1. Perturbation Theory

2. Perturbation theory for the square-well potential Hard spheres is one of the few cases in which we can find the radial distribution function with good accuracy. Can we use the results for hard spheres as starting point to develop models for other fluids? 2

3. Perturbation theory for the square-well potential Hard spheres is one of the few cases in which we can find the radial distribution function with good accuracy. Can we use the results for hard spheres as starting point to develop models for other fluids? This is achieved using perturbation theory, which is based on Taylor series expansions. 3

4. Perturbation theory for the square-well potential To discuss how it is done, consider, for example, the hard sphere and square-well potentials: Hard spheres: Square-well 4

5. Perturbation theory for the square-well potential Now, subtract these two potentials to define a perturbation: 5

6. Perturbation theory for the square-well potential We then write a potential as the sum of the reference system (hard spheres, here) and the perturbation: Parameter that allows the gradual transition from the reference potential to the potential of interest 6

7. Perturbation theory for the square-well potential Example: second virial coefficient of pure square-well fluids via perturbation theory. 7

8. Perturbation theory for the square-well potential To obtain the next terms in the series, let us recall how to compute the second virial coefficient: 8

9. Perturbation theory for the square-well potential The first derivative is: This derivative needs to be evaluated at 9

10. Perturbation theory for the square-well potential To obtain these relatively simple expressions, we used that: 10

11. Perturbation theory for the square-well potential The second derivative is: This derivative also needs to be evaluated at 11

12. Perturbation theory for the square-well potential To obtain these relatively simple expressions, we used that: 12

13. Perturbation theory for the square-well potential Up to second-order via perturbation theory, the expression for the second virial coefficient of a square-well pure fluid is: 13

14. Perturbation theory for the square-well potential Then: The rigorous result obtained in Chapter 8 was: 14

15. Perturbation theory for dense square-well fluids The starting point in the canonical ensemble is the equation for the Helmholtz energy: 15

16. Perturbation theory for dense square-well fluids Using a Taylor series: 16

17. Perturbation theory for dense square-well fluids Noting that the translational and internal parts of the partition function do not depend on the intermolecular potentials (except for long polymers where entanglement may be important): 17

18. Perturbation theory for dense square-well fluids Let us now assume pairwise additivity: 18

19. Perturbation theory for dense square-well fluids The first derivative becomes: 19

20. Perturbation theory for dense square-well fluids The first derivative becomes: 20

21. Perturbation theory for dense square-well fluids However: 21

22. Perturbation theory for dense square-well fluids Continuing: 22

23. Perturbation theory for dense square-well fluids Let us recall the radial distribution function is: For the pairwise additive hard sphere potential: 23

24. Perturbation theory for dense square-well fluids We obtain: 24

25. Perturbation theory for dense square-well fluids The Helmholtz energy truncated in the first order is: Then: 25

26. Perturbation theory for dense square-well fluids For the square-well fluid: From this relationship, it is possible to derive expressions for other thermodynamic properties, e.g.: 26

27. Perturbation theory for dense square-well fluids For the square-well fluid: 27

28. Perturbation theory for dense square-well fluids These results can be generalized to other fluids: The discussions of this and previous slides about first order perturbation constitute the first-order Barker- Henderson perturbation theory. 28

29. Second-order perturbation theory To improve accuracy, one more term can be considered in the Taylor expansion. The algebra is long (pages 266-268 in the textbook). The final expression for the Helmholtz energy is: 29

30. Second-order perturbation theory Using the second order term is not easy because it requires the three-body and four-body correlation functions: 30

31. Engineering applications of perturbation theory In one of the previous slides, we derived: Assuming the radial distribution function is density independent: 31

32. Engineering applications of perturbation theory Using the van der Waals excluded volume approximation and definitions for a and b: We obtain the van der Waals equation of state: 32

33. Engineering applications of perturbation theory Following a similar procedure but adopting the Carnahan Starling expression for the compressibility factor of hard spheres: We obtain: which is no longer a cubic equation of state. 33