Lennard–Jones (9,6) Gases with a Hard Core

(1)

First and Second-Phase Transitions of Gases at Isobaric Process;

Lennard–Jones (9,6) Gases with a Hard Core

Akira Matsumoto

Department of Molecular Sciences, Faculty of Science, Osaka Prefecture University, Gakuencho 1-1, Nakaku, Sakai, Osaka, 599-8531, Japan

Reprint requests to A. M.; E-mail:akibohn@nifty.com

Z. Naturforsch.69a, 665 – 672 (2014) / DOI: 10.5560/ZNA.2014-0060 Received August 6, 2014 / published online November 5, 2014

The thermodynamic functions for Lennard–Jones (9,6) gases with a hard core that are evaluated till the third virial coefficients, are investigated at an isobaric process. Some thermodynamic functions are analytically expressed as functions of intensive variables, temperature, and pressure. Some thermodynamic quantities for carbon dioxide are calculated numerically and drawn graphically. In critical states, the heat capacity diverges to infinity at the critical point while the Gibbs free energy, volume, enthalpy, and entropy are continuous at the critical point. In the coexistence of two phases, the boiling temperatures and the enthalpy changes of vaporization are obtained by numerical calcu- lations for 20 substances. The Gibbs free energy indicates a polygonal line; entropy, volume, and enthalpy jump from the liquid to gaseous phase at the boiling point. The heat capacity does not diverge to infinity but shows a finite discrepancy at boiling point. This suggests that a first-order phase transition at the boiling point and a second-order phase transition may occur at the critical point.

Key words:Lennard–Jones (9,6) Gases;T–PGrand Canonical Ensemble; Gibbs Free Energy;

Isobaric Process; Critical Constants; First and Second-order Phase Transitions.

1. Introduction

Properties of the first and second-order phase transitions at an isobaric process have been pointed out in the text book [1]. Recently, the behaviours in the neighbourhood of the critical point and the coexistence of gaseous and liquid phases have been illustrated, in- troducing some thermodynamic functions at the isobaric process for van der Waals [2], Lennard–Jones [3], Redlich–Kwong [4], and Square-well gases [5]. Espe- cially, all thermodynamic functions at the isobaric process contained the two intensive variables T and P, have been practically derived from the Gibbs free energy inT–Pgrand canonical ensemble [3,6]. From the viewpoint of an isobaric process, an attempt to inves- tigate the thermodynamic functions of the Lennard–

Jones (9,6) gases with a hard core at isobaric process is useful to evaluate the critical constants (T_c,P_c,V_c), the boiling temperatures, and enthalpy changes of vaporization. It may be significant to provide the properties of thermodynamic quantities by discussing the first and second-order phase transitions.

In this work, the expansion coefficients for the second virial coefficient of the Lennard–Jones (9,6) gases

with a hard core are analytically calculated. The pa- rametersε andσ for the Lennard–Jones (9,6) potential with a hard core for 20 substances are determined from the experimental data of second virial coefficients. The expansion coefficients for the third virial coefficient are calculated by using numerical integra- tion. The Gibbs free energy is analytically expressed as two intensive variables, temperature and pressure. Vol- ume, enthalpy, entropy, and heat capacity are included as variables. These thermodynamic quantities for carbon dioxide are determined through numerical calcula- tions and are graphically displayed at an isobaric process. The second-order phase transitions of Lennard–

Jones (9,6) gases at the critical point is discussed, and also the first-order phase transition at the boiling point is investigated.

2.T–PGrand Canonical Ensemble for Imperfect Gases

The partition function for imperfect gases in classical statistical mechanics [7,8] is represented by

Z(T,V,N) =

2πmkT h²

3N/2

(2)

· 1 N!exp

N

logV−B V − C

2V²

, (1)

whereBandCare the second and third virial coefficients, respectively. The partition function in theT–P grand canonical ensemble can be written in the following form by using the Laplace transform ofZ(T,V,N):

Y(T,P,N) =

2πmkT h²

3N/2

· 1 N!

Z ∞

0

exp

N

logV−B V − C

2V²−PV RT

dV.

(2)

IfNnow becomes very large, one can replace the inte- gral by the maximum value of the integrand as is done in statistical mechanics: that is, it is called the saddle point method. If the derivative of the integrand in (2) equals to zero, then we can obtain the cubic equation forVwhich is equivalent to the virial equation of state approximated till the third virial coefficient:

V³−RT

P V²−RT B

P V−RTC

P =0. (3) The Gibbs free energyG(T,P,N)is derived from the partition functionY(T,P,N)by the Legendre transfor- mation [3,6]:

G(T,P,N) =−kTlogY(T,P,N). (4) The Gibbs free energy per mol is expressed as

G(T,P) =−RT

"

CM+3

2logT+logV

−B V − C

2V²−PV RT

# (5)

with

C_M=3

2log2πmk h² −N

e

=−7.07228+logM,

(6) whereMis molecular weight.

We can exactly solve the cubic equation forV in (3).

SubstitutingV =x+RT/3P, the standard cubic equa- tion withoutx²-term is obtained:

x³+3Px−2Q=0. (7)

The coefficient P and Q in (7) are given, respectively, by

P=−

"

1 9

RT P

2

+RT B 3P

#

(8) and

Q= 1 27

RT P

3

+1 6

RT P

2

B +1

2 RT

P

C.

(9)

The discriminant for the cubic equation is given as D=− 1

128 RT

P 4

B²+ 1 27

RT P

4

C

− 1 27

RT P

3

B³+1 6

RT P

3

BC

+1 4

RT P

2

C².

(10)

If the pressurePis larger than the critical pressureP_c, then the discriminant is always positive, and the so- lution in (3) consists of a real root and two complex conjugated roots.

A real root is represented as V =RT

3P +√³ R₁+√³

R₂, (11)

where

R₁=Q+√

D (12)

and

R₂=Q−√

D (13)

in the case thatPin (7) is positive, that is the temperature is smaller than the critical temperatureT_c. Another real root is given as

V =RT 3P +√³

R₁−√³

R₂, (14)

where

R₁=Q+√

D (15)

and

R₂=−Q+√

D (16)

(3)

in the case that P in (7) is negative, that is the temperature is lager then the critical temperatureT_c. In the coexistence of gaseous and liquid phases, the discriminant is always negative and three real roots are expressed as, respectively,

V_G=RT 3P+2√

−Pcosϕ

3, (17)

V_L=RT 3P+2√

−Pcos

ϕ+2π 3

, (18)

and

V_M=RT 3P+2√

−Pcos

ϕ+4π 3

(19) with

ϕ=arccos Q

√−P³. (20) VGin (17) corresponds to the root for the gaseous state, V_Lin (18) to that of the liquid state, andV_Min (19) is the root in the region fromV_LtoV_G.

All thermodynamic quantities can be derived from the Gibbs free energy in the known way. Since the coefficient in the term(dV/dT)_Psatisfies the virial equation of state in (3), the volume is given as the form missing(dV/dT)_P,

V= ∂G

∂P

T

=V. (21)

V is equal to the positive root in (11) and (14) or (17) and (18). The entropy is obtained as

S=− ∂G

∂T

P

=R

"

C_M+3 2+3

2logT+logV−B V − C

2V²

−T 1

V dB dT + 1

2V² dC dT

#

. (22)

The enthalpy is expressed as H=−T²

∂

∂T G

T

P

=RT 3

2−T 1

V dB dT + 1

2V² dC dT

+PV.

(23)

The heat capacity at constant pressure is easily derived from the enthalpy:

C_P= dH

dT

P

=R

"

3 2−2T

1 V

dB dT + 1

2V² dC dT

+T² 1

V² dB dT + 1

V³ dC dT

dV dT

−T² 1

V d²B dT²+ 1

2V² d²C dT²

# +PdV

dT . (24)

If the gaseous and liquid phases coexist, the change of the Gibbs free energy between the gaseous and liquid phases at arbitrary temperature and pressure is obtained as

∆G=−RT

"

logV_G V_L−

1 V_G− 1

V_L

B

−1 2

1 V_G²− 1

V_L²

C− P

RT(V_G−V_L)

# . (25)

The enthalpy changes of vaporization are given as

∆H=−RT²

"

1 V_G− 1

V_L dB

dT +1

2 1

V_G²− 1 V_L²

dC

dT +P(V_G−V_L)

# .

(26)

This cubic equation determinesV_Min order to satisfy Maxwell’s rule betweenV_L andV_G [4,5]. Assuming that the volumeVL in the liquid phase is transformed intoV_Gin the gaseous phase at boiling temperatureT_B,

∆H(T_B,P)in (26) equals of vaporization atT_BandP.

Some thermodynamic quantities as functions of vari- ablesT andPmay be derived from (5) and (21) – (24), andVin (11), (14), (17), and (18) are denotedVLin the region ofT<T_BandV_GinT >T_B.

3. Expansion Coefficients for the Second and Third Virial Coefficients of Lennard–Jones (9,6) Gases with a Hard Core

The Lennard–Jones (9,6) potential with a hard core is defined as

U(r) =







∞, r<σ, 27

4 ε σ

r 9

−σ r

6

, r>σ. (27)

(4)

ε is the maximum energy attraction which occurs at r= (3/2)^1/3σ. Substitutingr=σx, the potential (27) is replaced by

U(x) =27

4 ε[x⁻⁹−x⁻⁶], x>1. (28) The second virial coefficients may be easily found, in classical statistics, from the well-known formula [8]

B=2 3πNAσ³

Z _∞

0

(1−exp[−βU(x)])x²dx,(29) where NA is the Avogadro number and β =1/kT. The second virial coefficients for Lennard–Jones (9,6) gases with a hard core are expressed as

B=b₀

"

1−

∞

∑

n=1

uⁿ n!

27 4

n

·

n k=0

∑

(−n)_k k!

1 2n+k−1

# ,

(30)

whereu=ε/kTandb₀=2/3πN_Aσ. The second virial coefficientsBare expressed as

B=b₀

∞

∑

n=0

b(n)uⁿ (31)

Table 1. Expansion coefficients of b(n) and c(n) for the Lennard–Jones (9,6) potential with a hard core.

n b(n) c(n)

0 1.00000 3.63193·10⁻¹

1 −3.37500 −1.28263

2 −7.59375·10⁻¹ 3.19830 3 −1.83064·10⁻¹ 2.14632·10⁻¹ 4 −3.74448·10⁻² −6.56577·10⁻¹ 5 −6.48084·10⁻³ −5.21069·10⁻¹ 6 −9.64977·10⁻⁴ −2.54040·10⁻¹ 7 −1.25701·10⁻⁴ −9.63884·10⁻² 8 −1.45326·10⁻⁵ −3.08382·10⁻² 9 −1.50916·10⁻⁶ −8.65874·10⁻³ 10 −1.42180·10⁻⁷ −2.18502·10⁻³ 11 −1.22545·10⁻⁸ −5.03409·10⁻⁴ 12 −9.73149·10⁻¹⁰ −1.07076·10⁻⁴ 13 −7.16365·10⁻¹¹ −2.12015·10⁻⁵ 14 −4.91419·10⁻¹² −3.93309·10⁻⁶ 15 −3.15572·10⁻¹³ −6.87073·10⁻⁷ 16 −1.90482·10⁻¹⁴ −1.13494·10⁻⁸ 17 −1.08458·10⁻¹⁵ −1.77885·10⁻⁸ 18 −5.84397·10⁻¹⁷ −2.65313·10⁻⁹ 19 −2.98840·10⁻¹⁸ −3.77504·10⁻¹⁰ 20 −1.45403·10⁻¹⁹ −5.13541·10⁻¹¹

in which the expansion coefficientsb(n)are given as b(n) =−1

n!

27 4

n n

∑

k=0

(−n)_k k!

1

2n+k−1. (32) b(n)are the expansion coefficients which are written as an analytic formula and have been numerically calculated as shown in Table1.

By means of a function f_{i j},

f_{i j}=exp[−βU_{i j}]−1, (33) the third virial coefficients are defined as

C=−N_A 3

Z Z

f₁₂f₁₃f₂₃dτ₁dτ₂, (34) where

dτ_i≡dx_idy_idz_i, (35)

(x_i,y_i,z_i) being the position of the moleculei.

The third virial coefficients may be transformed into, by the Kihara expansions [8,9],

C=−4π²N_A²

· Z _3/4

0 Z _b

a

Z _∞ 0

f12f13f23R⁵dR

dxd(y²), (36) wherea=1−p

1−y²,b=p

1−y²,R=r₁₂,(x²+ y²)R²=r²₁₃, and[(1−x²) +y²]R²=r²₂₃. The product of functions f can be written as the following sum:

f12f13f23=exp[−β(U₁₂+U₁₃+U₂₃)]

−1− {exp[−β(U₁₂+U₁₃)]−1}

− {exp[−β(U₁₂+U₂₃)]−1}

− {exp[−β(U₁₃+U₂₃)]−1}

+f₁₂+f₁₃+f₂₃.

(37)

The coefficients A(n,k) are functions of the vari- ablesξ andηin whichξ =R/r₁₂andη=R/r₂₃: A(n,k) = (1+ξ⁹+η⁹)^2/3

1+ξ⁶+η⁶ (1+ξ⁹+η⁹)^2/3

^n−k

−(1+ξ⁹)^2/3

1+ξ⁶ (1+ξ⁹)^2/3

^n−k

−(1+η⁹)^2/3

1+η⁶ (1+η⁹)^2/3

^n−k

−(ξ⁹+η⁹)^2/3

ξ⁶+η⁶ (ξ⁹+η⁹)^2/3

^n−k

+1+ξ⁶+η⁶,

(38)

(5)

and in the case thatn=1 andk=0, A(1,0) =log

p1+ξ⁹p 1+η⁹ p1+ξ⁹+η⁹ +ξ⁶log

p1+ξ⁹p ξ⁹+η⁹ ξ⁶p

1+ξ⁹+η⁹ +η⁶log

p1+η⁹p ξ⁹+η⁹ η⁶p

1+ξ⁹+η⁹ . (39)

The third virial coefficients for the Lennard–Jones (9,6) potential with a hard core are represented as

C=b²₀

∞ n=0

∑

c(n)uⁿ (40)

in which the coefficientsc(n)are given by the double integrals

c(n) =− 3 2n!

27 4

n

·

n

∑

k=0

(−n)_k k!

1 n+1/2k−1

· Z 3/4

0 Z b

a

A(n,k)dxd(y²),

(41)

and especially in the case forn=1, c(1) =−3

2 27

4

· Z 3/4

0 Z b

a

[A(1,0)−2A(0,0)]dxd(y²). (42)

-5.0 -4.5 -4.0 -3.5 -3.0

270 290 310 330 350 370

T/K

G/1000R

Fig. 1. Gibbs free energy for CO₂, plotted against temperature atPc=7.51 MPa andTc=308.7 K.

c(n) are the expansion coefficients which are calculated over again by double numerical integrations as shown in Table1.

4. Numerical Results

The parametersεandσfor the Lennard–Jones (9,6) potential with a hard core for 20 substances are determined from the experimental data of second virial coefficients [10] as shown in Table2. The critical con- Table 2. Parameters for the Lennard–Jones (9,6) potential with a hard core determined from experimental data of the second virial coefficients [10].

Substance (ε/k)/K σ/Å

Ne 27.715 2.93

Ar 92.732 3.63

Kr 130.326 3.90

Xe 178.521 4.28

N2 78.169 3.88

O2 94.688 3.62

F2 89.704 3.51

CO 85.083 4.00

NO 108.871 3.44

CO2 191.324 3.98

N2O 192.459 4.07

CF4 139.207 4.54

Si F4 161.378 4.65

SF6 202.729 5.07

CH4 118.582 3.99

C2H6 189.257 4.61

C3H8 229.331 5.15

C4H10 263.101 5.60

C2H4 174.596 4.43

C3H6 226.069 4.96

㻝㻜㻝㻝㻝㻞㻝㻟㻝㻠㻝㻡

270 290 310 330 350 370

T/K

S/R

Fig. 2. Entropy for CO₂, plotted against temperature atPc= 7.51 MPa andTc=308.7 K.

(6)

stantsTc,Pc, andVcare derived from the condition that the discriminant of the cubic (3) is zero at which there are triple roots [5]. Although the virial equation of state is approximated till the third virial coefficient, critical temperatures and pressures in Table3may be qualita- tively evaluated while critical volumes for all gases ex- Table 3. Critical constants for the critical temperatures (TC/K), critical pressures (P_C/MPa), and critical volumes (VC/cm³mol⁻¹). Boiling temperatures (TB/K) and enthalpy changes of vaporization (∆H/kJ mol⁻¹) at 0.1013 Mpa.

Present works (upper entries) and experimental results (lower entries) [11].

Substance T_C P_C V_C T_B ∆H

Ne 44.7 2.73 45.5 29.8 3.83

44.4 2.76 41.7 27.0 1.84

Ar 149.6 4.80 86.5 96.6 16.2

150.8 4.87 74.9 87.3 6.53

Kr 210.3 5.43 107.2 135.0 23.9

209.4 5.50 91.2 119.8 9.6

Xe 288.0 5.63 141.7 184.6 33.2

289.7 5.84 118. 165.0 13.0

N2 126.1 3.31 105.6 83.0 11.8

126.2 3.39 89.5 77.4 5.58

O2 152.8 4.94 95.8 98.5 16.7

154.6 5.05 73.4 90.2 6.8

F2 144.7 5.13 78.2 93.2 16.0

144.3 5.22 66.2 85.0 6.53

CO 137.3 3.29 115.7 90.4 12.8

132.9 3.50 93.1 81.7 6.0

NO 175.6 6.62 73.6 111.8 21.5

180.0 6.48 58. 121.4 13.8

CO2a 308.7 7.51 114.0 214.6 20.46

304.2 7.38 94.0 217.0 8.33

N2O 310.5 7.06 121.9 197.1 38.9

309.6 7.24 97.4 184.7 16.6

CF4 224.6 3.68 169.2 147.0 21.9

227.6 3.74 140.0 145.2 12.0

SiF4a 260.4 3.97 181.8 175.3 20.8

259.0 3.72 182.9 18.7

SF6a 327.1 3.85 235.6 224.7 23.2

318.7 3.76 198.0 222.5 17.1

CH4 191.3 4.62 114.8 123.8 20.3

190.6 4.60 99.0 111.7 8.2

C2H6 305.3 4.78 177.1 197.2 33.0

305.4 4.88 148. 184.5 14.7

C3H8 370.0 4.15 246.9 240.6 37.8

369.8 4.25 203. 231.1 18.8

C4H10 424.5 3.71 317.5 277.6 41.6

425.2 3.80 255. 272.7 20.4

C2H4 281.7 4.97 157.2 181.6 30.9

282.4 5.04 129. 169.4 13.5

C3H6 364.7 4.58 220.6 236.0 38.8

365.0 4.62 181. 225.4 18.4

aBoiling temperatures and the enthalpy changes for CO2, SiF4, and SF6are obtained at boiling pressures 0.518, 0.176, and 0.227 MPa, respectively.

cept neon are deviated from experimental results [11].

Numerical results obtained with these thermodynamic functions for carbon dioxide atP_c=7.51 MPa are displayed in Figures1–5. The Gibbs free energy consists of two slightly different tangents in both sides atT_c= 308.7 K while it is formed a continuous curve. Volume, entropy, and enthalpy are continuous atT_c=308.7 K, whereas these quantities show a sudden change in the neighbourhood of the critical temperature. The heat capacity at P_c =7.51 MPa diverges to infinity at T = 308.7 K as shown in Figure5. A singularity of the heat capacity, however, is found at the critical temperature.

This singularity suggests a phase transition. The gen- eralized diagrams of some thermodynamic quantities

0 50 100 150 200 250 300 350

270 290 310 330 350 370

T/K

V/R

Fig. 3. Volume for CO2, plotted against temperature atPc= 7.51 MPa andTc=308.7 K.

-300 -200 -100 0 100 200 300

270 290 310 330 350 370

T/K

H/2.5R

Fig. 4. Enthalpy for CO₂, plotted against temperature atPc= 7.51 MPa andTc=308.7 K.

(7)

0 500 1000 1500 2000 2500 3000

270 290 310 330 350 370

T/K

Cp/1000R

Fig. 5. Heat capacity at pressure constant for CO₂, plotted against temperature atPc=7.51 MPa andTc=308.7 K.

-100 -80 -60 -40 -20 0 20 40

170 190 210 230 250 270

K/K

S/R

Fig. 7. Entropy for CO₂, plotted against temperature atPc= 0.518 MPa andTB=214.6 K.

-9000 -7000 -5000 -3000 -1000 1000

170 190 210 230 250 270

T/K

H/2.5R

Fig. 9. Enthalpy for CO2, plotted against temperature atPc= 0.518 MPa andTB=214.6 K.

-3.8 -3.6 -3.4 -3.2 -3.0 -2.8

170 190 210 230 250 270

T/K

G/1000R

Fig. 6. Gibbs free energy for CO2, plotted against temperature atPc=0.518 MPa andTB=214.6 K.

-100 100 300 500 700 900

170 190 210 230 250 270

T/K

V/R

Fig. 8. Volume for CO2, plotted against temperature atPc= 0.518 MPa andTB=214.6 K.

-200 0 200 400 600 800 1000 1200 1400 1600 1800

170 190 210 230 250 270

T/K

Cp/1000R

Fig. 10. Heat capacity at pressure constant for CO2, plotted against temperature atPc=0.518 MPa andTB=214.6 K.

(8)

accompanying a second-order phase transition are typ- ically described in textbooks of physical chemistry [1].

Compared Figures1–5 with these diagrams [1], the behaviour in the neighbourhood of the critical temperature corresponds to a second-order phase transition.

In the equilibrium between the liquid and gaseous phases, the boiling temperatures TB at arbitrary pressure are easily found by the condition that ∆G= 0 in (25). Boiling temperatures are calculated at 0.1013 MPa and enthalpies∆Hof vaporization in (26) for all gases are obtained using these boiling temperatures. For CO2, SiF4, and SF₆, the boiling temperatures are evaluated at the boiling pressures, 0.518, 0.176, and 0.227 MPa, respectively. Boiling temperatures and the enthalpy changes of vaporization are shown in Table3.

Enthalpy changes∆Hof vaporization for all gases ex- cept SiF₄are not qualitative approaching experimental results [11]. Numerical results obtained with these thermodynamic functions for carbon dioxide are displayed in Figures6–10. The Gibbs free energy in Fig- ure6 indicates a polygonal line with a break at the boiling temperature. In the curves for entropy, volume, and enthalpy a jump is observed from the liquid to the gaseous phase at the boundary of the boiling point. As shown in Figure10, the heat capacity do not diverge to infinity but show a discontinuity at the boiling temperature. The heat capacity at 0.518 MPa shows a singularity at the boiling point. This singularity suggests a phase transition. The behaviour in the neighbourhood of the boiling pint corresponds to a first-order transition [1].

[1] P. W. Atokins, Physical Chemistry, 6th edn., Oxford Univ. Press, Oxford 1998, p. 153.

[2] A. Matsumoto, Z. Naturfosch.55a, 851 (2000).

[6] H. Takahasi, Proc. Phys.-Math. Soc. Japan 24, 60 (1942).

[7] J. E. Mayer and M. G. Mayer, Statistical Mechanics, John Wiley & Sons, Inc., New York 1940, pp. 263 and 277 – 297.

[8] J. O. Hirschfelder, C. F. Curtiss, and R. B. Bird, Molec- ular Theory of Gases and Liquids, John Wily &

Sons Inc., New York 1954, pp. 148 – 151, 162 – 166, 228 – 230, and 1119.

[9] T. Kihara, Rev. Mod. Phys.25, 831 (1953).

[10] J. H. Dymond and E. B. Smith, The Virial Coefficients of Pure Gases and Mixtures, Clarendon Press, Oxford 1980.

[11] R. C. Reid, J. M. Prausnitz, and T. K. Sherwood, The Properties of Gases and Liquids, 3rd edn., McGraw- Hill, New York 1977, Appendix.