this come so steep that a 1% error in experimental data are limited to must be done with care for often volume may cause 5% or more only the critical temperature, pres- the slight increase in the slope of error in pressure. Therefore, it is sure, and volume and one point on the vapor-pressure curve near the not to be expected that the equa- the vapor-pressure curve. Table 2 critical is overlooked. Second, one tion will predict closely above 1.5 gives a summary of the formulas may fix the critical volume by fit- times the critical density. This for evaluation of the constants in ting the data points on the critical region, which is really the com- the equation of state, together with isotherm as in step 5 in the pro- pressed-liquid region, has been the procedure for application to any cedure, which will be described given some consideration, and it is gas for which the P,, T,, V,, and later, for obtaining an equation of believed that to represent it proper- m can be obtained. state to fit data. When the best ly either the term b in the equation Another situation often arises possible fit of the critical isotherm will have to become a function of when it is desired to fit an equation has been obtained by adjusting the volume or more terms will have to of state to a rather complete set of critical volume and p, one may say be added to the equation. That b PVT data for a specific compound. that the critical volume has been might change with volume seems The procedure of application can determined by analytical means. reasonable when one recalls that be modified. Of course, to start, This is equivalent to fixing the the b in van der Waals’ equation one may obtain the equation by critical volume by a study of accounts for the effective volume using only the P,, T,, V,, and m; graphs of the isotherms and noting of the molecules themselves, and however, it is possible that a better the lowest temperature and point the effective volume of the mole- fit may be obtained by utilizing all at which a pressure-volume line cules might become smaller as the the data in the following procedure: undergoes a smooth inflection. gas is compressed. This is, in fact, 1. Select the best values of P,, T,, The Boyle-point temperature is the direction of the necessary V,, as would be done when the gen- worthy of further discussion in change in b to get agreement at eralized equation, is developed. this work. In the estimation of very high densities, for the equa- 2. Calculate 2, and read off from Boyle points for most compounds, tion at present tends to predict Figure 8 a value of B/Z, or p= the procedure was simply to ex- pressures too high. Decreasing b - 31.8832ze + 20.5332,. trapolate a large volume isometric will increase the denominators of 3. Calculate the temperature func- to the point where Z =PV/RT = 1. the higher power positive terms tions along the critical isotherm, i.e., This has two drawbacks: first, the and thus decrease the pressure. fz(T,), fs?T,), etc- extrapolated isometric instead of An interesting application of the 4. With the eauation established from 3, at the ciitical temperature being straight may have a slight question is contemplated for mix- calculate a series of pressures for dif- negative curvature, which will be tures. Ordinarily when one wishes ferent densities along this isotherm. important when the extrapolation to get an equation of state for mix- The maximum density to which this is carried over a wide range of tures, he averages the constants in calculation should be carried should temperature; second, the Boyle the equation of state for the pure not exceed about 1.5 times the critical point should be determined at zero gases involved according to some density, as above this density it is pressure. It is seen on the com- method of averaging such as arith- known that the error becomes large. pressibility chart, however, that metical mean or geometrical mean. 5. If the agreement of the calcu- the Boyle-point isotherm follows This process may be applied to the lated and experimentally measured pressures is not considered satisfac- Z = 1 up to fairly appreciable pres- proposed equation ; however, an tory in step 4 (it ought to be within sures, so that the procedure seems alternate process is permitted here. 1% if the data are reliable), it is justified. Furthermore, as the equa- The fundamental constants that de- likely that an improvement can be tion is not very sensitive to the termine the equation, P,, T,, V,, made by adjusting V, and P. This Boyle point except at high tempera- and m, may be averaged at the may not seem quite right to adjust a tures, it seems that Figure 3, which start and the equation of state for data point such as V, to get an equa- correlates these extrapolated Boyle the mixture determined by the tion of state; however, usually V, points with the critical tempera- same process as would be used for cannot be determined experimentally a pure component. It is expected with a very high degree of accuracy ture, is sufficiently accurate for and it is quite possible that the value most purposes. to try this out in the near future. first chosen is somewhat in error. As As better critical data become Comparison of this equation with usually the critical temperature and available, it is expected that some other equations has been deliberate- pressure are known with greater of the correlations will change. ly avoided, largely because of the precision than the critical volume, it

Vol. 1, No. 2 A.1.Ch.E. Journal Page 149 TABLE2.-sUMMARY OF THE FORMULAS AND PROCEDURES FOR EVALUATINGTHE is not recommended that these be ARBITRARYCONSTANTS IN THE EQUATIONOF STATE changed unless the experimental values have been determined or pre- Equation of state sented with a considerable lack of precision. Also as V, is changed, the - 5.475TjT, - 5.47.52'/TC value of 2, will change and a dif- RT Az+BzT+Cze A3 +B3 T+ Cse ferent (3 will be predicted from Figure p=---+V-b +- 8. One need not limit the values of B (V-b)2 (V-b)3 to this graph, however, for if the selection of another (3 value which B5 T does not fit the graph will give a bet- + A4 + ter fit of the data, that is the one to (V - b)4 (V - b)5 use. From the nature of the equation (3 must lie between 3 and 4 to retain the correct inflection at the critical point. As a guide in predicting the Formulas (in order of evaluation) effects of adjusting (3 and V,, it has been found by trial that increasing B b=V--- P V, P,v, C3 = - (V, - b) Cz (58) tends to decrease all pressures on the where 2, = ~ 152, R Tc critical isotherm except at the critical point itself; whereas, increasing V, - 5.475 tends to increase all pressures on the Az = f2 (T,) - BzT, - Cze critical isotherm. (57) 6. When the critical isotherm is fz ( Tc)= 9Pc(V, - b)2_ 3.8R Tc(V, - b) represented accurately by the equation, read off a value of T, and T'/T, (3 5) from Figures 3 and 5, or T'1 T, - 0.67512, + 0.9869 7. Obtain slope m by reference to the data for the critical isometric. Usually this can be done easily by reading off one high temperature point on the critical isometric and calculating m = (P-P,) / (T-T,) . 8. Evaluate all the constants in the BI = m (V,-b)3- R (V,-b)* temperature functions so that the f4(Tc)= 12Pc (Vc-b)4 equation is completely determined. 9. Calculate a few points at high -3.4RTc (Vc-b)3 (37) -B2 (V,-b) - --B5 (59) temperatures and low temperatures (Vc- b)z (near the saturation curve) for several isometrics, the latter being taken for example at densities about 2570 f5(Tc)= 0.8RTc (Vc-b)4 - Ii.475 greater than the critical density, = f3 - AI (T,) B3T, - C3e about two thirds the critical density, - 3P, (Vc-bI5 (38) (60) about one quarter the critical density, and about one tenth or one twentieth the critical density. Of course, the exact densities will be those available from the data. 10. If the calculated points do not agree with the data with the desired degree of precision, the values of m, T,, and T' may be adjusted. The effect of changing these values is approximately as follows. Increasing m increases the pressure at higher temperatures for densities near the criti- -5,475 TBIT, cal. Increasing the Boyle temperature -fz (T,) - bRTe - Cz(e - e - 5.475) Bz = (56) T, decreases the pressure at higher TB- T, temperatures for low densities. In- creasing T' increases the pressure at higher temperature for medium densities at about half the critical density Procedure for application of the fore- 5. Read off the third derivative while it decreases the pressure for going formulas to obtain the equation characteristic (3 from Figure 8, or temperatures near the saturation of state for a given compound (3 zz - 31.8832,' + 20.5332,. curve. There is one other parameter 1. Select the best values of T,, P, which may be changed in the equa- and V,. In case any of these are not 6. Calculate the reduced tempera- tion, the constant in the exponential known or are in doubt, refer to Hou- term, which is given as 5.475, but it ture and pressure of a vapor-pressure might be shifted a little in a particu- gen and Watson(l0) for procedures point and place on Figure 7. By in- for estimating. lar case. In general increasing this constant has about the same effect as 2. Calculate 2, = P,V ,IRT,. terpolation estimate the value of M. 3. Read off the Boyle temperature and from this calculate dpldT = rn = increasing T'. 11. It is good practice to check on T, from Figure 3. - MP,/ T,. 4. Read off the temperature from the calculation of the constants by T' substituting V, into the equation 7. With P,, T,, V,, rn, T,, T', and B Figure 5, or3= - 0.67512, + obtained at the end of step 3 to see T' now fixed, substitute in the foregoing whether it produces the critical pres- 0.9869. formulas. sure. The procedure should be re-

Page 150 A.1.Ch.E. Journal June, 1955 peated at the end of step 8. These are the University of Michigan for T not considered complete checks on the financial assistance in the form of T = reduced property, T, z7,fc work, but they reveal most of the a Faculty Research Grant. D errors which might be introduced into P - A,etc. the calculations. ,--Y, NOTATION In conclusion the new equation LITERATURE CITED should have some value where one A, B, and C = functions of specific 1. Beattie, J. A., and 0. C. Bridge- wishes accurate PVT representa- volume man. Proc. Am. Acad. Arts Sci., 63, 229-308 (1928). tion from a minimum of data or 4, B,, (32, -43, B3, C3, A,, and B5= where one wishes to represent some characteristic constants 2, Beattie, J. A., N. Poffenberger, data with an empirical equation, C, = heat capacity at constant pres- and C. H. Hadlock, J. Chem. Php., Even at this time the equation is sure 3, 96-7 (1935). being employed to calculate the 3. Beattie, J. A., W. C. Kay, and J. thermodynamic properties of a H = enthalpy Kaminsky, J. Am. Chem. SOC.,59, compound for which the PVT data M = slope of reduced vapor pres- 1589-90 (1937). are meager. The equation permits sure curve at critical Doint = 4. Benedict, M., J. Am. Chem. SOC., all the usual differentiations and 59, 2224-33 (1937). integrations required for such 5. Benedict, M., G. B. Webb, and L. thermodynamic treatment. For ex- P = pressure C. Rubin, J. Chem. Phys., 8, 334 (1940) ; 10, 747 (1942). ample, the changes in enthalpy and R = universal gas constant entropy may be given in terms of 6. Deschner, W. W., and G. G. the ideal-gas heat capacities and S = entropy Brown Znd. Eng. Chem., 32, 836- the equation of state. T =temperature in absolute scale 40 (1940). 7. Dodge, B. F., “Chemical Engineer- T, = absolute temperature at Boyle ing Thermodynamics,” McGraw- point Hill Book Company, Inc., New T’= absolute temperature for York (1944). which the slope at P,=O of 8. Farrington, P. S., and B. H. Sage, the isotherm on the com- Znd. Eng. Chem., 41, No. 8, 1734-7’ pressibility chart equals the (1949). slope of the line joining the 9. Gornowski, E. J., E. H. Amick, critical point and (Z= 1, and A. N. Hixson, Znd. Eng. P, = 0) Chenz., 39, No. 10, 1348-52 (1947). 10. Hougen, 0. A., and K. M. Watson, V=specific volume or mole vol- “Chemical Process Principles,” ume Part 11, John Wiley & Sons, Inc., PV New York (1947). 2 = compressibility factor = - RT 11. Joffe, J., J. Am. Chem. Soc., 69, b = a characteristic constant for a No. 3, 540 (1947). given substance 12. Keenan, J. H., and F. G. Keyes, “Thermodynamic Properties of e = base of natural logarithm, Steam,” John Wiley & Sons, Inc., 2.71828 New York (1936). 13. Maass, O., and S. N. Naldrett, and fl, f2, f3,f4, f5 = temperature functions Can. J. Res., B18, 103,108 (1940). 14. Marchman, H., W. Prengle, Jr., fl(Tc>,fz(Tc>, f3(Tc), f4(T,), and and R. L. Motard, Znd. Eng. f5 (T,)= temperature functions Chenz., 41, No. 11, 2658-60 (1949). evaluated at T = T, 15. Michels, A., and C. Michels, Proc. m = slope of the critical isometric Rog. SOC.,A153, 201-24 (1935). on pressure-temperature dia- 16. Michels, A., B. Blaisse, and C. gram Michels, Proc. Roy SOC.,A160, 358 (1937). =-M- PC 17. Myers, J. Research Natl. Bur. (g) Standards, 29, 168 (1942). v=vc TC + c2 18. Onnes, H. K., and A. T. Van Urk, Commun. Kamerlingh Onnes (V-b Lab., Univ. Leiden, No. 169d Greek Letters (1924). 19. Perry, J. H., “Chemical En- ct = residual volume = -RT - v engineers’ Handbook,” 3 ed., Mc- P Graw-Hill Book Company, Inc., P = a characteristic constalnt in New York (1950). ACKNOWLEDGMENT 20. Plank, R., Forsch. Gebiete In- genieurw., 7, 161 (1936). The authors wish to express their p = Joule-Thompson coefficient = 21. Reamer, H. H., B. H. Sage, and gratitude to the following persons, W. N. Lacey, Znd. Eng. Chem., 41, who assisted with many of the NO. 3, 482-4 (1949). calculations: H. F. Barry, R. A. 22. Reamer, H. H., B. H. Sage, and Farran, K. H. Gharda, G. A. Gryka, Wf1 W. N. Lacey, Znd. Eng. Chem., 42, H. -4. O’Hern, I. P. Patel, R. G. No. 1, 140-3 (1950). Reimus, R. B. Roof, and P. P. Vora. Subscripts 23. Thodos, G., Znd. Eng. Chem., 42, Appreciation is also expressed to c = the value at critical point, e.g., No. 8, 1514-26 (1950). the Rackham Graduate School of Presented at A.I.ChE. San Fralzcisco meeting, T,, P,, V,, and 2, 1953.

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