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Home , Isobaric process, Isothermal process, Van der Waals equation

Thermodynamic Functions at of van der Waals

Akira Matsumoto Department of Material Sciences, College of Integrated Arts and Sciences, Osaka Prefecture University, Sakai, Osaka, 599-8531, Japan Reprint requests to Prof. A. Matsumoto; Fax. 81-0722-54-9927; E-mail: [email protected]

Z. Naturforsch. 55 a, 851-855 (2000); received June 29, 2000

The thermodynamic functions for the van der Waals equation are investigated at isobaric process. The Gibbs free is expressed as the sum of the and PV, and the in this case is described as the implicit function of the cubic equation for V in the van der Waals equation. Furthermore, the is given as a function of the reduced , and volume, introducing a reduced . Volume, , , capacity, thermal expansivity, and isothermal are given as functions of the reduced temperature, pressure and volume, respectively. Some thermodynamic quantities are calculated numerically and drawn graphically. The , thermal expansivity, and isothermal compressibility diverge to infinity at the critical point. This suggests that a second-order transition may occur at the critical point.

Key words: Van der Waals Gases; Isobaric Process; Second-order .

1. Introduction the properties of the determinant. In the case of these functions, a singularity will occur thermodynamically There is a continuing problem concerning the ther- at a critical point, while the volume V(T, P) becomes modynamic behaviours in a supercritical state. The a complicated expression as a root of the cubic equa- van der Waals equation as one of many empirical tion. Introducing a reduced equation of state obey- state equations, can be given some qualitative illustra- ing the principle of corresponding states, the Gibbs tions of observed P-V isotherm, the critical state and free energy can be replaced by a function of reduced the phase equilibrium between and states. temperature, pressure and volume. Some thermody- There seems to be a problem associated with the fact namic quantities are written in a form, applicable that thermodynamic quantities have been expressed to all substances, which does not require the spe- quantitatively as functions of temperature and pres- cific constants, a and b, for any gas. From the view- sure as intensive variables, while the van der Waals point of an isobaric process, this may be significant equation has been applied to investigate deviations to provide the properties of thermodynamic quanti- from the equation of the ideal state of a gas by virial ties in the supercritical state and the second-order expansion [1,2]. phase transition of van der Waals gases at the critical The Gibbs free energy can be described as the sum point. of the Helmholtz free energy, A(T, V), and PV. The In this , the Gibbs free energy is obtained Gibbs free energy, however, must essentially be repre- from the Helmholtz free energy in comparison with sented by the variables temperature and pressure. The the imperfect gases in classical volume V in the Gibbs free energy equation always [1], and also this free energy becomes a function of satisfies the cubic equation for V in the van der Waals reduced temperature, pressure and volume. The vol- equation and becomes an implicit function of tem- ume, enthalpy, entropy, heat capacity, thermal expan- perature and pressure. The determinant of the cubic sivity and isothermal compressibility are included as equation is naturally zero at the critical point. Some variables. These thermodynamic quantities are deter- thermodynamic functions that contain the derivative mined through numerical calculations and are graph- of V(T, P) with respect to T or P depend strongly on ically displayed for an isobaric or .

0932-0784 / 2000 / 0900-865 $ 06.00 © Verlag der Zeitschrift für Naturforschung, Tübingen • www.znaturforsch.com 852 Thermodynamic Functions at Isobaric Process of van der Waals Gases • A. Matsumoto

The second-order phase transition of van der Waals If the temperature, pressure and volume of a gas are gases at the critical point is discussed. expressed in terms of Tc, Pc and Vc, respectively, then T = 9TC, P = irPc, and V = (f)Vc. The reduced 2. Thermodynamic Functions equation of state obeying the principle of correspond- ing states is easily derived from the van der Waals A van der Waals gas is defined by the equation of equation: state RT a (V) P = (1) V - b V2' By applying 9,7r, and 0, the Gibbs free energy can be The Helmholtz free energy A{T, V) [3] is ex- written as pressed with the temperature function (p(T) as G(0, tt) = -RT 9 [C + j ln 9 + ln(0 - I) (8) RF a C A(T, V) = — --]iV + V(T). (2) 37T(f). In classical statistical mechanics [1], the partition function of an imperfect gas is represented by The constant C in (8) can be rewritten as / lirmkT \3N/2 Zy „ „ Z(T, V, N) = ( —2 J TT7 = Z«ZV. (3) 3 2irmkTc NA V h N\ C = - ln -rl ln — + ln Vc (9) 2 h e Z-d is the partition function of an with the variable of T, and Zy is the configurational partition = -7.07228 + - ln MTC + In Vc, function with the variable V. The first and second term in (3) correspond to the free energy Zy and ZKj, where M is the molecular weight, Tc is in units of respectively. The free energy in (2) is rewritten as 3 K and Vc in units of cm . The Gibbs free energy, fl,21 as a function of the molecular weight and the critical constants of a particular gas, is not applicable to all . ^ , mr3, 2-KmkT , Na A(T,V) = -RT[- In—^--ln —* (4) substances. The reduced volume in this free energy, is ob- tained by the cubic equation for 0 in (7) as + ln(V - b) + RTV , 1 80 , 3 1 + + (10) where NA is Avogadro's number and e the base of 3 5LX 7T TT the natural logarithm. We can arrive at the Gibbs free Similarly, as mentioned in the case of V for (6), the energy by adding to PV: root of cf) in (10) is written as an implicit function of 9 and 7T. The Gibbs free energy in (8), therefore, can be G(T, P) = A(T, V) + PV (5) described as a function of 9 and 7r. All thermodynamic functions can be derived from the Gibbs free energy _r3 2ixmkT NA Tr , = -RT - ln —— ln — + ln(V - b) in the known way. 2 bl e We can exactly solve the cubic equation for a PV, Substituting (p = x + a, the standard cubic equation without x2 can be replaced by + RTV ~ ~RT ' x}+px + q = 0. (11) The volume in (5) always satisfies the roots of the cubic equation for V in (1): The coefficients p and q in (11) result from

ab v-M » + f)V + £v = 0. (6) a=-(l + —), (12) ~P 9 TT Thermodynamic Functions at Isobaric Process of van der Waals Gases • A. Matsumoto 853

^ j 3) The heat capacity at constant reduced pressure is eas- P .?.-(«*-I) 3 7T ily derived from the enthalpy, and ap i dH _ Q T 3ft 1 (22) Q = -- = a3 + —. (14) 2 2-7T 27T 3 6 d(j) The determinant of the cubic Eq. (11) is defined as = R[2 + ^Tj3(d9)"1 D = P3+Q2 (15) where the derivative of 0 with respect to 0 can be a3 3a2 3ft 1 1 given as 7T 47T2 27T2 7T3 47T2 dd> 8 dx = (23) If 7T > 1 or 0 > 1, then the determinant is always <7d 6 ^ 9tx d ft positive, and a real root and two complex conjugated roots are given. Especially, the triple real roots are found at D = 0. A real root is given by 1 (24) da ^ da + 2y/D da = a + .X = ft + y/R\ + \/R2, (16) where 1 1 d D R\ = Q + \[T) (17) 'r2 ^a 2 y/D d a J and dQ/da and dD/da can easily be obtained. The thermal R2 = Q-VD. (* 8) expansivity at constant reduced pressure is defined by

Since the coefficient in the term (dcj)/dO) satisfies the n 1 ,3V\ 1 n^ reduced equation of state in (10), the volume is given Q = 7(5T)p = 2t0(d?)' (25) as the form missing (dcfr/dO)^, The isothermal compressibility is defined by (19)

V dP Pq (p d 7T (j) becomes the real root in (16). The entropy is ob- tained as where the derivative of (f) with respect to 7r can be given as

80 dx (27) d 7T 9t:2 d 7T = + ^ + + i)]. with da: Id Q Id D The entropy in (20) involves the constant C. How- d 7T 3 L 3 d7T + r~7=-]—2y/D d7r ) (28) ever this entropy, which is dependent on C, becomes generally applicable to all substances. The enthalpy is obtained as 1 (d Q 1 d D ffii d 7r 2\fD d 7r 3 G d G (21) H = 1W(T)]p = ~ lde(J)]' where

9 3tr<£ d Q _ SO 2 3 )+ 3a .1 = RTd-2, PTT + -TT")- d 7T 97t2 27t 27r2 2ir2 854 Thermodynamic Functions at Isobaric Process of van der Waals Gases • A. Matsumoto

o

0.5 1.0 1.5 0.0 0.5 1.0 1.5 2.0

0.5 1.0 1.5

Fig. 2. Heat capacities, plotted against the reduced temper- ature at various reduced . A: 7r = 1.0; B: 7r = 1.5; C: 7r = 2.0; D: tt = 2.5.

0 5 10 15 2 0

A Fig. 1. Volume (a), enthalpy (b), Gibbs free energy (c) and entropy (d), plotted against the reduced temperature at ir = 1. C= 10.30 in (9) for Ar. and 2 1 d D 3a 3a -) (30) d 7T 9-7T2 v 7r 27T2 27T2

3 2 a 3 a 3a 1 1 0.0 0.5 1 0 1.5 2.0

7T2 27T3 7T3 7T4 TT3

According to (23) or (27), the heat capacity, thermal Fig. 3. Thermal expansivities, plotted against the reduced expansivity and isothermal compressibility diverge to temperature at various reduced pressures. A: 7r = 1.0; B: 7r = infinity at the critical point, whereas the Gibbs free 1.5; C: tt = 2.0; D: tt = 2.5. energy, entropy, volume, and enthalpy, which are ex- pressed as the form missing (d0/d0)„., are continuous results obtained with these thermodynamic functions at this point. are displayed in Figs. 1 - 4. The Gibbs free energy depends mainly on the function C6 and decreases 3. Numerical Results linearly, except for small 9. The volume, entropy and enthalpy are continuous at 9 = 1, whereas these quan- In the above-mentioned cases, most thermody- tities show a sudden change in the neighborhood of namic functions are written as functions of the vari- the critical point. As shown in Fig. 2, the heat ca- ables 9, tt, and 4> and are generally applicable to all pacity at 7T = 1 diverges to infinity at 9 = 1, and substances. In contrast, the Gibbs free energy and the the other heat capacities at various reduced pressures entropy are not generally applicable as these are de- have a maximum value when 7r is larger than 1. The pendent on C in (9). C is required as the practical values become smaller than the maximum when 7r value of a particular gas to calculate numerically the is increased. The heat capacities are converging to Gibbs free energy and the entropy. Now, for example, 5/2 R when 9 is increased. The thermal expansiv- when the critical data for Ar [4] are substituted for ity at 7T = 1, also, diverges to infinity at 9 = 1, and C. we obtain 10.30 as the value for Ar. Numerical these quantities at various reduced pressures show a 855 Thermodynamic Functions at Isobaric Process of van der Waals Gases • A. Matsumoto

15 tendency similar to that of the heat capacities, as shown in Figure 3. The isothermal compressibility at 7t = 1 diverges to infinity at 9 = 1, as shown in Fig- 10 ure 4. If 9 is larger than 1, then the isothermal com- o pressibility is monotonically decreasing, as shown in CL uH Figure 4. Over all, the heat capacity, thermal expansiv- ity and isothermal compressibility show a singularity 5 at the critical point. This singularity suggests a phase transition. The generalized diagrams of some ther- modynamic quantities accompanying a second-order 0 phase transition are typically described in textbooks 0 0 of physical [4], Compared with these dia- n grams [4], those in Figs. 1 and 2 (A) correspond to a second-order phase transition. Furthermore, the re- Fig. 4. Isothermal , plotted against the re- sults in Fig. 2 agree qualitatively with the cubic graphs duced pressure at two reduced . A: 6 = 1.0; B: 9= 1.5. for heat capacities of pure [5].

[1] J. E. Mayer and M. G. Mayer, Statistical Mechanics. [3] J. C. Slater, Introduction to Chemical Physics. Mc- John Wiley & Sons, Inc., New York 1940, pp. 262 and Graw-Hill, New York 1939, p. 187. 263. [4] P. W. Atkins, 6th Ed., Oxford [2] L. D. Landau and E. M. Lifshitz, Statistical Physics. University, Oxford 1998, pp. 33 and 153. Part 1. 3rd Ed., revised and enlarged by E. M. Lifshitz [5] R. W. Shaw, T. B. Brill, A. A. Clifford, C. A. Eckert, and P. Pitaevskii. Translated from the Russian by J. B. and E. U. Franck, C&EN, Dec. 23, 26 (1991). Sykes and M. J. Kearsley, Pergamon Press, Oxford 1950, pp. 232 - 235.