Equations of State
Compiled by:
Gan Chin Heng / Shermon Ong
07S06G / 07S06H
How are states represented?

Diagrammatically (Phase diagrams)
Pressure
Solid
Liquid
Critical point
Triple point
Gas
Temp
How are states represented?

Mathematically
 Using
equations of state
 Relate state variables to describe property of
matter
 Examples of state variables
Pressure
 Volume
 Temperature

Equations of state

Mainly used to describe fluids
 Liquids
 Gases

Particular emphasis today on gases
ABCs of gas equations
Law
B
Boyle’s Law
C
Charles’ Law
A
Avogadro’s
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

dm3 at 298K
 22.4 dm3 at 273K
 24.0
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
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)
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
Assumptions
Ideal gas particles occupy
negligible volume
Ideal gas particles have
negligible intermolecular
interactions
But sadly assumptions
fail…Nothing is ideal in this world…
It’s downright
squeezy here
Real gas particles DO occupy
finite volume
Real gas particles have
considerable intermolecular
interactions
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?
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

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

So how?

A Dutch physicist named Johannes Diderik
van der Waals devised a way...
Johannes Diderik van der Waals
November 23, 1837
– March 8, 1923
 Dutch
 1910 Nobel Prize in
Physics

So in 1873…
I can approximate
the behaviour of
fluids with an
equation
Scientific
community
ORLY?
YARLY!
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)
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)
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]
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
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)
Van der Waals Equation
Combining both derivations…
 We get the Van der Waals Equation

2

n 
 p + a    [V-nb] = nRT
 V  

OR

a 
 p + 2  [Vm -b] = RT
Vm 

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
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
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
Critical temperature and
associated constants
Critical temperature?
Given a p-V plot of a
real gas…
 At higher temperatures
T3 and T4, isotherm
resembles that of an
ideal gas

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
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
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
VDW equation and critical
constants

Using VDW equation,
we can derive the
following

a 
 p + 2  [Vm -b] = RT
Vm 

RT
a
p=
- 2
Vm -b Vm
VDW equation and critical
constants

At Tc, Vc and Pc, it’s a
point of inflexion on pVm graph
 dp 

 0
 dVm T
d p
0

2 
 dVm T
2
VDW equation and critical
constants
 dp 
RT
2a
 3

 
2
(Vm  b) Vm
 dVm T
 d2 p 
2 RT
6a

 4

2 
3
 dVm T (Vm  b) Vm
Rearranging...
a
8a
Vm,c = 3b; p c =
; Tc =
2
27b
27Rb
p c Vm,c
3
Zc =
=
RTc
8
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
Compressibility Factor
Compressibility Factor
Recall Z plot?
 Z = pVm / RT; also
called the
compressibility factor
 Z should be 1 at all
conditions for an ideal
gas

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
Compressibility Factor
 At
intermediate p, Z <
1
 Attractive forces
dominant
 More significant for
gases with significant
IMF
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

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

Virial Equations
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?
Virial Equations

Form
= 1 + B/Vm + C/Vm2 + D/Vm3 + …
 pVm/RT = 1 + B’p + C’p2 + D’p3 + …
 pVm/RT

B,B’,C,C’,D & D’ are virial coefficients
 Temperature
dependent
 Can be derived theoretically or experimentally
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
Summary
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
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
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
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