Van Der Waals equation of state for gases and liquids Sunghoon Hong Seoul National University

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1 Van Der Waals equation of state for gases and liquids Sunghoon Hong Seoul National University

2 1910 Nobel Prize Van Der Waals equation Explain state of gases and liquids "The Nobel Prize in Physics 1910". Nobelprize.org. Nobel Media AB Web. 29 Oct 2017.

3 1910 Nobel Lecture

4 Table of Contents Why Van Der Waals(VDW) equation is needed? Incentive to VDW equation Derivation of VDW equation Properties of VDW equation Usage of VDW gases

5 Why Van Der Waals equation is needed? We cannot describe the free falling perfectly only using F=mg We cannot explain the state of gases just using PV=nRT

6 Assumptions of PV=nRT All gas particles have no masses and volumes No interaction between particles All collisions are perfectly elastic collision.

7 Why Van Der Waals equation is needed? PV=nRT is different from real gases In experiment, more exact model is needed

8 Incentive to VDW equation First incentive came to VDW when studying about heat Interested in Maxwell velocity distribution

9 Incentive to VDW equation Gases and liquids have no essential difference Continuity exists between two states

10 Incentive to VDW equation Take care about gases proper volumes and molecule interaction Derived new equation for gases

11 Incentive to VDW equation Begin to consider molecular interaction Developed the recent version of VDW equation

12 Derivation of VDW equation Start from P = RT v where v = V/n Let the volume of particles as b, then equations become P = RT v b

13 Derivation of VDW equation For interaction of particles, let Helmholtz free energy as F, then F = U TS df = du TdS = pdv TdS = F V T dv + F T V dt so P int = F V T 1 V 2

14 Derivation of VDW equation We can set proportional factor as a, so P becomes P = RT v b a v 2 If we organize this equation with RT, it becomes P + a v 2 v b = RT

15 Derivation using partition function Partition function of N molecules : Z N = ( 1 V )( N! 3 ) N λ th λ th means thermal wavelength Replace volume V by V nb Add a Boltzmann factor e β( an2 V )

16 Derivation using partition function Z N = 1 N! (V nb λ 3 ) N e βan2 /V th Helmholtz Free Energy : F = k B TlnZ N P = F V T = nrt V nb n2 a V 2 -> P + a v 2 v b = RT

17 Properties of VDW equation It predicts critical behavior of gases/liquids Isothermal line becomes S-line below the critical temperature For T < T c, there exist the area that ( V ) P T> 0 so compressibility κ T is negative

18 Critical Temperature Start from P = nrt n2 a V nb V 2 For critical point, we need ( P V ) T= 0, ( 2 P V 2) T= 0 ( P V ) T= ( 2 P V 2) T= nrt + 2n2 a = 0 V nb 2 V 3 2nRT (V nb) 3 6n2 a V 4 = 0

19 Critical Temperature RT = 2na(V nb)2 V 3 RT = 3na(V nb)3 V 4 from ( P V ) T= 0 from ( 2 P V 2) T= 0 2V = 3V 3nb -> V c = 3nb

20 Critical Temperature Put V c = 3nb into RT = 2na(V nb)2 V 3 T c = 8a 27Rb with P c = a 27b 2 P cv c nrt c = 3 8 independent of a and b, κ T = 1 V ( V P ) T

21 Critical Temperature in VDW s lecture First, he believed b is invariable parameter Derivative of b also needed parameter

22 Critical Temperature in VDW s lecture Critical temperature and pressure in VDW s lecture He find that it always same for same a and b value.

23 Properties of VDW equation Line phase coexistence End at the critical point. The p-t phase diagram for a VDW gas Stephen J. Bundell and Katherine M. Blundell, Concepts in Thermal Physics, second edition, Department of physics, University of Oxford, UK, 2010.

24 P-T Phase Diagram Fixed pressure, stable lowtemp state is liquid Stable high-temp state is gas No sharp phase boundary for T > T c The p-t phase diagram for a VDW gas Stephen J. Bundell and Katherine M. Blundell, Concepts in Thermal Physics, second edition, Department of physics, University of Oxford, UK, 2010.

25 Properties of VDW equation Gibbs function of VDW gas G = f T RTln V b a V + PV (for 1 mole gas) Phase coexistence occurs between points B 1, B 2 The Maxwell construction for VDW gas Stephen J. Bundell and Katherine M. Blundell, Concepts in Thermal Physics, second edition, Department of physics, University of Oxford, UK, 2010.

26 The Maxwell construction G p B2, T B = G p B1, T + 2 B1 Vdp G p B2, T = G p B1, T as Figure, so B 2 B1 Vdp = 0 The Maxwell construction for VDW gas Stephen J. Bundell and Katherine M. Blundell, Concepts in Thermal Physics, second edition, Department of physics, University of Oxford, UK, 2010.

27 The Maxwell construction Phase coexistence when the two shaded areas are same. Dashed line Maxwell construction : liquid and vapor are in equilibrium The Maxwell construction for VDW gas Stephen J. Bundell and Katherine M. Blundell, Concepts in Thermal Physics, second edition, Department of physics, University of Oxford, UK, 2010.

28 Usage of VDW gases VDW equation used to research nitrogen s behavior Liquid nitrogen can be realized due to VDW equation

29 Usage of VDW gases Research about real gases properties progressed increasingly Contribute to researching about intermolecular forces Significant tool for studying phase transition

30 Conclusion VDW equation fits to real gases and liquids more than PV=nRT VDW equation includes molecular interactions and real volume of gases VDW gases have properties such that critical temperature, Maxwell construction and characteristic phase diagram

31 References [1] "The Nobel Prize in Physics 1910". Nobelprize.org. Nobel Media AB Web. 29 Oct [2] van der Waals; J. D. (1873). Over de Continuiteit van den Gas- en Vloeistoftoestand. University of Leiden. [3] Stephen J. Bundell and Katherine M. Blundell, Concepts in Thermal Physics, second edition, Department of physics, University of Oxford, UK, [4] R. K. Pathria & Paul D. Beale, Statistical Mechanics, third edition, Academic press, 2011.

32 References [5] Maxwell, J.C. (1890). "LXIX. Van der Waals on the Continuity of the Gaseous and Liquid States". In Niven, W. D. The scientific papers of James Clerk. [6] van der Waals; J. D. (1873). Over de Continuiteit van den Gas- en Vloeistoftoestand. University of Leiden.

Chemistry. Lecture 10 Maxwell Relations. NC State University

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