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 ap- proaches 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 pro- posed, 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 experi- mental 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,.