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

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

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 A B C
Avogadro’s Law
Boyle’s Law
Charles’ Law

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 T is in Kelvins
Volume proportionate to temperature, i.e. V  T
T is in Kelvins
Note the extrapolated lines (to be explained later)

9. pV = nRT 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. But sadly assumptions fail…Nothing is ideal in this world…
It’s downright squeezy here

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. I can approximate the behaviour of fluids with an equation
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:

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:
Ethane:
Helium:

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 B,B’,C,C’,D & D’ are virial coefficients
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