STATES OF MATTER 
properties of gases and liquids (fluids) and solids
24.
Deviations from ideal gas behaviour, intermolecular forces, Van der Waals
equation, compressibility factors and the critical pressure and critical
temperature of a gas
Doc Brown's
chemistry revision notes:
Helpful for UK
advanced level chemistry students aged ~1618, IB courses and US grades 1112 K12 honors.
24. Non–ideal real gas behaviour and Van der Waals Equation
24a. The deviations of
a gases from ideal
behaviour and their causes

Certain postulates in the
kinetic theory of gases (see
section 17) are far from true in real gases, particularly at
higher pressures and a lower temperatures.

This can be clearly seen
in the diagram on the right.

If the gases conformed to
the ideal gas law equation PV=nRT, the product PV should be constant with increasing
pressure at constant temperature, clearly this is not the case.

It can also be seen
that the greatest deviation from ideal behaviour always tend to occur at
higher pressures (right diagram) and often at lower temperatures
(see the compressibility factor diagram
further down the page, and
both positive and negative deviation occur).

Several points in the
theoretical kinetic particle model cannot be ignored in 'real gases'.

The actual
volume of the
molecules (V_{molecules}) is significant at high pressures i.e. the free space for
random particle movement (V_{ideal}) is less than it appears from volume measurements.

V_{real} = V_{ideal}
+ V_{molecules}

At very high pressures
therefore, the value of PV becomes greater than the ideal value and
presumably outweighs the intermolecular force of attraction factor which would
tend to increase the closer the molecules are and decrease P (see forces
arguments next).

The deviation from ideal
gas behaviour due to the molecular volume factor will generally increase with
(i) the greater the pressure and (ii) the larger the volume of the molecule
(~increasing M_{r}).

Intermolecular
forces
always exist i.e. instantaneous dipole – induced dipole forces (Van der Waals
forces) between ANY molecules and at high pressures the molecules are closer
together and so attraction is more likely to occur. As a particle hits the
container side there is an imbalance of the intermolecular forces which act in
all directions in the bulk of the gas. Just as the particle is about to hit
the surface there will be a net greater attraction towards the bulk of the gas
as the molecule, so reducing its impact force i.e. reduces its 'ideal'
pressure (p_{ideal})
by an amount (p_{reduction}).

p_{real} = p_{ideal}
– p_{reduction}

At lower temperatures
when the KE of the molecules are at their lower values, the intermolecular
forces can have more of an effect in reducing P, so the PV value is less
than the ideal value. The effect becomes less as the temperature increases
(graph aboveright) and also as the pressure becomes
much higher when the molecule volume factor outweighs the intermolecular force
factor.

These intermolecular
forces will increase the bigger the molecule (~increasing number of electrons) and the more polar the molecule
where permanent dipole – permanent dipole forces can operate in addition to
the instantaneous dipole – induced dipole forces.

Also, the lower the
temperature, the kinetic energies are lower so its more likely that neighbouring molecules can
affect each other. The reduction in p_{ideal} also increases
with increasing pressure too, since the molecules will be on average closer
together.

There is direct
experimental evidence for the effects of intermolecular forces in gases from
adiabatic expansion or compression situations. Adiabatic means to effect a
change in a system fast enough to avoid heat transfer to or from the
surroundings. e.g.

(i) If a gas at high
pressure is suddenly released through a small nozzle it rapidly cools on
expansion into the lower pressure zone. The reason for the cooling is that
in order to expand the intermolecular forces must be overcome by energy
absorption, an endothermic process. The change is so rapid that the source
of heat energy can only come from the kinetic energy of the gas molecules
themselves, so the gas rapidly cools. This is observed when a carbon dioxide
fire extinguisher is used, just for a second bits of solid CO_{2}
can be seen, which rapidly vaporise. However, it proves that the gas was
rapidly cooled from room temperature to –78^{o}C!

(ii) When you rapidly
pump air into a bicycle tyre the gas warms up because the molecules are
forced closer together so the intermolecular forces can operate more
strongly, this, just like bond formation, is always an exothermic process.

Therefore generally
speaking for any gas the lower its pressure and the higher its temperature,
the more closely it will be 'ideal', i.e. closely obey the ideal gas equation PV=nRT
etc. Also the smaller the molecular mass or the weaker the intermolecular
forces, the gas will be closer to ideal behaviour.

However, for any gas at
a particular P and T, its all a question of what factor outweighs the others.

Note that both positive
and negative deviation from ideal gas behaviour can occur and there will be
situations where the different causes of non–ideal behaviour cancel each other out.

Check out the graphs at the start of this age

The measurement and
predictions of gas behaviour is very important in industrial processes and so
many mathematical developments have been devised to accurately describe the
real behaviour of gases.

The
Van der Waals equation
is one of the earliest and simplest equations to model real gas behaviour.
24b. The Van der Waals equation of state

Equations such as the Van
der Waals equation for real non–ideal gases attempt to take into account the volume occupied by the
molecules and the intermolecular forces between them. The idea is to
incorporate 'corrective' terms to reproduce or model real gas P–V–T behaviour
with a modified equation of state.

The Van der Waals equation
for one mole of gas can be most simply stated in as

(i) (p + a') (V – b') = RT

This is called an 'equation
of state' and is the simplest version for a nonideal gas.

The term a' represents the
extra pressure the gas would exert if it behaved ideally.

In real gases the
intermolecular forces are imbalanced at the point of impact on the container
wall, with a net attraction in the direction of the bulk of the gas.

In the
bulk of the gas, each molecules is subjected to the same 'time averaged'
attractions in all directions, but heading for the container wall it is
considered to be 'dragged back a bit' by attraction with the bulk of the gas
surrounding it on all sides bar the surface of impact, which is therefore
reduced in force. (see also intermolecular forces
discussion)

The term b' represents the
volume that the molecules occupy, so V–b' represents the actual volume of free
space the molecules can move in. (see also
molecule
volume discussion)

For n moles of gas the Van
der Waals equation is ...

(ii)
[p + (an^{2}/V^{2})] (V
– nb) =
nRT

a and b are the Van der
Waal equation constants.

The factor n^{2}/V^{2} is
related to the gas density, the more dense the gas (i.e. moles/volume), at
higher pressures, the more intense will be
the intermolecular attractive force field effects.

Dividing through by n,
using the (V – nb) term, gives the alternative version ...

(iii)
[p + (an^{2}/V^{2})] [(V/n)
– b)]
= RT

(iv)
p = [nRT/(V – nb)] – (an^{2}/V^{2})

(v) p = [RT/(V/n – b)] – (an^{2}/V^{2})

For 1 mole of gas the equation
simplifies to

A selection of a and b
Van der Waal's constants are given below.
Data 
Van der Waals constants 
critical values of the gas 
Gas 
a (Pa m^{6} mol^{–2}) 
b (m^{3} mol^{–1}) 
pressure
p_{c} (Pa) 
temp. T_{c} (K) 
air, av M_{r}(mix)
~ 29 
0.1358 
3.64 x 10^{–5} 
3.77 x 10^{6} 
133 K 
ammonia, M_{r}(NH_{3}) = 17 
0.4233 
3.73 x 10^{–5} 
11.3 x 10^{6} 
406 K 
butane, M_{r}(C_{4}H_{10}) =
59 
1.466 
12.2 x 10^{–5} 
3.78 x 10^{6} 
425 K 
carbon dioxide, M_{r}(CO_{2})
= 44 
0.3643 
4.27 x 10^{–5} 
7.39 x 10^{6} 
304 K 
dichlorodifluoromethane, Freon CFC–11, M_{r}(CCl_{2}F_{2})
= 121 
1.078 
9.98 x 10^{–5} 
4.12 x 10^{6} 
385 K 
helium, M_{r}(He) = 4 
0.00341 
2.34 x 10^{–5} 
0.23 x 10^{6} 
5 K 
hydrogen, M_{r}(H_{2}) = 2 
0.0247 
2.65 x 10^{–5} 
1.29 x 10^{6} 
33 K 
nitrogen, M_{r}(N_{2}) = 28 
0.1361 
3.85 x 10^{–5} 
3.39 x 10^{6} 
126 K 
water, M_{r}(H_{2}O) = 18 
0.5507 
3.04 x 10^{–5} 
22.1 x 10^{6} 
647 K 

The constant a
varies considerably from gas to gas because of the wide variety of
intermolecular forces e.g. very low for helium and non–polar hydrogen (2 e's
each, just instantaneous dipole–induced dipole forces), to much higher a
values for larger polar molecules like water or dichlorodifluoromethane (more
electrons and extra permanent dipole–permanent dipole intermolecular forces).

The constant b
varies less, and not unexpectedly, just tends to rise with increase in
molecule size.

Critical values of gas
behaviour.

Critical temperature T_{c}

This is the maximum
temperature at which a substance can exist as a liquid. Above T_{c},
only the gaseous state can exist, however great the density or pressure! It
might be truer to say that above T_{c}, the gaseous and liquid state
become indistinguishable as the meniscus just disappears!

Critical pressure p_{c}

Generally speaking the
critical values for a gas/liquid increase with increase in intermolecular
forces e.g. due to increase in molecular mass or increasing polarity of
molecule.
24c. Compressibility factors
The compressibility
factor z, is defined as the ratio PV/nRT.
Since PV = nRT for an
ideal gas, then z = 1 for an ideal gas.
z varies with pressure or
temperature for any gas, see the
PV versus P graph in
start of this page, which gives an indication of how z might vary with
pressure at a given temperature).
Clearly from the graph on the right
for methane, z can be at least as high as 2, and, at least as low as 0.6, showing considerable deviation from ideal gas behaviour, particularly at low
temperatures (influence of intermolecular forces stronger) and high
pressures (where the effect of both actual molecule volume and
intermolecular forces are important).
See more detailed
discussion at start of this page.
As the pressure becomes lower and/or
temperature higher, the gas becomes more ideal in terms of its physical
behaviour and particularly 'ideal' as the pressure tends towards zero.
Known values of z can be used to
calculate the real P–V values for a non–ideal gas.
z = pV/nRT, pV = znRT,
p = znRT/V
and V = znRT/p
24d. The Critical Point –
The Critical Temperature and Critical Pressure for a gas
Question! If you increase
the pressure of a gas it can change into a liquid. But, increasing the pressure,
also increases the temperature, so shouldn't the gas remain a gas?

Gases can be converted
to liquids by compressing the gas at a suitable temperature and this is done
commercially at as lower temperature as possible e.g. liquefaction of air to
fractionally distil off nitrogen and oxygen or liquefying petroleum gas.

Gases become more
difficult to liquefy as the temperature increases because the kinetic
energies of the particles that make up the gas also increase and the
intermolecular forces have less influence i.e. more easily overcome.

When you increase the pressure of a gas
you force the molecules closer together and if the extra intermolecular
force is strong enough liquefaction occurs. Remember the force of electrical
attraction is proportional to the numerical +ve charge multiplied by the
–ve charge divided by the distance squared.

However when you
compress a gas it can heat up. This is because heat is generated by the
increased intermolecular interaction (remember bond formation is also
exothermic) but here its just weak molecule association due to the
intermolecular attractive forces.

BUT liquefaction =
condensation and is an exothermic process, so heat must be removed to effect the
state change of gas ==> liquid. If the temperature is low enough and the
heat is dispersed liquefaction can still happen.

If it is too hot it
would stay as a gas. So liquefaction conditions are all about temperature,
pressure and heat transfer i.e. the ambient conditions.

However, above a certain
temperature called the critical temperature (T_{c})
you cannot get a liquid with a 'surface', what you get is an extremely dense
gas that is close to being a liquid but not quite!

The critical temperature
of a substance is the temperature at and above which vapour of the substance
cannot be liquefied, no matter how much pressure is applied.

The critical pressure
(P_{c}) of a substance is the minimum pressure required to
liquefy a gas at its critical temperature i.e. the critical pressure is the
vapour pressure at the critical temperature.

The vapourliquid
critical point denotes the conditions above which distinct liquid and gas
phases do not exist and a meniscus no longer exists!

The point at the critical temperature and critical
pressure is called the critical point of the substance.
Learning objectives
Know that the principal
reasons for the deviation of gaseous behaviour from the ideal gas laws,
at most temperatures and pressures is due to the influence of
intermolecular attractive forces between the particles in the gas.
Know that significant
deviations also occur at high pressure (again due to above) AND the
particles occupy an actual volume  they are not points of zero volume!
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