Chapter 3
Real gases
Chemical Thermodynamics : Georg Duesberg
Chapter11111
Chapter 1 :1Slide
: Slide
1
1
Real Gases
•  Perfect gas: only contribution to energy is KE of molecules
•  Real gases: Molecules interact if they are close enough, have a
potential energy contribution.
•  At large separations, attractions predominate (condensation!)
•  At contact molecules repel each other (condensed states have volume!)
Ideal (Isotherms)
F
A
p
2
Thermo
meter
Pressure
gauge
Chemical Thermodynamics : Georg Duesberg
Real (CO2)
Deviations from ideality can be described by the COMPRESSION
FACTOR, Z (sometimes called the compressibility).
Z = pV/(nRT) = pVm/(RT) For ideal gases Z = 1
Pressure region
I (very Low)
Molecules have large separations -> no interactions ->
Ideal Gas Behavior: Z =1
II (moderate)
Molecules are close -> attractive forces apply -> The gas
occupies less volumes as expected from Boyles law: Z<1
III (high)
Molecules compressed highly -> repulsive forces dominate
-> hardly further decrease in volume Z>1
Chemical Thermodynamics : Georg Duesberg
Chapter33333 1 : Slide
3
Microscopic interpretation: Leonard Jones Potential
When p is very high, r is small so shortrange repulsions are important. The gas
is more difficult to compress than an
ideal gas, so Z > 1.
When p is very low, r is large and
intermolecular forces are negligible, so
the gas acts close to ideally and Z ∼ 1.
At intermediate pressures attractive
forces are important and often Z < 1.
Chapter 1 : Slide 4
Chemical Thermodynamics : Georg Duesberg
Real Gases: What happens if we press down the piston
( at 20 °C, gas: carbon dioxide)
A – B perfect gas behavior (isotherm)
B – C slight deviation from perfect
behavior – less pressure than expected
C – D – E no change in pressure reading
over further compression – but
increasing amount of liquid observed
E – F : steep in crease in P, only liquid
visible (At contact molecules repel each
other condensed states have volume!)
The line C – D – E is the vapour pressure
of a liquid at this tempeature
5
Chemical Thermodynamics : Georg Duesberg
Deviations from ideality can be described by the COMPRESSION FACTOR,
Z (sometimes called the compressibility).
Z = pV/(nRT) = pVm/(RT) For ideal gases Z = 1
•  Attractive forces vary with nature of gas
•  At High Pressures repelling forces dominate
Z=
Chemical Thermodynamics : Georg Duesberg
Deviations from ideality can be described by the COMPRESSION
FACTOR, Z (sometimes called the compressibility).
Z = pV/(nRT) = pVm/(RT) For ideal gases Z = 1
•  At Low Temperatures the attractive regime is pronounced
•  higher Temperature ->faster motion -> less interaction
Z=
7
Chemical Thermodynamics : Georg Duesberg
Boyle Temperatur
The temperature at which this occurs
is the Boyle temperature, TB, and then
the gas behaves ideally over a wider
range of p than at other temperatures.
Each gas has a characteristic TB, e.g. 23
K for He, 347 K for air, 715 K for
CO2.
The compression factor approaches 1 at low pressures, but does so with different
slopes. For a perfect gas, the slope is zero, but real gases may have either positive
or negative slopes, and the slope may vary with temperature.
At the Boyle temperature, the slope is zero and the gas behaves perfectly over a
wider range of conditions than at other temperatures.
Chapter 1 : Slide 8
Chemical Thermodynamics : Georg Duesberg
Virial Equation of State
Most fundamental and theoretically sound Polynomial expansion Viris (lat.):
force (Kammerling Onnes 1901)
Virial coefficients:
p Vm = RT (1 + B’p + C’p2 + ...)
i.e.
p Vm = RT (1 + B/Vm + C/Vm2 + ...)
This is the virial equation of state and B and
C are the second and third virial coefficients.
The first is 1.
B and C are themselves functions of
temperature, B(T) and C(T).
Usually B/Vm >> C/Vm2
B = 0 at Boyle temperature
Also allow derivation of exact correspondence between virial coefficients
and intermolecular interactions
Real gas – Van der Waals equation.
ideal gas : PV = nRT
(P + x )(V − y ) = nRT
2
⎛
⎞
⎛
⎞
⎜ P + a⎛⎜ n ⎞⎟ ⎟(V − nb ) = nRT or P = RT − ⎜ a ⎟
⎜ V ⎟
⎜
⎟
V
V
−
b
⎝
⎠
m
⎝ m ⎠
⎝
⎠
2
Johannes Diderik van der Waals
got the Noble price in physics in 1910
Chemical Thermodynamics : Georg Duesberg
Chapter1010101010 1 :
Slide 10
Real gas – Van der Waals equation: b
1.  The molecules occupy a significant fraction of the volume.
-> Collisions are more frequent.
-> There is less volume available for molecular motion.
Real gas molecules are not point masses
(Vid = Vobs - const.) or Vid = Vobs - nb
–  b is a constant for different gases
Very roughly, b ∼ 4/3 πr3 where r is the molecular radius.
Other explanation:
What happens if we reduce T to zero. Is volume of the gas, V, going to become
zero? We can set P ≠ 0. By the ideal gas law we would have V = 0, which cannot
be true. We can correct for it by a term equal to the total volume of the gas
molecules, when totally compressed (condensed) nb. Now at T = 0 and P ≠ 0
we have V = nb.
P(V − nb) = nRT
Chapter 1 : Slide 11
Chemical Thermodynamics : Georg Duesberg
Real gas – Van der Waals equation: a
2) There are attractive forces between real molecules, which reduce
the pressure: p ∝ wall collision frequency
and
p ∝ change in momentum at each collision.
Both factors are proportional to concentration, n/V, and p is reduced
by an amount a(n/V)2, where a depends on the type of gas.
[Note: a/V2 is called the internal pressure of the gas].
Real gas molecules do attract one another (Pid = Pobs + constant)
Pid = Pobs + a (n / V)2 a is also different for different gases
2
n
a
2
V
Chemical Thermodynamics : Georg Duesberg
a describes attractive force between pairs
of molecules. Goes as square of the
concentration (n/V)2 .
Van der Waals equation of state
2
⎛
⎞
⎛
⎞
n
⎛
⎞
⎜ P + a⎜ ⎟ ⎟(V − nb ) = nRT or P = RT − ⎜ a ⎟
⎜ V ⎟
⎜
⎟
V
V
−
b
⎝
⎠
m
⎝ m ⎠
⎝
⎠
6
2
2
3
2
1
Substance
a/(atm dm mol− )
b/(10− dm mol− )
Air
1.4
0.039
Ammonia, NH3
4.169
3.71
Argon, Ar
1.338
3.20
Carbon dioxide, CO2
3.610
4.29
Ethane, C2H6
5.507
6.51
Ethene, C2H4
4.552
5.82
Helium, He
0.0341
2.38
Hydrogen, H2
0.2420
2.65
Nitrogen, N2
1.352
3.87
Oxygen, O2
1.364
3.19
Xenon, Xe
4.137
5.16
If 1 mole of nitrogen is
confined to 2l and is at
P=10atm what is Tideal
and TVdW?
Tip: R =0.082dm3atmK-1mol-1
•  Parameters depend on the gas, but are taken to be independent of T.
•  a is large when attractions are large, b scales in proportion to molecular size
13
(note
units)
Chemical Thermodynamics : Georg Duesberg
CONDENSATION or
LIQUEFACTION
This demonstrates that there
are attractive forces between
gas molecules, if they are
pushed close enough together.
E.G. CO2 liquefies under
pressure at room temperature.
Above 31 0C no amount of
pressure will liquefy CO2: this
is the CRITICAL
TEMPERATURE, Tc.
Chapter 1 : Slide 14
Chemical Thermodynamics : Georg Duesberg
Carbon dioxide: a typical pV diagram for a real gas:
Experimental isotherms of carbon
dioxide at several temperatures. The
`critical isotherm', the isotherm at the
critical temperature, is at 31.04 °C. The
critical point is marked with a star.
Tc, pc and Vm,c are the critical constants
for the gas.
The isotherm at Tc has a horizontal
inflection at the critical point
dp/dV = 0
and d2p/dV2 = 0.
Chemical Thermodynamics : Georg Duesberg
Chapter 1 :
Critical Point
At the critical temperature the densities of the liquid and gas become equal - the
boundary disappears. The material will fill the container so it is like a gas, but may
be much denser than a typical gas, and is called a 'supercritical fluid'. The
isotherm at Tc has a horizontal inflection at the critical point
dp/dV = 0
and d2p/dV2 = 0.
Consider 1 mol of gas, with molar volume V, at
the critical point (Tc, pc, Vc)
0 = dp/dV
= -RTc(Vc-b)-2 + 2aVc-3
0 = d2p/dV2
= 2RTc(Vc-b)-3 - 6aVc-4
The solution is
Vc = 3b,
pc = a/(27b2),
Tc = 8a/(27Rb).
Chemical Thermodynamics : Georg Duesberg
16
Critical Point drying
Applications: TEM sample prep, porous materials, MEMS
Chemical Thermodynamics : Georg Duesberg
Chapter1717171717 1 :
Slide 17
CNT
Chemical Thermodynamics : Georg Duesberg
Metal contacts on CNT
Chemical Thermodynamics : Georg Duesberg
Etch
Chemical Thermodynamics : Georg Duesberg
Freely suspended CNT
Etch
Chemical Thermodynamics : Georg Duesberg
TEM electron beam
Freely suspended CNT
Etch
Chemical Thermodynamics : Georg Duesberg
Protective resist
Etch
Chemical Thermodynamics : Georg Duesberg
Protective resist
Etch
Chemical Thermodynamics : Georg Duesberg
Chemical Thermodynamics : Georg Duesberg
Chemical Thermodynamics : Georg Duesberg
Chemical Thermodynamics : Georg Duesberg
Chemical Thermodynamics : Georg Duesberg
Suspended on contacted individual CNTs –
Platform for combined investigations
Structure and Electronic
Properties can be
related:
Individual tubes or
bundels?
What kinds of CNT
(MWCNT, SWCNT,
(n,m), peapods..)
Chemical Thermodynamics : Georg Duesberg
Combined TEM and Raman investigations on individual
SWCNTs
Chemical Thermodynamics : Georg Duesberg
Maxwell Construction
Below Tc calculated vdW isotherms have oscillations that are unphysical. In the
Maxwell construction these are replaced with horizontal lines, with equal areas
above and below, to generate the isotherms. (The line is the vapour pressure of a
liquid at this temperature, or liquid-vapor equilibrium)
31
Chemical Thermodynamics : Georg Duesberg
,
Features of vdW equation
•  Reduces to perfect gas equation at high T
and V
•  Liquids and gases coexist when
attractions ≈ repulsions
•  Critical constants are related to
coefficients.
•  Flat inflexion of curve when T=Tc.
•  Can derive (by setting 1st and 2nd
derivatives of equation to zero)
expression for critical constants
•  Vc = 3b,
•  pc = a/27b2,
•  Tc =8a/27Rb
•  Can derive expression for the Boyle
Temperature
•  TB = a/Rb
32
Chemical Thermodynamics : Georg Duesberg
Features of vdW equation
•  Can derive (by setting 1st and 2nd derivatives of equation to
zero) expression for critical constants
⎛ ∂ 2 p ⎞
⎛ ∂p ⎞
i.e., ⎜ ⎟
= 0 and ⎜⎜ 2 ⎟⎟
= 0 we have,
⎝ ∂v ⎠T =Tc
⎝ ∂v ⎠T =Tc
− RT
2a
⎛ ∂p ⎞
∴ ⎜ ⎟ =
+ 3
2
v
⎝ ∂v ⎠T (v − b)
P=
RT
a
− 2
v−b v
⎛ ∂ 2 p ⎞
2 RT
6a
and ⎜⎜ 2 ⎟⎟ =
− 4
3
⎝ ∂v ⎠T (v − b ) v
At critical points the above equation reduces to
− RT
2
(v − b )
33
Chemical Thermodynamics : Georg Duesberg
2a
+ 3 =0
v
and
2 RT
(v − b )3
6a
− 4 =0
v
Features of vdW equation
By dividing those equations and simplifying we get
vc
b=
3
We also can say
RTc
a
Pc =
− 2
v−b v
Substituting for b and solving for ‘a’ from 2nd derivative we get,
a = 9RTcvc
Substituting these expressions for a and b in equation (P(c) and
solving for vc, we get
3RTc
vc =
8 pc
RTc
∴b =
8 pc
2
2
c
⎛ 27 ⎞ R T
and a = ⎜ ⎟
⎝ 64 ⎠ pc
Note: Usually constants a and b for different gases are given.
Chemical Thermodynamics : Georg Duesberg
Chapter3434343434 1 :
Slide 34
Critical constants
pc
atm
Vc
Zc
cm3/mol
Tc
K
TB
K
Ar
48.0
75.3
150.7
0.292
411.5
CO2
72.9
94.0
304.2
0.274
714.8
He
2.26
57.8
5.2
0.305
22.64
O2
50.14
78.0
154.8
0.308
405.9
The general Van der Waals
pVT surface
Chemical Thermodynamics : Georg Duesberg
The principle of corresponding states
Gases behave differently at a given pressure and temperature, but they
behave very much the same at temperatures and pressures normalized with
respect to their critical temperatures and pressures. The ratios of pressure,
temperature and specific volume of a real gas to the corresponding critical
values are called the reduced properties.
Define reduced variables
pr = p/pc
Tr = T/Tc
Vr = Vm/Vm,c
Van der Waals hoped that different gases confined to the same Vr at
the same Tr would have the same pr.
Chapter 1 : Slide 36
Chemical Thermodynamics : Georg Duesberg
Proof: rewrite Van der Waals equation for 1 mol of gas, p = RT/(V-b)a/V2, in terms of reduced variables:
RTr Tc
a
pr pc =
− 2 2
Vr Vc − b Vr Vc
Substitute for the critical values:
pra
RTr 8a
a
=
− 2 2
2
27b
27Rb(Vr 3b − b) Vr 9b
8Tr
3
pr =
− 2
3Vr − 1 Vr
Thus all gases have the same reduced
equation of state (within theVan derWaals
approximation).
Also:
Zc =pcVc/RTc = 3/8 =0.375
Chemical Thermodynamics : Georg Duesberg
Thus
Principle of Corresponding States
With reduced variables, different gases fall on the same curves ->
Degree of generality ( principle of corresponding states)
According to this law, there is a
functional relationship for all
substances, which may be
expressed mathematically as vR
= f (PR,TR). From this law it is
clear that if any two gases have
equal values of reduced pressure
and reduced temperature, they
will have the same value of
reduced volume. This law is
most accurate in the vicinity of
the critical point.
Compression factor plotted using reduced
variables. Different curves are different TR
The compressibility factor of any gas is a function of only two properties, usually temperature
and pressure so that Z1 = f (TR, PR) except near the critical point. This is the basis for the
generalized compressibility chart. The generalized compressibility chart is plotted with Z versus
PR for various values of TR. This is constructed by plotting the known data of one or more gases
and can be used for any gas.
It may be seen from the chart that the value of the compressibility factor at
the critical state is about 0.25. Note that the value of Z obtained from Van
der waals’ equation of state at the critical point,
Pc vc 3 which is higher than the actual value.
Zc =
=
RTc 8
The following observations can be made from the generalized compressibility
chart:
Ø  At very low pressures (PR <<1), the gases behave as an ideal gas regardless
of temperature.
Ø  At high temperature (TR > 2), ideal gas behaviour can be assumed with
good accuracy regardless of pressure except when (PR >> 1).
Ø  The deviation of a gas from ideal gas behaviour is greatest in the vicinity of
the critical point.
There are many other equations of state for real gases
1)  The Berthelot Equation.
Replace Van der Waals' "a" with a temperature dependent term, a/T:
[p + a/(Vm2T)] [Vm - b] = RT
2)  The Dieterici Equation : [p exp(a/VmRT)] [Vm - b] = RT
3) Redlich-Kwong
4) Peng-Robinson
p=
p=
RT
A
− 1/ 2
Vm − B T Vm (Vm + B)
RT
α
−
Vm − β Vm (Vm + β ) + β (Vm − β )
Redlich-Kwong, Peng-Robinson are quantitative in region where gas liquefies
Berthelot,Dieterici and others with more than ten parameters can give good
fits…
with four free parameters, you can describe an elephant. With five his tail is
Chapter 1 : Slide 41
rotating …
Chemical Thermodynamics : Georg Duesberg
Summary: Real gases
•  REAL GASES: the COMPRESSION FACTOR and
INTERMOLECULAR FORCES.
•  pV diagrams: LIQUEFACTION and the CRITICAL POINT.
•  BOYLE TEMPERATURE
•  The VAN DER WAALS approximate equation of state p = RT/(Vm-b) a/Vm2 is more realistic for REAL GASES. There are other equations of
state which work well e.g The VIRIAL EQUATION
REDUCED VARIABLES and the PRINCIPLE OF CORRESPONDING
STATES
Pressure region
I (very Low)
Molecules have large separations -> no interactions ->
Ideal Gas Behavior: Z =1
II (moderate)
Molecules are close -> attractive forces apply -> The gas
occupies less volumes as expected from Boyles law: Z<1
III (high)
Molecules compressed highly -> repulsive forces
dominate -> hardly further decrease in volume Z>1
Chapter 1 : Slide 42
Chemical Thermodynamics : Georg Duesberg
Real gas – Van der Waals equation.
For nitrogen a=0.14 and b=3.87x10-5. If 1.0 mole of nitrogen is
confined to 2.00l and is at P=10atm what is Tideal and TVdW?
PV = nRT PV / nR = T
10 × 2/1× 0.082 = 244
2
⎛
⎞
n
⎛
⎞
⎜ P + a⎜ ⎟ ⎟(V − nb ) = nRT
⎜
⎝ V ⎠ ⎟⎠
⎝
2
⎛
⎞
n
⎛
⎞
⎜ P + a⎜ ⎟ ⎟(V − nb ) / nR = T
⎜
⎝ V ⎠ ⎟⎠
⎝
2
⎛
⎞
1
⎛
⎞
⎜1 + 0.14⎜ ⎟ ⎟(2 − 1× 0.0387 ) / 1× 0.082 = 240
⎜
⎝ 2 ⎠ ⎟⎠
⎝
Under these conditions the temperature only changes by ~1%.
Chapter 1 : Slide 43
Chemical Thermodynamics : Georg Duesberg