The properties of gases

1
The properties of gases

• ECHE 710
• 9/3/03

2
Ideal gas equation of state
3
Extensive and intensive properties

• Volume
• Mass
• Energy
• Enthalpy
• Entropy

• Temperature
• Pressure
• Chemical potential
• Molar volume
• Molar entropy

4
Properties of ideal gases
T -gt 0 as v-gt 0 T lim (P-gt0) Pv/R
Pv 22.414 L.atm as P-gt0 H2 (g) crosses N2 (g)
diamonds CO2 (g) circles T 273.15 K
5
The compressibility factor
300 K
methane
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Compressibility factor, Z

• At low T ? attractive forces are dominant, vreal
  lt videal ? Pv/RT lt1 ? Z lt 1
• At high temperatures, molecules have high kinetic
  energies, repulsive forces are dominant, ? vreal
  gt videal ? Pv/RT gt1
• ? Z gt 1

7
The van der Waals EOS

• Is an extension of the ideal gas EOS to account
  for intermolecular interactions.
• What happens when v-gt8?
• a gt attractive forces
• b gt molecular size

8
Z from van der Waals
9
The Redlich-Kwong EOS
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The Peng-Robinson EOS
a is a function of temperature
Ethane, 400K Experimental (solid) V der Waals
(dot-dashed) RK (long-dashed) PR (short-dashed)
11
A cubic EOS can describe the Liquid and Vapor
phases
Experimental P-V isotherms CO2, Tc 30.99 oC
12
Calculated PV isotherms of CO2
RK
V. Der Waals
13
Subcritical isotherm

• GADL experimental curve areas of the loop below
  and above DL are equal.
• AB metastable (SH vapor) CD (SC liquid)
• BC unstable (dp/dv) gt0
• Cubic eqn.

14
Conditions at the critical point

• Can be solved to obtain a and b in terms of Tc,
  Pc, and vc
• vc3b
• Pc a/(27b2)
• Tc 8a/(27bR)

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The critical compressibility factor, Zc

• From van der Waals
• Pcvc/(RTc) Zc 0.375
• From Redlich-Kwong
• Pcvc/(RTc) Zc 0.33333
• From Peng-Robinson
• Pcvc/(RTc) Zc 0.30740
• gt Zc constant gt should be the same for all
  substances (see Table 2.5)

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Law of corresponding states
17
Law of corresponding states
It is a universal equation, for all substances.
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Virial equation of state
What are the limits as v -gt 8 and as P -gt 0 ? See
Table 2.6
19
Second virial coefficients

• Experimentally from the slope of Z vs. P
• T-dependence
• Boyle temperature gt B2v(T) 0, repulsive and
  attractive interactions cancel each other.

20
Pair interactions between molecules

• For non polar molecules, U(r) is a good
  approximation.

u(r) 0 far all r, what is B2v?
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General form of u(r), nonpolar molecules

• u(r) -gt -c6/r6 ? attractive forces
• u(r) -gt cn/rn ? repulsive forces
• (n 12)
• The Lennard-Jones potential

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The LJ 12-6 potential

• B2v(T) can be calculated numerically from the LJ
  potential

23
Law of corresponding states

• Defining T kBT/e and x r/s

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Reduced 2nd virial coefficient
When P is sufficiently small to neglect the 3rd
virial coefficient
25
Short-range attractive interactions

• Permanent dipoles
• Induced dipoles
• Instantaneous dipole-dipole interaction (London
  attraction)

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Dipole-dipole interactions
Permanent dipole-permanent dipole
Induced dipole (See table 2.8)
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Dipole-induced dipole interactions
This interaction is always attractive
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London dispersion forces

• Interactions of electron clouds that generate
  instantaneous dipoles

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Total (short-range) attractive forces

• C6 udipole-dipole uinduced udispersion

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Simpler intermolecular potentials

• Hard sphere (a) and Square-well (b) potentials

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2nd virial coeff. for the HS potential
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2nd virial coeff. for the square-well potential

• Nitrogen, calculated, exp. points (circles)

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2nd virial coeff. for cubic EOS
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HW 2 (due Monday, Sept. 8)

• PROBLEMS
• 2.8 2.13 2.15 2.20
• 2.24 2.25 2.29 2.33
• 2.46 2.47 2.58 2.59