Phase Equilibrium Calculations by Equation of State v2

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1 Bulletin of Research Center for Comuting and Multimedia Studies, Hosei University, 7 (3) Published online (htt://hdl.handle.net/4/89) Phase Equilibrium Calculations by Equation of State v Yosuke KATAOKA and Yuri YAMADA Deartment of Chemical Science and Technology, Faculty of Bioscience and Alied Chemistry, Hosei University, 3-7- Kajino-cho, Koganei, Tokyo , Jaan yosuke.kataoka.7t@stu.hosei.ac.j SUMMARY Practical methods for hase equilibrium calculations by the equation of state for a erfect solid and liquid (v) are described using examles of worksheets. Using the Microsoft Excel worksheets, the ressure, Gibbs energy and other thermodynamic quantities of a molecular system are obtained as functions of temerature and volume. Some examles of VBA (Visual Basic for Alications) rograms are given to obtain volume and other thermodynamic roerties as a function of temerature under a given ressure. The thermodynamic consistency test is erformed as a function of temerature at a given ressure, where the difference between the heat caacity under a constant ressure and that under a constant volume should equal to an exression consisting of the thermal exansion coefficient, temerature, volume and isothermal comressibility. KEY WORDS: hase equilibrium, equation of state, erfect solid and liquid, thermodynamic consistency. INTRODUCTION Solid-liquid-gas transition is observed in ordinal substances []. The hase transition is a result of intermolecular interactions at a finite temerature []. This henomenon is calculated by the equation of state (EOS) for these hases [, ]. An equation of state for a erfect solid is roosed [3, 4] using a simlified model of a face-centered cubic lattice of a ure substance consisting of sherical molecules. The Lennard-Jones interaction is assumed as follows []: 6 σ σ ur () 4ε = r r () The constant ε is the deth of the otential well and σ is the searation at which u(σ) =. Constants ε and σ are used as the units of energy and length, resectively. The van der Waals EOS [] is assumed for a liquid hase and the three hase equilibrium is derived in revious studies [, 6]. The ressure EOS is written as follows: NkT N = a () V Nb V The first term on the right hand side (RHS) of the equation is the reulsion effect in a -D hard shere system []. The symbol N indicates the number of molecule in the system. The Boltzmann constant is written as k. The symbols, V and T are the ressure, volume and temerature, resectively. This reulsion term is modified by Carnahan-Starling for a 3-D hard shere system [7]: 3 kt + η+ η η a V (, T) =, 3 v ( η) v (3) 3 V π σ HS v, η, N 6 v where a is the van der Waals coefficient and η is the acking fraction. The internal energy of the van der Waals EOS is exressed as follows [, 8]: 3 an UVT (, ) = NkT. (4) V Coyright 3 Hosei University Received 9 March 3 Published online 3 June 3

2 . MAIN CHARACTER OF EQUATION OF STATE FOR A PERFECT SOLID Pressure and internal energies U of each hase, i.e., solid and liquid, are written as functions of temerature and volume [9]. The internal energy of the EOS for a erfect solid is the sum of the averaged kinetic energy and the otential energy [9]: 3 UVT (, ) = NkT+ E,ext ( V,K), () The ressure EOS for a erfect solid is given as [9]: (V, T ) = NkT! V E (V, K) $,ext # " V & + 6σ 3 NkT. (6) % V T The entroy S is exressed as a function of volume and temerature by the standard thermodynamic rocedure [, 3,, 9]. The most imortant characteristic of the resent EOS is that it is comosed of the erfect gas term, K term and an additional term roortional to the temerature T. The last term is not included in the internal energy in the resent aer. (V,T ) = (erfect gas) + (K) + (T > K) (7) 3. CRITICAL POINT The solid-gas critical oint is obtained from a lot of ressure vs. volume V as shown in Fig.. This calculation is shown with an examle of VBA rograms []. To run the rogram, macros is chosen in Microsoft Excel as described in Figs. and 3. Another rogram for calculating the critical oint in the liquid hase is attached []. A lot of vs. v in the liquid hase is obtained by running this rogram as shown in Fig. 4. T= ε/k, v /(ε/σ 3 ) d/dv, v. -. (d/dv)/(ε/σ 6 ) (V/N)/σ 3 Fig. Plots of and its derivative vs. V in the solid hase at T = ε/k. Fig. Develoer - Macros Coyright 3 Hosei University Bulletin of RCCMS, Hosei Univ., 7

3 3 /(ε/ s 3 ) Fig. 3 Macro name Run. ddv/(e/s^6). 8, vl/(e/s^3) =.48979ε/k (V/N)/(σ 3 ) Fig. 4 Plots of and its derivative vs. V in the liquid hase at T = ε/k. (d/dv)/(ε/σ 6 ). CALCULATION OF PHASE TRANSITION POINT The ressure and Gibbs energy er molecule G/N are tabulated as a function of volume V at a given temerature T as in Table. An examle of the worksheets for this urose is given in ref. []. The G/N vs. lot is shown in Fig. for the solid hase. Fig. 6 shows the G/N vs. lot for the liquid hase. An overlaed figure of both lots (Fig. 7) is obtained by the coy and aste function in Microsoft Excel. If the calculated oints are sarse as in Fig. 7, more oints are necessary to fix the hase transition oint recisely. The melting temerature, vaor ressure and sublimation vaor ressure are calculated and comared with exerimental and simulation results in a revious aer [9]. Table Examle of a table for the hase transition oint calculation. E E,vsolidext G,vsolidext,vLiquid/(e G,vLiquid/ T/(e/k) s^3/v V /s^3 ended ended /s^3) e,,.7.e-3.e E E+ 7.43E-4-3.E+.E-.E+ 7.9E E+ 6.8E-3 -.4E+.E-.E+ 4.E- -.8E- 6.9E-3-9.3E-.E-.E+ 4.9E- -.8E- -.E- -.67E+ 3.E- 3.33E E- -6.3E- -3.3E- -.E+ 4.E-.E+ -4.E- -.E+ -.93E- -3.9E+.E-.E+ -.E+ -.97E+ -8.E E+ 6.E-.67E+ -.97E+ -4.6E+ -.E+ -4.6E+ 7.E-.43E+ -.87E+ -6.E+ -8.7E E+ 8.E-.E+ -3.E+ -6.E+ -.E- -3.E+ 9.E-.E+ -.E+ -3.9E+.7E E-.E+.E+.4E+.77E+.73E+ 3.37E+.E+ 9.9E-.96E+.6E+.4E+.3E+ Coyright 3 Hosei University Bulletin of RCCMS, Hosei Univ., 7

4 4 T=. 7ε/k - /(ε/σ 3 ) Fig. G/N vs. in the solid hase at T =.7 ε/k. T=. 7ε/k - /(ε/σ 3 ) Fig. 6 G/N vs. in the liquid hase at T =.7 ε/k T=. 7ε/k /(ε/σ 3 ) Fig. 7 Overlaed figure of G/N vs. lots in the solid and liquid hases at T =.7 ε/k. 6. THERMODYNAMIC QUANTITIES AT A GIVEN PRESSURE The equation for a given ressure must be solved to discuss the thermodynamic roerties under a constant ressure. (V,T ) = (8) This is solved by Goal Seek in Microsoft Excel [3]. In ractice, a reasonable trial volume should be given for each hase. v σ 3 (gas), v.σ 3 (liquid), v.σ 3 (solid) The tyical volume is shown below: The obtained results for volume are given in Fig. 8 for =. ε/σ 3. The entroy er molecule S/N is lotted against temerature T in Fig. 9. G/N vs. T is lotted in Fig.. Some VBA rograms are attached for calculations of volume and other thermodynamic quantities at a given ressure [4, ]. (9) Coyright 3 Hosei University Bulletin of RCCMS, Hosei Univ., 7

5 (V/N)/σ 3 =. ε/σ Fig. 8 Volume er molecule vs. T at =. ε/σ 3. (S/N)/k =. ε/σ Fig. 9 S/N vs. T at =. ε/σ =. ε/σ Fig. G/N vs. T at =. ε/σ 3. Finally, an examle of thermodynamic consistency tests is shown. in thermodynamics: The following equation must be satisfied C C v = α TV κ T (9) The heat caacity under a constant ressure C, heat caacity under a constant volume C v, exansion coefficient α and isothermal comressibility κ T are the tyical notations used in thermodynamics. This equation is satisfied within numerical errors as shown in Figs. and. Fig. shows the absolute value of the difference between the LHS and the RHS of the equation normalized by the LHS. The increment of temerature ΔT is. ε/k in the numerical differentiation by temerature for the calculation of the exansion coefficient. " V % " $ ' ΔV % $ ' () # T & # ΔT & The increment of ressure Δ is. ε/σ 3 in the solid and liquid hases for the calculation of isothermal comressibility. " V % " $ ' ΔV % $ ' () # & # Δ & T T Coyright 3 Hosei University Bulletin of RCCMS, Hosei Univ., 7

6 6 In the case of the gas hase, the increment of ressure Δ is -6 ε/σ 3. The numerical error is large only for the region where the exansion coefficient, isothermal comressibility and heat caacity under a constant ressure have very large values as seen in Figs. and. (C- Cv)/Nk =. ε/σ 3 Lines:(C-Cv)/N Dot s: α TV/κN Fig. Thermodynamics quantity (C C v ) / N and α TV κ T N vs. T at =. ε/σ3. abs(di f)/(c- Cv) =. ε/σ Fig. Relative error (C C v ) / N α TV κ T N / " (C C ) / N $ # v % vs. T at =. ε/σ3. ACKNOWLEDGMENT The authors would like to thank the Research Center for Comuting and Multimedia Studies of Hosei University for the use of comuter resources. REFFERENCES [] P. W. Atkins, Physical Chemistry, Oxford Univ. Press, Oxford (998). [] D. A. McQuarrie, Statistical Mechanics, Harer Collins (976). [3] Y. Kataoka and Y. Yamada, J. Comut. Chem. Jn.,, 98-4 (). htts:// [4] Y. Kataoka and Y. Yamada, Thermodynamics on Perfect Solid by Equation of State, in Jaanese, Bulletin of Research Center for Comuting and Multimedia Studies, Hosei University, 6, 7- (). htt://hdl.handle.net/4/784 [] Y. Kataoka and Y. Yamada, J. Comut. Chem. Jn.,, 8-88 (). htts:// [6] Y. Kataoka and Y. Yamada, Phase Transition in Perfect Liquid and Perfect Solid, in Jaanese, Bulletin of Research Center for Comuting and Multimedia Studies, Hosei University, 6, (). htt://hdl.handle.net/4/79 [7] N. F. Carnahan and K. E. Starling, J. Chem. Phys., 3, 6-63 (97). [8] Y. Kataoka and Y. Yamada, J. Comut. Chem. Jn., 8, 97-4 (9). htts:// [9] Y. Kataoka and Y. Yamada, in ress, J. Comut. Chem. Jn. [] PvSolidEOS.xlsm, htt:// [] PvLiquidEOS.xlsm, htt:// [] EOSv_(T=.7).xlsx, htt:// [3] EOSv_(GoalSeek).xlsm, htt:// Coyright 3 Hosei University Bulletin of RCCMS, Hosei Univ., 7

7 7 [4] vsolid(v,t)=.xlsm, htt:// [] PVLiquid(V,T)=.xlsm, htt:// Coyright 3 Hosei University Bulletin of RCCMS, Hosei Univ., 7