Equations of State

1. Equations of State Compiled by: Gan Chin Heng / Shermon Ong 07S06G / 07S06H

2. Pressure Solid Liquid Critical point Triple point Gas Temp How are states represented? • Diagrammatically (Phase diagrams)

3. How are states represented? • Mathematically • Using equations of state • Relate state variables to describe property of matter • Examples of state variables • Pressure • Volume • Temperature

4. Equations of state • Mainly used to describe fluids • Liquids • Gases • Particular emphasis today on gases

5. ABCs of gas equations • Avogadro’s Law • Boyle’s Law • Charles’ Law • A • B • C

6. Avogadro’s Law • At constant temperature and pressure • Volume of gas proportionate to amount of gas • i.e. V  n • Independent of gas’ identity • Approximate molar volumes of gas • 24.0 dm3 at 298K • 22.4 dm3 at 273K

7. Boyle’s Law • At constant temperature and amounts • Gas’ volume inversely proportionate to pressure, i.e. V  1/p • The product of V & p, which is constant, increases with temperature

8. Charles’ Law • At constant pressure and amounts • Volume proportionate to temperature, i.e. V  T • T is in Kelvins • Note the extrapolated lines (to be explained later)

9. Combining all 3 laws… • V  (1/p)(T)(n) • V  nT/p • Rearranging, pV = (constant)nT • Thus we get the ideal gas equation: pV = nRT

10. Assumptions

11. It’s downright squeezy here But sadly assumptions fail…Nothing is ideal in this world…

12. Failures of ideal gas equation • Failure of Charles’ Law • At very low temperatures • Volume do not decrease to zero • Gas liquefies instead • Remember the extrapolated lines?

13. Failures of ideal gas equation • From pV = nRT, let Vm be molar volume • pVm = RT • pVm / RT = 1 • pVm / RT is also known as Z, the compressibility factor • Z should be 1 at all conditions for an ideal gas

14. Failures of ideal gas equation • Looking at Z plot of real gases… • Obvious deviation from the line Z=1 • Failure of ideal gas equation to account for these deviations

15. So how? • A Dutch physicist named Johannes Diderik van der Waals devised a way...

16. Johannes Diderik van der Waals • November 23, 1837 – March 8, 1923 • Dutch • 1910 Nobel Prize in Physics

17. So in 1873… I can approximate the behaviour of fluids with an equation Scientific community ORLY? YARLY!

18. Van der Waals Equation • Modified from ideal gas equation • Accounts for: • Non-zero volumes of gas particles (repulsive effect) • Attractive forces between gas particles (attractive effect)

19. Van der Waals Equation • Attractive effect • Pressure = Force per unit area of container exerted by gas molecules • Dependent on: • Frequency of collision • Force of each collision • Both factors affected by attractive forces • Each factor dependent on concentration (n/V)

20. Van der Waals Equation • Hence pressure changed proportional to (n/V)2 • Letting a be the constant relating p and (n/V)2… • Pressure term, p, in ideal gas equation becomes [p+a(n/V)2]

21. Van der Waals Equation • Repulsive effect • Gas molecules behave like small, impenetrable spheres • Actual volume available for gas smaller than volume of container, V • Reduction in volume proportional to amount of gas, n

22. Van der Waals Equation • Let another constant, b, relate amount of gas, n, to reduction in volume • Volume term in ideal gas equation, V, becomes (V-nb)

23. Van der Waals Equation • Combining both derivations… • We get the Van der Waals Equation

24. Van der Waals Equation -> So what’s the big deal? • Real world significances • Constants a and b depend on the gas identity • Relative values of a and b can give a rough comparison of properties of both gases

25. Van der Waals Equation -> So what’s the big deal? • Value of constant a • Gives a rough indication of magnitude of intermolecular attraction • Usually, the stronger the attractive forces, the higher is the value of a • Some values (L2 bar mol-2): • Water: 5.536 • HCl: 3.716 • Neon: 0.2135

26. Van der Waals Equation -> So what’s the big deal? • Value of constant b • Gives a rough indication of size of gas molecules • Usually, the bigger the gas molecules, the higher is the value of b • Some values (L mol-1): • Benzene: 0.1154 • Ethane: 0.0638 • Helium: 0.0237

27. Critical temperature and associated constants

28. Critical temperature? • Given a p-V plot of a real gas… • At higher temperatures T3 and T4, isotherm resembles that of an ideal gas

29. Critical temperature? • At T1 and V1, when gas volume decreased, pressure increases • From V2 to V3, no change in pressure even though volume decreases • Condensation taking place and pressure = vapor pressure at T1 • Pressure rises steeply after V3 because liquid compression is difficult

30. Critical temperature? • At higher temperature T2, plateau region becomes shorter • At a temperature Tc, this ‘plateau’ becomes a point • Tc is the critical temperature • Volume at that point, Vc = critical volume • Pressure at that point, Pc = critical pressure

31. Critical temperature • At T > Tc, gas can’t be compressed into liquid • At Tc, isotherm in a p-V graph will have a point of inflection • 1st and 2nd derivative of isotherm = 0 • We shall look at a gas obeying the Van der Waals equation

32. VDW equation and critical constants • Using VDW equation, we can derive the following

33. VDW equation and critical constants • At Tc, Vc and Pc, it’s a point of inflexion on p-Vm graph

34. VDW equation and critical constants

35. VDW equation and critical constants • Qualitative trends • As seen from formula, bigger molecules decrease critical temperature • Stronger IMF increase critical temperature • Usually outweighs size factor as bigger molecules have greater id-id interaction • Real values: • Water: 647K • Oxygen: 154.6K • Neon: 44.4K • Helium: 5.19K

36. Compressibility Factor

37. Compressibility Factor • Recall Z plot? • Z = pVm / RT; also called the compressibility factor • Z should be 1 at all conditions for an ideal gas

38. Compressibility Factor • For real gases, Z not equals to 1 • Z = Vm / Vm,id • Implications: • At high p, Vm > Vm,id, Z > 1 • Repulsive forces dominant

39. Compressibility Factor • At intermediate p, Z < 1 • Attractive forces dominant • More significant for gases with significant IMF

40. Boyle Temperature • Z also varies with temperature • At a particular temperature • Z = 1 over a wide range of pressures • That means gas behaves ideally • Obeys Boyle’s Law (recall V  1/p) • This temperature is called Boyle Temperature

41. Boyle Temperature • Mathematical implication • Initial gradient of Z-p plot = 0 at T • dZ/dp = 0 • For a gas obeying VDW equation • TB = a / Rb • Low Boyle Temperature favoured by weaker IMF and bigger gas molecules

42. Virial Equations

43. Virial Equations • Recall compressibility factor Z? • Z = pVm/RT • Z = 1 for ideal gases • What about real gases? • Obviously Z ≠ 1 • So how do virial equations address this problem?

44. Virial Equations • Form • pVm/RT = 1 + B/Vm + C/Vm2 + D/Vm3 + … • pVm/RT = 1 + B’p + C’p2 + D’p3 + … • B,B’,C,C’,D & D’ are virial coefficients • Temperature dependent • Can be derived theoretically or experimentally

45. Virial Equations • Most flexible form of state equation • Terms can be added when necessary • Accuracy can be increase by adding infinite terms • For same gas at same temperature • Coefficients B and B’ are proportionate but not equal to each other

46. Summary

47. Summary • States can be represented using diagrams or equations • Ideal Gas Equation combines Avagadro's, Boyle's and Charles' Laws • Assumptions of Ideal Gas Equation fail for real gases, causing deviations • Van der Waals Gas Equation accounts for attractive and repulsive effects ignored by Ideal Gas Equation

48. Summary • Constants a and b represent the properties of a real gas • A gas with higher a value usually has stronger IMF • A gas with higher b value is usually bigger • A gas cannot be condensed into liquid at temperatures higher than its critical temperature

49. Summary • Critical temperature is represented as a point of inflexion on a p-V graph • Compressibility factor measures the deviation of a real gas' behaviour from that of an ideal gas • Boyle Temperature is the temperature where Z=1 over a wide range of pressures • Boyle Temperature can be found from Z-p graph where dZ/dp=0

50. Summary • Virial equations are highly flexible equations of state where extra terms can be added • Virial equations' coefficients are temperature dependent and can be derived experimentally or theoretically