Real Gases

The ideal gas equation of state is not sufficient to describe the P,V, and T behaviour of most real gases.

Most real gases depart from ideal behaviour at deviation from • low temperature

• high pressure

Low Temperatures

The variation of the potential energy of two molecules on their separation.

High positive potential energy (little separation)• Repulsive interactions

Intermediate separations• attractive interactions dominate

Large separations (on the right) • the potential energy is zero and there is no

interaction between the molecules.

High Pressures

Real gas molecules do attract one another

(Pid = Pobs + constant) Real gas molecules are not point

masses

(Vid = Vobs - const.)

Vid = Vobs - nb

• b is a constant for different gases

Pid = Pobs + a (n / V)2

• a is also different for different gases

Ideal gas Law Pid Vid = nRT

2

2

mmobs

obs

V

abV

RTP

nRTnbVVn

aP

Critical temperature (Tc) - the temperature above which a gas cannot be liquefied

Critical pressure (Pc) – the minimum pressure that needs to be applied at Tc to bring about liquefaction

The compression factor

RTPV

Z m

For a perfect gas, the slope is zero

Boyle temperature• the slope is zero

and the gas behaves perfectly over a wider range of conditions than at other temperatures.

At the critical point

375.0Z ;27

8a

27

aP ;3

c

2c

c

ccc

c

RTVP

RbT

bbV

Boyle temperature - for a van der Waal's gas, the Boyle temperature (TB) is written

Rba

TB

The reduced state variables are defined

cr

cr

cr

TT

T

PP

P ;VV

V

Re-write the Van der Waal’s in terms of reduced variables

38

313

3278

27

2

2

rr

r

r

rm

rr

TV

VP

bVVRb

aTT

baP

P

All substances obey the same equation of state in terms of the reduced variables.

Degree of generality.

The chemical potential of a real gas is written in terms of its fugacity

Jo

J fRT ln

The activity coefficient (J) relates the activity to the concentration terms of interest.

In gaseous systems, we relate the fugacity (or activity) to the ideal pressure of the gas via

JJJ fP

The fugacity (f) represents the chemical potential of a real gas.

Define the fugacity coefficient

= f / P For a real gas

PflnRT)bar 1 ,T()T(

Comparing the chemical potential of the real gas to the chemical potential of an ideal gas at the same pressure

PPlnRT

PflnRTidreal

The fugacity coefficients are obtained from the compression factors (Z) as shown below

P

0

dPP

1ZlnRT