Atkins’ Physical Chemistry Eighth Edition Chapter 1 The Properties of Gases Copyright © 2006 by...

Atkins’ Physical ChemistryEighth Edition

Chapter 1The Properties of Gases

Copyright © 2006 by Peter Atkins and Julio de Paula

Peter Atkins • Julio de Paula

Consider a case in which two gases, A and B, are in a container of volume V at a total pressure PT

PA = nARTV

PB = nBRTV

nA is the number of moles of A

nB is the number of moles of B

PT = PA + PB XA = nA

nA + nB

XB = nB

nA + nB

PA = XA PT PB = XB PT

Pi = Xi PT mole fraction (Xi) = ni

nT

Dalton’s Law of Partial Pressures

An “Ideal Gas”An “Ideal Gas”

Assumptions:

• Gas molecules do not exert any force (attractive or repulsive) on each other

• i.e., collisions are perfectly elastic

• Volume of molecules themselves is negligible compared to volume of container

• i.e., the molecules are considered to be points

• An ideal gas “obeys” PV = nRT

• i.e., calculated value ≈ experimental value

Real Gases

Assumptions made in the kinetic-molecular model:

negligible volume of gas molecules themselves

no attractive forces between gas molecules

These breakdown at high pressure and/or low temperature.

Fig 1.13 Variation of the potential energy of two moleculeson their separation.

Attractions between electrons and nuclei

Repulsions between electrons

Repulsions between nuclei

Fig 1.14 Variation of the compression factor, Z, with pressure for several gases at 0 oC

Compression factor:

perfect

real

V

VZ

om

m

• For a perfect gas Z = 1under all conditions

• At high P: Z >1 (large Vm)

• At lower P: Z < 1 for most gases(attractive forces predominate)

Fig 1.15 Experimental isothermsof CO2 at several temperatures

• At high Vm and high P real isotherms ≈ perfect isotherms

• ∴ perfect gas law can be expressed as a virial equation of state:

• i.e., PV = nRT treated as a powerseries expansion in V

...)V

C

V

B1(RTPV

2mm

m

...)V

C

V

B1(RTPV

2mm

m

Fig 1.15 Experimental isothermsof CO2 at several temperatures

• Tc is the critical temperature ≡ at T ≥ Tc gas will not form two phases when compressed

• The critical point is defined with critical constants ≡ Tc, Pc, Vc

• Above the critical point a supercritical fluid exists

Supercritical COSupercritical CO22

The low critical temperature and critical pressure for CO2 make supercritical CO2 a good solvent for extracting nonpolar substances (like caffeine)

Diagram of a supercritical fluid extraction process

The van der Waals EquationThe van der Waals Equation

) (V − nb) = nRTn2a(P + V2

Eqn 1.21a

Fig 1.8 A region of a P-V-TFig 1.8 A region of a P-V-T surface of a perfect gassurface of a perfect gas

Fig 1.17 A region of a P-V-TFig 1.17 A region of a P-V-T surface of a vdW gassurface of a vdW gas

1T

T

c

When van der Waals equation fails:

Principle of Corresponding States

• To compare properties of systems choose a fundamentalproperty of the same kind and set up a relative scale

• Critical constants are characteristic of each gas

• Introduce dimensionless reduced variables:

•

• e.g., gases at the same Vr and Tr will exert the same Pr

• Called the Principle of Corresponding States (approximation)

cr

c

mr

cr T

TT

V

V V

P

PP

Fig 1.19 Compression factors of four gases plotted using reduced variables