Physical Chemistry Gaseous State Notes:
.Real gases and their Liquefication
.Deviation from Ideal Gas behaviour
.van der Waals equation
.Critical points and Critical constants
.Equation of corresponding states
.Liquefication of gases
Intermolecular forces
.Questions and answers
Structure
3.1 Introduttion
Objcctivcs
3.2 Deviation from Ideal (;as Rehaviour
3.3 van der Waals Equation
3.4 Critical Phenomena
3.5 Critical Point and ('ritical.Cohstants
Critical Conztanrs and van der ?riaah ('on\r:~nlh
Delcrminarion of Critical Con\[;!'nb
Tc\t ior van dcr Wqalz Equatio I
3.6 Equation of Corresponding States
3.7 Liyuetaction of Gases
Lindr's Mcthod
CI:~udc's Method
3.8 Intermolecular Forces
van dcr W a d \ Forcch
Tol:~l1nrcrd;rion Energy
H ~ J r o ~ cBonding
n
Efl'ccr of Molecul;~l.Inrcraction., on Ph>\ic'al Propertiez
3.9 Summary
3.10 Terminal Questions
3.1 1 Answers
The gas laws developed*in Unit 2 are based on certain assumptions regarding molecules and
thc~rinteraction with each other. Some of these assumptions are not valid under all
conditions: the gases obey ideal gas laws only :rt [ow pressures and high temperatures. To
start with,' thc deviation of the real gases from ideal gas behavio'ur will be discussed in this
unit. The features of the isotherms at different temperatures will he explained. Afterwards,
van 3er Waals e&ation will be deduced. This will be follnwed by a discussion on critical
phenomena and critical constants. The relationships between critical constants and van der
Wa:lls constants will be derived. The principle of corresponding states will be explained.
After this, the methods of liquefaction of gases will be outlined. Finally the nature of
intermolecular forces and their effect on gases will be discussed. The study of intermolecular
forces will help you understand the properties or liquids and'solids which we will take up in
units 4 and 5, respectively.
Objectives
After studying th~sunit, you should be able to :
state the difference in behaviour between real and ideal gascs,
deduce van dcr Waals equation,
define the terms critical temperature. critical pressure and critical volume.
derive the relationships between the critical constants and van der Waals constants.
state and discuss the princ~pleof corresponding states,
state the principles of liquefaction methods,
explain the nature of intermolecular forces, and
discuss the effect of intermolecular forces on the condensation of gases into liqc~itl\and
solids.
3.2 DEVIATION FROM IDEAL GAS BEHAVIOUR
An ideal gas is a hvpothetical concept. The reg11gases obcy ideal gas laws only at low
pressures and h~ghtemperatures.-Betore going into the reasons for the deviation from ideal
gas behaviour, let us study the behaviour of gases at different pressures and temperatures.
, Behaviour of-Real &ses
Experimentally, the behaviour of a gas-can be studied by measuring its pressure. volume.
For real gases, rhe value of z is tempehture and the number of moles. If it behaves ideally, its compressibility factor. I,
greater than or less than unity. , which is defined by Eq. 3.1 must be equal to I .
If z deviates from the value of unity, the gas is said to deviate from ideal behaviour. In
Fig. 3.1, z is plotted against pressure for several gases. W e notice that all gases approach Ideal
behaviour at !ow pressures. This is inferred from the fact that z approaches unity at low
pressure for all gases.
T o illustrate the effect of temperature, z is plotted against pressure for nitrogen gas at three
temperatures in Fig. 3.2. Note that the curve at high temperature (673 K) approaches ideal
0 200 400 600 800
gas behaviour much more'than the curves at lower temperatures (203 K and 293 K). This is
p!atm
true of all the gases. T o sum,up, the gases behave ideallj at low pressures and at high
Fig. 3.1 : Plots of z against p for
temperatures. .
several gases. van der Waals derived an equation of state for explaining the experimental facts of the
behaviour of gases. We shall study this in the next'section.
3.3 VAN DER WAALS EQUATION
The origin of the deviations from ideal gas behaviour lies in two faulty assumptions of the
kinetic theory of gases ( d i m m e d in Unit 2 ) . Firstly, the volume of a molecule is.by no
0.61 I I I I means negligible a$. cannot be ignored under all conditions. Secondly. there certainly '
0 200.400 600 800 exists intermolecular interaction between molecules at close distances. van der Waals
~latm modified the ideal gas equation by taking into account the above shortcomings.
Fig. 3.2 : Plots of z against p for
nitrogen gas at three temperatures Volume Correction : van der Waals realised that the molecules of a real gas have definite
volume. Therefore. the entire volume (V) of the container is not available for the free
movement of the gas molecules.The \olume available for the motion af the molecules can
be given by ( V - nb), where n is the number of moles of the gas and '6'the. correction in
volume for one mole of the gas. The quantity 'h' is known as co-volume.
The van der Waals constant 'b: is
equal to the excluded volume of one Hence, corrected volume = V,,,,! = V - nb ... (3.2)
mole of a gas. It can be shown that
'b'is equal to four times the actua! Pressure Correction : van der ~ a a l applied
s pressure correction by taking into account the
volume of the molecules. The intern~oiecularforces. The pressure of a gas is due to the collision of the gas molecules on
constant b'has the oniu, m' mol-' thc walls of its container. Consider two identical molecules in a gas such that one is
somewhere in the middle of the container aqd the othei-just strikes the wall (Fig. 3,3).
It can be seen that a molecule.in the middle o.f tho containcr is attracted on all sides uy the
othcr molecules surrounding it. However, in cas; of a molecule which just strikes the wall.
there is a net backward drag on the molecule and it will strike the wall with a somewhat
k a k e n e d impaci. ~ e n ' c cthe
, observed pressure (p)of a gas will be less than the pressure
exerted by an ideal gas. A pressure correction is, therefore. t o be applied. The correction .
Fig. 3.3 :The attraction experienced
term in pressure { A p ) is proportional to two factors, vi7..
by the molecules 01 a gas.
@ the number of molecules striking the wall per unit aica and
@ ihe number of molecules attracting a molecule from behind.
Each of the above factors is proportional to the concentra!ion of the gas.
i.e.. Ap cc (concentration)"
Number of moles ( n )
But the concentration of the gas = --
Liquefaction of gases (sec. 3.7)
Volume of the container ( 6
clearly indicates the presence of Hcnce, it can be written that,
forces of attraction among gaseous
rnolec~~les.
The cttnstanr 'o'has the units
Pa rn6 rnol-'
where 'u' is a parameter characteristic of a gas. Henge the corrected pressure @,) is given
by,
, If the corrected pressure and the corrected volume of the gas are substituted in the ideal gas Renl Cnwr and their Liquefartlon
equation (Eq. 2.9), we obtain
n'a
@ + ~ ) ( v - n b ) = n R ~ ... (3.5)
This equation is known as van der Waals equation. Since for one mole of a gas, V =-, V,,,
(i.e.. molar volume) and n = 1, hence. Eq. 3.5 becomes
van der Waals cquatlon (Eq. 3.5 or 3.6) is quite important and is applicable over a much
wider range of p - V- T data than the ideal gas equation. The' quantities 'a' and 'b' are
called the van'der Waals constants or parameters. The values'of 'a' and 'b' are obtained
empiri<ally by fitting in experimental p- V- Tdata to Eq. 3.5. It may be pointed that 'b' is
a measure of the molecular size and 'a' is related to the intermolecular interaction. Table 3.1
gives the values of the parameters 'a' and 'b' of some selected gases. It can be seen that 'b'
increases as the size of the molecule increases whereas 'a' has large value for an easily
compressible gas. The values of the critical consrants p , V, and T, are also given in Table
3.1 and their significance will be dealt with in Sec. 3.5.
Table 3.1 : van der Waals Parameters and Critical Constants of Some Gases
Gas ( 1
'a'/~a,m~rnol-'(lo6 X !b'/m'mol-'1 lo-' X p./Pa ] lo6 X ~c/rn'mol-~ T,fl(
He. 1 0.003457 1 23.70 1 2.20 .1 57.8 1 5.21
To help you use Table 3.1, the
H2
actual values of the parameters for
methane are given below
0.
a = 0.2283 Pa m \ n o ~ - ~
N2 b = 42.78 X 1 0 - b ' mol-'
p, = 46.41 X 10' Pa:
CO? V,,= 99.0 X lo-" m3 mol-'
T, = 191.1 K .
Explanation of the Behaviour of Gases using van der Waals Equation :
Many a times, either one or both the correction terms could become negligible. Let us study
these cases.
When 'b' is negligible
If 'b' is very &all, then Eq. 3.6 becomes,
This shows that under these conditions, p ~ , ,will
, be less than R T o r z will be less than
unity. Eq. 3.8 will Lie valid for substances like water vapo,lr for which 'a' is large and '6'is
co~nparativelysmall (See Tab!e 3.1). Also for gases such as N2, CH4 and C 0 2 (Fig. 3.1) at
moderately low pressures, V, is large such that (V* - b ) is early equal to V,. Hence, Eq.
3..8 is applicable for such gases at moderatety low pressure*.
When 'a' is negligible '
If 'a' 1s negligible, we have
p(V,,, - b ) = RT ... (3.9)
i.e., pV, = R T pb +
Hence, pV,,, will be greater that R T o r z will be greater than unity. Particularly this is'true
for hydrogen (Fig. 3.1) and noble gases for which the value of 'a' is small. This is also true
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