Virial hal denklemlerinin uygulanabilirlik sınırlarının araştırılması

Abstract

Termodinamikte, pVT ilişkisini veren pek çok bal denklemi var- dır. Bunlardan bir kısmı tamamen deneysel, bir kısmı ise teorik veya yarı teorik hal denklemleridir. Bu hal denklemlerinden biri de virial hal denklemidir. Bu denklemin diğerlerine göre en önemli üstünlüğü, teorik bir denklem olmasından kaynaklanmaktadır. Denkle min katsayıları, maddenin potansiyel fonksiyonu ile yakından ilişki lidir. Denklem, hacım ile ters üssel, basınç ile ise üssel olarak artan bir seri açılımından ibarettir. P*V BCD Z = = 1 4 -- 4 - - - 4 - - 4... (1) P*V Z = = 14 B'-p 4 C'*p2 + D'*p3 4... (2) R*T Burada B, C, D,... katsayıları sırasıyla ikinci, üçüncü, dördüncü, vs. virial katsayılardır. Bu katsayılar, basınç ve yoğunluktan ba ğımsız olup, sadece sıcaklığın bir fonksiyonudur. Gas karışımları için ise, bileşime de bağımlıdır, ikinci virial katsayı, iki mole külün etkileşimini, üçüncü virial katsayı üç molekülün etkileşimini vs. hesaba katan katsayılardır. Virial hal denklemi, genel olarak serinin üçüncü teriminden sonrası ihmal edilerek kullanılmaktadır. Bu denklemler ve üçüncü virial katsayının da ihmal edilmesi sonucu oluşan lineer denklemle rin, hangi sıcaklık ve basınç sınırları içinde uygulanabilir olduğu nun saptanması, çalışmanın esasını teşkil etmektedir. Bu amaçla, bir bilgisayar programı oluşturularak metan, argon, azot, etilen, karbon monoksit ve karton dioksit gazlarının pVT verilerinden yarar lanılmıştı.

Many equations of state have teen proposed and each year additional ones appeal" in the literature, but most of them are either totally or at least partially empirical. All empirical equations of state are based on more or less arbitrary assumptions which are not generally valid. The constants which appeal" in a gas- phase equation of state reflect the nonideality of the gas; there is a need for such constants because of intera»lecular forces. There fore, to establish the composition dependence of the constants it is important that the constants in an equation of state have a clear physical significance. For reliable results, it is desirable to have a theoretically meaningful equation of state such that mixture properties may be related to pure-component properties with a minimum of arbitrariness. The equation of state for gases has a sound theoretical found ation and is free of arbitrary assumptions. The virial equation gives the compressibility factor as a power series in the reciprocal molar volume 1/V: P*V BCD Z= = 1 4 - - + - - + -- 4... (1) In this equation, B is the second virial coefficient, C is the third virial coefficient, D is the fourth, and so on. All the virial coefficients are independent of pressure or density and for pure components, they are functions only of the temperature. The unuque advantage of the virial equation follows because there is a theore tical relation between the virial coefficient and the intermolecular potential. Further, in a gaseous mixture, the virial coefficients depend on the composition in an exact and simple manner. The compressibility factor is sometimes written as a power series in the pressure: P*V Z = = 14 R»T 4 C'-p2 4 B'*p3 4. (2) where the coefficients B', C, D',... depend on temperature tat are Independent of pressure or density. There are relationships between equation (1) and (2). These relationships are as follows: B' = C* = D' = B R«T C-B2 (R*T)2 D - 3»B*C 4 2»BS (R*T)3 (3) (4) (5) The second virial coefficient B is generally evaluated from low pressure pVT data by definition B = limit r b2 f- >0 L ^P (6) Similarly, the third virial coefficient must also be evaluated from pVT date at low pressures; it is defined by C = limit H* o- >o L hj* l1 n (7) For many different gases, it has been observed that equation (1), when truncated after the third term (i.e., when D and all higher virial coefficients are neglected), gives a good represent ation of the compressibility factor to about one half the critical density and a fair representation nearly to the critical density. viii For hi^ıer densities, the virial equation is, at present, of little practical interest. Experimental as well as theoretical methods are not as yet sufficiently developed to obtain useful quantitative results for fourth and higher virial coefficients. The significance of the virial coefficients lies in their direct relation to intermolecular forces. In an ideal gas, the molecules exert no forces on one another. In the real world, no ideal gas exists, but when the mean distance between molecules becomes very large, all gases tend to behave as ideal gases. This is not surprising since intermolecular forces diminish rapidly with increasing intermolecular distance and therefore forces between molecules at lot-? density are extremely weak. However, as the density rises, molecules come into closer proximity with one another and, as a result, interact more frequently. The purpose of the virial coefficients is to take this interactions into account. The physical significance of the second virial coefficient is that it takes into account deviations from ideal behavior which result from interactions between two molecules. Similarly, the third virial coefficient takes into account deviations from ideal behavior which result from interactions of three molecules. The physical signifi cance of each higher virial coefficient follows in an analogous manner. Virial coefficients can be theoretically calculated with poten tial functions. Some of these functions are: Ideal-Gas Patential, Hard-Sphere Potential, Sutherland Potential, Lennard- Jones Potential Mie Potential, The Square-Well Potential, The Exp. -6 Potential, The Kihara Potential, The Stockmayer Potential, and so on. There is also some correlations to calculate second and third virial coefficients. For second virial coefficients, The Pitser- Curl-Tsonopoulos correlation is available: B»pc where.Pc r T -. r T = f(0) + wfU).Tc l Tc J L Tc (8) r T -, 0.330 0.1385 0.0121 0.000607 f(0) = 0.1445 - - - - (9) L tJ Tr T2r T3r T»r ix fd) L TcJ 0.331 0,423 0.008 0637 + TE T3j T»i (10) «here Tr=T/Tc. For third virial coefficients, it is difficult to establish an accurate correspondind-states correlation because of the scarcity of good experimental results. A correlation of reduced third virial coefficients was established by using reliable experimental data for argon, methane, and ethane. The correlation has the following form: C C2 C3 - = ci 4 4 4 Vs Tr T»r C5 C8 C4 4 4. r T5 r Tr *exp(- ) 10 C8 C? 4 10 ] 1 - exp( ) Tr J Constant ck is related to the acentric factor "w" by ck = Akl 4 Ak2*w 4 Aka-w2 (k=l,2,...,8) Aki (1=1,2,3) are constants. (11) (12) Equation (11) holds for the reduced temperature range 0. 66STr<3. 13 but only for acentric factors between aero and 0,1, The purpose of this study was to examine the investigation of the applicability limit's of the virial equations. The virial equ ations used in this work are: Z = Z = P-V R-T P-T R*T B = 14 T 14 B'*p (13) (14) x Z = = 1 4 + (15) R*T V V2 = 1+ B'»p 4 C'p2 (16) R-T The following equations are developed for estimating the temperature dependency of the second and virial coefficients. b B = a 4 4 d*log(T) (17) n r C = m 4 4 4 s*exp(T0.5) (18) Tp log(T) $e developed two methods for estimating the second and third virial coefficients. In this work, the pVT data were taken fro» the literature. The gases choosed for this research are: Methane, argone, nitrogen, ethylne, carbon monoxide and carbon dioxide. The proposed methods are simple and compares favourably with the second and third virial coefficients data reported in the lite rature.