gaseous state - theory

Gaseous State 225
The state of matter in which the molecular forces of attraction
between the particles of matter are minimum, is known as gaseous state. It
is the simplest state and shows great uniformity in behaviour.
Characteristics of gases
(1) Gases or their mixtures are homogeneous in composition.
(2) Gases have very low density due to negligible intermolecular
forces.
(3) Gases have infinite expansibility and high compressibility.
(4) Gases exert pressure.
(5) Gases possess high diffusibility.
(6) Gases do not have definite shape and volume like liquids.
(7) Gaseous molecules move very rapidly in all directions in a random
manner i.e., gases have highest kinetic energy.
(8) Gaseous molecules collide with one another and also with the walls
of container with perfectly elastic collisions.
(9) Gases can be liquified, if subjected to low temperatures (below
critical) or high pressures.
(10) Thermal energy of gases >> molecular attraction.
(11) Gases undergo similar change with the change of temperature
and pressure. In other words, gases obey certain laws known as gas laws.
Measurable properties of gases
(1) The characteristics of gases are described fully in terms of four
parameters or measurable properties :
(i) The volume, V, of the gas.
(ii) Its pressure, P
(iii) Its temperature, T
(iv) The amount of the gas (i.e., mass or number of moles).
(2) Volume : (i) Since gases occupy the entire space available to
them, the measurement of volume of a gas only requires a measurement of
the container confining the gas.
(ii)Volume is expressed in litres (L), millilitres (mL) or cubic
centimetres )( 3
cm or cubic metres )( 3
m .
(iii) mLL 10001  ; LmL 3
101 
 ; 333
1011 mdmL 

LmLcmdmm 3636333
101010101 
(3) Mass : (i) The mass of a gas can be determined by weighing the
container in which the gas is enclosed and again weighing the container
after removing the gas. The difference between the two weights gives the
mass of the gas.
(ii)The mass of the gas is related to the number of moles of the gas
i.e.
moles of gas (n)
M
m

massMolar
gramsinMass
(4) Temperature : (i) Gases expand on increasing the temperature. If
temperature is increased twice, the square of the velocity )( 2
v also
increases two times.
(ii)Temperature is measured in centigrade degree )( Co
or celsius
degree with the help of thermometers. Temperature is also measured in
Fahrenheit (Fo
).
(iii) S.I. unit of temperature is kelvin (K) or absolute degree.
273 CK o
(iv) Relation between F and Co
is
9
32
5


oo
FC
(5) Pressure : (i) Pressure of the gas is the force exerted by the gas
per unit area of the walls of the container in all directions. Thus, Pressure
(P)
)(Area
)(onAccelerati)(Mass
)Area(
)Force(
a
am
A
F 

(ii) Pressure exerted by a gas is due to kinetic energy
)
2
1
( 2
mvKE  of the molecules. Kinetic energy of the gas molecules
increases, as the temperature is increased. Thus, Pressure of a gas 
Temperature (T).
(iii) Pressure of a pure gas is measured by manometer while that of
a mixture of gases by barometer.
(iv) Commonly two types of manometers are used,
(a) Open end manometer; (b) Closed end manometer
Gaseous state
Chapter
6

226 Gaseous State
(v) The S.I. unit of pressure, the pascal (Pa), is defined as 1
newton per metre square. It is very small unit.
212
111 
 smkgNmPa
(vi) C.G.S. unit of pressure is dynes 2
cm .
(vii) M.K.S. unit of pressure is 2
/ mkgf . The unit 2
/ cmkgf
sometime called ata (atmosphere technical absolute).
(viii) Higher unit of pressure is bar, KPa or MPa.
KPaKNmNmPabar 10010010101 2255
 
(ix) Several other units used for pressure are,
Name Symbol Value
bar bar Pabar 5
101 
atmosphere atm Paatm 5
1001325.11 
Torr Torr
PaPaTorr 322.133
760
101325
1 
millimetre of
mercury
mm Hg PaHgmm 322.1331 
(x) The pressure relative to the atmosphere is called gauge pressure.
The pressure relative to the perfect vacuum is called absolute pressure.
Absolute pressure = Gauge pressure + Atmosphere pressure.
(xi) When the pressure in a system is less than atmospheric
pressure, the gauge pressure becomes negative, but is frequently designated
and called vacuum. For example, 16 cm vacuum will be
bar80.0013.1
76
1676


.
(xii) If ‘h’ is the height of the fluid in a column or the difference in
the heights of the fluid columns in the two limbs of the manometer, d is the
density of the fluid
)/6.13/106.13( 333
cmgmKgHg  and g is the gravity, then
pressure is given by, dghPP  atmgas
(xiii) Two sets of conditions are widely used as 'standard' values for
reporting data.
Condition T P Vm
(Molar volume)
S.T.P./N.T.P. 273.15 K 1 atm 22.414 L
S.A.T.P*
. 298.15 K 1 bar 24.800 L
* Standard ambient temperature and pressure.
Boyle's law
(1) In 1662, Robert Boyle discovered the first of several relationships
among gas variables (P, T, V).
(2) It states that, “For a fixed amount of a gas at constant
temperature, the gas volume is inversely proportional to the gas pressure.”
Thus, VP /1 at constant temperature and mass
or VKP / (where K is constant)
or KPV  or KVPVP  2211 (For two or more gases)
(3) Graphical representation of Boyle's law : Graph between P and V
at constant temperature is called isotherm and is an equilateral (or
rectangular) hyperbola. By plotting P versus V/1 , this hyperbola can be
converted to a straight line. Other types of isotherms are also shown below,
(4) At constant mass and temperature density of a gas is directly
proportional to its pressure and inversely proportional to its volume.
Thus,
V
Pd
1
 






d
V
mass

or K
V
V
P
P
d
d
 .......
1
2
2
1
2
1
(5) At altitudes, as P is low d of air is less. That is why
mountaineers carry oxygen cylinders.
Charle's law
(1) French chemist, Jacques Charles first studied variation of volume
with temperature, in 1787.
(2) It states that, “The volume of a given mass of a gas is directly
proportional to the absolute temperature )273(  Co
at constant
pressure”.
Thus, TV  at constant pressure and mass
or )15.273)((  CtKKTV o
, (where k is constant),
T
V
K  or K
T
V
T
V

2
2
1
1
(For two or more gases)
(3) If Ct o
0 , then 0VV 
hence, 15.2730  KV

15.273
0V
K 
]1[
15.273
1]15.273[
15.273
00
0
tV
t
Vt
V
V v






where v is the volume coefficient,
13
0
0
10661.3
15.273
1 


 C
tV
VV o
v
Thus, for every o
1 change in temperature, the volume of a gas
changes by 






273
1
15.273
1
of the volume at Co
0 .
(4) Graphical representation of Charle's law : Graph between V and
T at constant pressure is called isobar or isoplestics and is always a straight
line. A plot of V versus )( Ct o
at constant pressure is a straight line cutting
the temperature axis at Co
15.273 . It is the lowest possible
temperature.
T1< T2< T3
T3
T2
T1
P
PV
O
log P
O
log 1/V
T1< T2< T3
T3
T2
T1
V or 1/d
P
O
T1< T2< T3
T3
T2
T1
1/V or d
P
O

Gaseous State 227
(5) At constant mass and pressure density of a gas is inversely
proportional to it absolute temperature.
Thus,
VT
d
11
 






d
mass
V
or K
V
V
T
T
d
d
 ......
1
2
1
2
2
1
(6) Use of hot air balloons in sports and meteorological observations
is an application of Charle's law.
Gay-Lussac's law (Amonton's law)
(1) In 1802, French chemist Joseph Gay-Lussac studied the variation
of pressure with temperature and extende the Charle’s law so, this law is
also called Charle’s-Gay Lussac’s law.
(2) It states that, “The pressure of a given mass of a gas is directly
proportional to the absolute temperature )273(  Co
at constant
volume.”
Thus, TP  at constant volume and mass
or )15.273)((  CtKKTP o
(where K is constant)
T
P
K  or K
T
P
T
P

2
2
1
1
(For two or more gases)
(3) If Ct o
0 , then 0PP 
Hence, 15.2730  KP

15.273
0P
K 
]1[
15.273
1]15.273[
15.273
00
0
tP
t
Pt
P
P 






where P is the pressure coefficient,
13
0
0
10661.3
15.273
1 


 C
tP
PP o
P
Thus, for every o
1 change in temperature, the pressure of a gas
changes by 






273
1
15.273
1
of the pressure at Co
0 .
(4) This law fails at low temperatures, because the volume of the
gas molecules be come significant.
(5) Graphical representation of Gay-Lussac's law : A graph between
P and T at constant V is called isochore.
Avogadro's law
(1) According to this law, “Equal volumes of any two gases at the
same temperature and pressure contain the same number of molecules.”
Thus, nV  (at constant T and P)
or KnV  (where K is constant)
or K
n
V
n
V
 .......
2
2
1
1
Example,
litren
litre
litres
volumes
moles
litren
litre
litre
volume
mole
litren
litre
litres
volumes
moles
gOHgOgH
1
1
2
2
2
2
2/1
2/1
1
1
1
2
1
1
2
2
2
2 )(2)()(2 
(2) One mole of any gas contains the same number of molecules
(Avogadro's number 23
1002.6  ) and by this law must occupy the
same volume at a given temperature and pressure. The volume of one mole
of a gas is called molar volume, Vm
which is 22.4 L 1
mol at S.T.P. or
N.T.P.
(3) This law can also express as, “The molar gas volume at a given
temperature and pressure is a specific constant independent of the nature
of the gas”.
Thus, mV specific constant 1
4.22 
 molL at S.T.P. or N.T.P.
Ideal gas equation
(1) The simple gas laws relating gas volume to pressure, temperature
and amount of gas, respectively, are stated below :
Boyle's law :
V
P
1
 or
P
V
1
 (n and T constant)
Charle's law : TV (n and P constant)
Avogadro's law : nV  (T and P constant)
If all the above law's combines, then
P
nT
V 
or
P
nRT
V  ( R Ideal gas constant)
or nRTPV 
This is called ideal gas equation. R is called ideal gas constant. This
equation is obeyed by isothermal and adiabatic processes.
(2) Nature and values of R : From the ideal gas equation,
eTemperaturmole
VolumePressure



nT
PV
R
eTemperaturmole
LengthForce
eTemperaturmole
Volume
Area
Force






eTemperaturmole
energyorWork

 .
R is expressed in the unit of work or energy 11 
Kmol .
Since different values of R are summarised below :
11
0821.0 
 KmolatmLR
T(k)
1/d
or V
O
0oC t(oC)–273.15oC
1/d or V
22.4 L mol–1 = V0
T(k)
P
O
P
T
V1
V2
V3
V4
V1< V2< V3< V4
O

228 Gaseous State
11
3143.8 
 Kmoljoule (S.I. unit)
11
3143.8 
 KmolNm
113
3143.8 
 KmoldmKPa
113
3143.8 
 KmolcmMPa
1119
10189.5 
 KmoleV
11
99.1 
 Kmolcal
(3) Gas constant, R for a single molecule is called Boltzmann
constant (k)
11
23
7
10023.6
10314.8 


 egreedmoleergs
N
R
k
1116
1038.1 
 greedemolergs
or 1123
1038.1 
 egreedmoljoule
(4) Calculation of mass, molecular weight and density of the gas by
gas equation
RT
M
m
nRTPV 







)(gasoftheweightMolecular
)(gasofthemass
M
m
n

PV
mRT
M 
RT
PM
d  






V
m
d
or
R
M
P
dT
 , 
R
M
Constant
( M and R are constant for a particular gas)
Thus,
P
dT
or
2
22
1
11
T
Td
P
Td
 = Constant
(For two or more different temperature and pressure)
(5) Gas densities differ from those of solids and liquids as,
(i) Gas densities are generally stated in g/L instead of 3
/ cmg .
(ii)Gas densities are strongly dependent on pressure and
temperature as, Pd  T/1
Densities of liquids and solids, do depend somewhat on temperature,
but they are far less dependent on pressure.
(iii) The density of a gas is directly proportional to its molar mass.
No simple relationship exists between the density and molar mass for liquid
and solids.
(iv) Density of a gas at STP
4.22
massmolar

)( 2Nd at STP 1
25.1
4.22
28 
 Lg ,
)( 2Od at STP 1
43.1
4.22
32 
 Lg
Dalton's law of partial pressures
(1) According to this law, “When two or more gases, which do not
react chemically are kept in a closed vessel, the total pressure exerted by the
mixture is equal to the sum of the partial pressures of individual gases.”
Thus, .........321total  PPPP
Where ,......,, 321 PPP are partial pressures of gas number 1, 2, 3
.........
(2) Partial pressure is the pressure exerted by a gas when it is
present alone in the same container and at the same temperature.
Partial pressure of a gas
Total1
Total1
1 )(fractionMole
mixturethein)(molesofnumberTotal
)(gasofthemolesofNumber
)( PX
n
Pn
P 


(3) If a number of gases having volume ......,, 321 VVV at pressure
........,, 321 PPP are mixed together in container of volume V, then,
V
VPVPVP
P
.....332211
Total


or
V
RT
nnn .....)( 321  )( nRTPV 
or
V
RT
n .....)( 321 nnnn 
(4) Applications : This law is used in the calculation of following
relationships,
(i) Mole fraction of a gas )( 1X in a mixture of gas
Total
1 )(gasaofessurePartial pr
P
P

(ii) % of a gas in mixture 100
)(gasaofessurePartial pr
Total
1

P
P
(iii) Pressure of dry gas collected over water : When a gas is
collected over water, it becomes moist due to water vapour which exerts its
own partial pressure at the same temperature of the gas. This partial
perssure of water vapours is called aqueous tension. Thus,
urwater vapoTotalgasmoistgasdry or PPPP 
or  gasmoistgasdry PP Aqueous tension (Aqueous tension is
directly proportional to absolute temperature)
(iv) Relative humidity (RH) at a given temperature is given by,
ofwaterssureVapour pre
airinofwateressurePartial pr
RH .
(5) Limitations : This law is applicable only when the component
gases in the mixture do not react with each other. For example, 2N and
2O , CO and 2CO , 2N and 2Cl , CO and 2N etc. But this law is not
applicable to gases which combine chemically. For example, 2H and 2Cl ,
CO and 2Cl , 3NH , HBr and HCl, NO and 2O etc.
(6) Another law, which is really equivalent to the law of partial
pressures and related to the partial volumes of gases is known as Law of
partial volumes given by Amagat. According to this law, “When two or more
gases, which do not react chemically are kept in a closed vessel, the total
volume exerted by the mixture is equal to the sum of the partial volumes of
individual gases.”
Thus, ......321Total  VVVV
Where ,......,, 321 VVV are partial volumes of gas number 1, 2, 3.....
Graham's law of diffusion and Effusion
(1) Diffusion is the process of spontaneous spreading and
intermixing of gases to form homogenous mixture irrespective of force of
gravity. While Effusion is the escape of gas molecules through a tiny hole
such as pinhole in a balloon.

Gaseous State 229
 All gases spontaneously diffuse into one another when they are
brought into contact.
 Diffusion into a vacuum will take place much more rapidly than
diffusion into another place.
 Both the rate of diffusion of a gas and its rate of effusion depend
on its molar mass. Lighter gases diffuses faster than heavier gases. The gas
with highest rate of diffusion is hydrogen.
(2) According to this law, “At constant pressure and temperature,
the rate of diffusion or effusion of a gas is inversely proportional to the
square root of its vapour density.”
Thus, rate of diffusion
d
r
1
)(  (T and P constant)
For two or more gases at constant pressure and temperature,
1
2
2
1
d
d
r
r

(3) Graham's law can be modified in a number of ways as,
(i) Since, 2  vapour density (V.D.) = Molecular weight
then,
1
2
1
2
1
2
2
1
2
2
M
M
d
d
d
d
r
r




where, 1M and 2M are the molecular weights of the two gases.
(ii) Since, rate of diffusion
diffusionfortakenTime
diffusedgasaofVolume
)( r
then,
1
2
22
11
22
11
2
1
/
/
/
/
d
d
tw
tw
tV
tV
r
r

(a) When equal volume of the two gases diffuse, i.e. 21 VV 
then,
1
2
1
2
2
1
d
d
t
t
r
r

(b) When volumes of the two gases diffuse in the same time, i.e.
21 tt 
then,
1
2
2
1
2
1
d
d
V
V
r
r

(iii) Since, pr  (when p is not constant)
then,
1
2
2
1
2
1
M
M
P
P
r
r
 








M
P
r
(4) Rate of diffusion and effusion can be determined as,
(i) Rate of diffusion is equal to distance travelled by gas per unit
time through a tube of uniform cross-section.
(ii) Number of moles effusing per unit time is also called rate of
diffusion.
(iii) Decrease in pressure of a cylinder per unit time is called rate of
effusion of gas.
(iv) The volume of gas effused through a given surface per unit time
is also called rate of effusion.
(5) Applications : Graham's law has been used as follows,
(i) To determine vapour densities and molecular weights of gases.
(ii)To prepare Ausell’s marsh gas indicator, used in mines.
(iii) Atmolysis : The process of separation of two gases on the basis
of their different rates of diffusion due to difference in their densities is
called atmolysis. It has been applied with success for the separation of
isotopes and other gaseous mixtures.
Kinetic theory of gases
(1) Kinetic theory was developed by Bernoulli, Joule, Clausius,
Maxwell and Boltzmann etc. and represents dynamic particle or microscopic
model for different gases since it throws light on the behaviour of the
particles (atoms and molecules) which constitute the gases and cannot be
seen. Properties of gases which we studied earlier are part of macroscopic
model.
(2) Postulates
(i) Every gas consists of a large number of small particles called
molecules moving with very high velocities in all possible directions.
(ii)The volume of the individual molecule is negligible as compared
to the total volume of the gas.
(iii) Gaseous molecules are perfectly elastic so that there is no net loss of
kinetic energy due to their collisions.
(iv) The effect of gravity on the motion of the molecules is
negligible.
(v) Gaseous molecules are considered as point masses because they
do not posses potential energy. So the attractive and repulsive forces
between the gas molecules are negligible.
(vi) The pressure of a gas is due to the continuous bombardment on
the walls of the containing vessel.
(vii) At constant temperature the average K.E. of all gases is same.
(viii) The average K.E. of the gas molecules is directly proportional
to the absolute temperature.
(3) Kinetic gas equation : On the basis of above postulates, the
following gas equation was derived,
2
3
1
rmsmnuPV 
where, P = pressure exerted by the gas
V = volume of the gas
m = average mass of each molecule
n = number of molecules
u = root mean square (RMS) velocity of the gas.
(4) Calculation of kinetic energy
We know that,
K.E. of one molecule 2
2
1
mu
K.E. of n molecules PVmnu
2
3
2
1 2
 )
3
1
( 2
mnuPV 
n = 1, Then K.E. of 1 mole gas RT
2
3
 )( RTPV 
JoulesTT 47.12314.8
2
3
 .
KT
N
RT
N 2
3
2
3
)numberAvogadro(
moleperK.E.Average







 constantBoltzmann
N
R
K
This equation shows that K.E. of translation of a gas depends only
on the absolute temperature. This is known as Maxwell generalisation. Thus
average K.E.  T.
If KT 0 (i.e., )15.273 Co
 then, average K.E. = 0. Thus,
absolute zero (0K) is the temperature at which molecular motion ceases.
Molecular collisions

230 Gaseous State
(1) The closest distance between the centres of two molecules taking
part in a collision is called molecular or collision diameter (). The
molecular diameter of all the gases is nearly same lying in the order of
cm8
10
.
(2) The number of collisions taking place in unit time per unit
volume, called collision frequency (z).
(i) The number of collision made by a single molecule with other
molecules per unit time are given by, nuZA av.
2
2
where n is the number of molecules per unit molar volume,
3
23
0
0224.0
1002.6)number(Avogadro 
 m
V
N
n
m
(ii)The total number of bimolecular collision per unit time are given
by, 2
.av
2
2
1
nuZAA 
(iii) If the collisions involve two unlike molecules, the number of
bimolecular collision are given by,
2/1
2 )(
8 




 

BA
BA
ABAB
MM
MM
RTZ 
where,
2
BA
AB




AM , BM are molecular weights )( 0mNM 
(iv) (a) At particular temperature; 2
pZ 
(b) At particular pressure; 2/3
 TZ
(c) At particular volume; 2/1
TZ 
(3) During molecular collisions a molecule covers a small distance
before it gets deflected. The average distance travelled by the gas molecules
between two successive collision is called mean free path ().
)(timeunitpermoleculesinglebymadecollisionsofNo.
)time(unitpertravelleddistanceAverage av
AZ
u
 .
2
avr.
2
av
2
1
2  nnu
u

(4) Based on kinetic theory of gases mean free path,
P
T
 . Thus,
(i) Larger the size of the molecules, smaller the mean free path, i.e.,
2
(radius)
1

(ii) Greater the number of molecules per unit volume, smaller the
mean free path.
(iii) Larger the temperature, larger the mean free path.
(iv) Larger the pressure, smaller the mean free path.
(5) Relation between collision frequency (Z) and mean free path ()
is given by,

rmsu
Z 
Molecular speeds or velocities
(1) At any particular time, in the given sample of gas all the
molecules do not possess same speed, due to the frequent molecular
collisions with the walls of the container and also with one another, the
molecules move with ever changing speeds and also with ever changing
direction of motion.
(2) According to Maxwell, at a particular temperature the
distribution of speeds remains constant and this distribution is referred to
as the Maxwell-Boltzmann distribution and given by the following
expression,
dcue
RT
M
n
dn RTMu 22/
2/3
0
..
2
4
2










where, 0dn Number of molecules out of total number of
molecules n, having velocities between c and dcc  , ndn /0 Fraction of
the total number of molecules, M = molecular weight, T = absolute
temperature. The exponential factor RTMu
e 2/2

is called Boltzmann factor.
(3) Maxwell gave distribution curves of molecular speeds for 2CO
at different temperatures. Special features of the curve are :
(i) Fraction of molecules with two high or two low speeds is very small.
(ii) No molecules has zero velocity.
(iii) Initially the fraction of molecules increases in velocity till the
peak of the curve which pertains to most probable velocity and thereafter it
falls with increase in velocity.
(4) Types of molecular speeds or Velocities
(i) Root mean square velocity (urms
) : It is the square root of the mean
of the squares of the velocity of a large number of molecules of the same
gas.
n
uuuu
u n
rms
22
3
2
2
2
1 .....

MmN
RT
MmN
PV
urms




)(
3
)(
3
00 d
P
m
kT
M
RT 333

where k = Boltzmann constant
0N
R

(a) For the same gas at two different temperatures, the ratio of
RMS velocities will be,
2
1
2
1
T
T
u
u

(b) For two different gases at the same temperature, the ratio of
RMS velocities will be,
1
2
2
1
M
M
u
u

(c) RMS velocity at any temperature Cto
may be related to its
value at S.T.P. as,
d
tP
ut
273
)273(3 
 .
(ii)Average velocity )( avv : It is the average of the various velocities
possessed by the molecules.
n
vvvv
v n
av
......321 
 ;
m
kT
M
RT
vav

88

(iii) Most probable velocity )( mp : It is the velocity possessed by
maximum number of molecules of a gas at a given temperature.
Molecular diameter

Ump
Uav
Urms
Ump
1800 K (T3)
Ump
300 K (T1) T1<T2<T3
1500 K (T2)
Fractionofmolecules
Molecular speed

Gaseous State 231
d
P
M
PV
M
RT
mp
222

(5) Relation between molecular speeds or velocities,
(i) Relation between rmsu and avv : rmsav uv  9213.0
or avrms vu  085.1
(ii)Relation between mp and rmsu : rmsmp u 816.0
or mprmsu  224.1
(iii) Relation between mp and avv : mpavv  128.1
(iv) Relation between mp , avv and rmsu
mp : avv : rmsu
M
RT2
:
M
RT

8
:
M
RT3
2 :

8
: 3
1.414 : 1.595 : 1.732
1 : 1.128 : 1.224 i.e., rmsavmp uv 
Real and Ideal gases
(1) Gases which obey gas laws or ideal gas equation )( nRTPV  at
all temperatures and pressures are called ideal or perfect gases. Almost all
gases deviate from the ideal behaviour i.e., no gas is perfect and the concept
of perfect gas is only theoretical.
(2) Gases tend to show ideal behaviour more and more as the
temperature rises above the boiling point of their liquefied forms and the
pressure is lowered. Such gases are known as real or non ideal gases. Thus,
a “real gas is that which obeys the gas laws under low pressure or high
temperature”.
(3) The deviations can be displayed, by plotting the P-V isotherms of
real gas and ideal gas.
(4) It is difficult to determine quantitatively the deviation of a real
gas from ideal gas behaviour from the P-V isotherm curve as shown above.
Compressibility factor Z defined by the equation,
ZnRTPV  or RTPVnRTPVZ m // 
is more suitable for a quantitative description of the deviation from
ideal gas behaviour.
(5) Greater is the departure of Z from unity, more is the deviation
from ideal behaviour. Thus, when
(i) 1Z , the gas is ideal at all temperatures and pressures. In case
of 2N , the value of Z is close to 1 at Co
50 . This temperature at which a
real gas exhibits ideal behaviour, for considerable range of pressure, is
known as Boyle's temperature or Boyle's point )( BT .
(ii) 1Z , the gas is less compressible than expected from ideal
behaviour and shows positive deviation, usual at high P i.e. RTPV  .
(iii) 1Z , the gas is more compressible than expected from ideal
behaviour and shows negative deviation, usually at low P i.e. RTPV  .
(iv) 1Z for 2H and He at all pressure i.e., always shows
positive deviation.
(v) The most easily liquefiable and highly soluble gases ),( 23 SONH
show larger deviations from ideal behaviour i.e. 1Z .
(vi) Some gases like 2CO show both negative and positive
deviation.
(6) Causes of deviations of real gases from ideal behaviour : The
ideal gas laws can be derived from the kinetic theory of gases which is
based on the following two important assumptions,
(i) The volume occupied by the molecules is negligible in
comparison to the total volume of gas.
(ii) The molecules exert no forces of attraction upon one another. It
is because neither of these assumptions can be regarded as applicable to
real gases that the latter show departure from the ideal behaviour.
Vander Waal's equation
(1) To rectify the errors caused by ignoring the intermolecular forces
of attraction and the volume occupied by molecules, Vander Waal (in 1873)
modified the ideal gas equation by introducing two corrections,
(i) Volume correction (ii) Pressure correction
(2) Vander Waal's equation is obeyed by the real gases over wide
range of temperatures and pressures, hence it is called equation of state for
the real gases.
(3) The Vander Waal's equation for n moles of the gas is,
nRTnbV
V
an
P 








moleculesofsizefinite
forcorrectionVolume
attractionmolecularfor
correctionPressure
2
2
][
a and b are Vander Waal's constants whose values depend on the
nature of the gas. Normally for a gas ba  .
(i) Constant a : It is a indirect measure of magnitude of attractive
forces between the molecules. Greater is the value of a, more easily the gas
can be liquefied. Thus the easily liquefiable gases (like
)2232 COSHNHSO  have high values than the permanent gases
(like )222 HeHON  .
Units of 'a' are : atm. 22 
molL or atm. 26 
molm or
24 
molmN (S.I. unit).
(ii) Constant b : Also called co-volume or excluded volume,






 3
00
3
4
44 rNvNb 
It's value gives an idea about the effective size of gas molecules.
Greater is the value of b, larger is the size and smaller is the compressible
volume. As b is the effective volume of the gas molecules, the constant value
of b for any gas over a wide range of temperature and pressure indicates
that the gas molecules are incompressible.
Units of 'b' are : 1
molL or 13 
molm (S.I. unit)
(iii) The two Vander Waal's constants and Boyle's temperature )( BT
are related as,
bR
a
TB 
(4) Vander Waal's equation at different temperature and pressures
(i) When pressure is extremely low : For one mole of gas,
nRT
PV
Z 
CO2
O2
N2
CH4
P
H2
He

232 Gaseous State
RTbV
V
a
P 





 )(2
or 2
V
ab
Pb
V
a
RTPV 
(ii)When pressure is extremely high : For one mole of gas,
PbRTPV  ;
RT
Pb
RT
PV
 1 or
RT
Pb
Z  1
where Z is compressibility factor.
(iii) When temperature is extremely high : For one mole of gas,
RTPV  .
(iv) When pressure is low : For one mole of gas,
RTbV
V
a
P 





 )(2
or 2
V
ab
V
a
PbRTPV 
or
VRT
a
RT
PV
 1 or
VRT
a
Z  1
(v) For hydrogen : Molecular mass of hydrogen is small hence value
of 'a' will be small owing to smaller intermolecular force. Thus the terms
V
a
and 2
V
ab
may be ignored. Then Vander Waal's equation becomes,
PbRTPV  or
RT
Pb
RT
PV
 1
or
RT
Pb
Z  1
In case of hydrogen, compressibility factor is always greater than
one.
(5) Merits of Vander Waal's equation
(i) The Vander Waal's equation holds good for real gases upto
moderately high pressures.
(ii) The equation represents the trend of the isotherms representing
the variation of PV with P for various gases.
(iii) From the Vander Waal's equation it is possible to obtain
expressions of Boyle's temperature, critical constants and inversion
temperature in terms of the Vander Waal's constants 'a' and 'b'.
(iv) Vander Waal's equation is useful in obtaining a 'reduced
equation of state' which being a general equation of state has the advantage
that a single curve can be obtained for all gases when the equation if
graphically represented by plotting the variables.
(6) Limitations of Vander Waal's equation
(i) This equation shows appreciable deviations at too low
temperatures or too high pressures.
(ii)The values of Vander Waal's constants a and b do not remain
constant over the entire ranges of T and P, hence this equation is valid only
over specific range of T and P.
(7) Other equations of state : In addition to Vander Waal's equation,
there are also equations of state which have been used to explain real
behaviour of gases are,
(i) Clausius equation : RTbV
cVT
a
P 






 )(
)( 2
. Here 'c' is
another constant besides a, b and R.
(ii) Berthelot equation : RTbV
TV
a
P 





 )(2
.
(iii) Wohl equation : 2
)()( V
c
bVV
a
bV
RT
P 




(iv) Dieterici equation : RTVa
e
bV
RT
P /
. 

 .
The expression is derived on the basis of the concept that molecules
near the wall will have higher potential energy than those in the bulk.
(v) Kammerlingh Onnes equation : It is the most general or
satisfactory expression as equation expresses PV as a power series of P at a
given temperature.
......32
 DPCPBPAPV
Here coefficients A, B, C etc. are known as first, second and third
etc. virial coefficients.
(a) Virial coefficients are different for different gases.
(b) At very low pressure, first virial coefficient, A = RT.
(c) At high pressure, other virial coefficients also become important
and must be considered.
The critical state
(1) A state for every substance at which the vapour and liquid states
are indistinguishable is known as critical state. It is defined by critical
temperature and critical pressure.
(2) Critical temperature (Tc
) of a gas is that temperature above
which the gas cannot be liquified however large pressure is applied. It is
given by,
Rb
a
Tc
27
8

(3) Critical pressure (Pc
) is the minimum pressure which must be
applied to a gas to liquify it at its critical temperature. It is given by,
2
27b
a
Pc 
(4) Critical volume (Vc
) is the volume occupied by one mole of the
substance at its critical temperature and critical pressure. It is given by,
bVc 3
(5) Critical compressibility factor (Zc
) is given by,
375.0
8
3

c
cc
c
RT
VP
Z
A gas behaves as a Vander Waal’s gas if its critical compressibility
factor )( cZ is equal to 0.375. A substance is the gaseous state below cT is
called vapour and above cT is called gas.
Degrees of freedom of a gaseous molecule
(1) The motion of atoms and molecules is generally described in
terms of the degree of freedom which they possess.
(2) The degrees of freedom of a molecule are defined as the
independent number of parameters required to describe the state of the
molecule completely.
(3) When a gaseous molecule is heated, the energy supplied to it
may bring about three kinds of motion in it, these are,
(i) The translational motion (ii) The rotational motion
(iii) The vibrational motion.
This is expressed by saying that the molecule possesses translational,
rotational and vibrational degrees of freedom.
(4) For a molecule made up of N atoms, total degrees of freedom =
3N. Further split up of these is as follows,
Translational Rotational Vibrational
For linear molecule : 3 2 3N – 5
For non-linear molecule : 3 3 3N – 6
Specific and Molar heat capacity of gases
(1) Specific heat (or specific heat capacity) of a substance is the
quantity of heat (in calories, joules, kcal, or kilo joules) required to raise the
temperature of 1g of that substance through Co
1 . It can be measured at
constant pressure )( pc and at constant volume )( vc .
(2) Molar heat capacity of a substance is the quantity of heat
required to raise the temperature of 1 mole of the substance by Co
1 .

Gaseous State 233
 Molar heat capacity = Specific heat capacity  Molecular weight,
i.e.,
McC vv  and McC pp  .
(3) Since gases upon heating show considerable tendency towards
expansion if heated under constant pressure conditions, an additional energy
has to be supplied for raising its temperature by Co
1 relative to that
required under constant volume conditions, i.e.,
vp CC  or )(expansion,ondoneWork RVPCC vp 
where, pC molar heat capacity at constant pressure; vC
molar heat capacity at constant volume.
(4) Some useful relations of Cp
and Cv
(i) JcaloriesRCC vp 314.82 
(ii) RCv
2
3
 (for monoatomic gas) and xCv 
2
3
(for di and
polyatomic gas), where x varies from gas to gas.
(iii) 
v
p
C
C
(Ratio of molar capacities)
(iv) For monoatomic gas caloriesCv 3 whereas,
caloriesRCC vp 5
(v) For monoatomic gas, 66.1
2
3
2
5
)( 
R
R
C
C
v
p
 .
(vi) For diatomic gas 40.1
2
5
2
7
)( 
R
R
C
C
v
p

(vii) For triatomic gas 33.1
6
8
)( 
R
R
C
C
v
p

Liquefaction of gases
(1) A gas may be liquefied by cooling or by the application of high
pressure or by the combined effect of both. The first successful attempt for
liquefying gases was made by Faraday.
(2) Gases for which the intermolecular forces of attraction are small
such as 2H , 2N , Ar and 2O , have low values of cT and cannot be
liquefied by the application of pressure are known as “permanent gases”
while the gases for which the intermolecular forces of attraction are large,
such as polar molecules 3NH , 2SO and OH2 have high values of cT
and can be liquefied easily.
(3) Methods of liquefaction of gases : The modern methods of cooling
the gas to or below their cT and hence of liquefaction of gases are done by
Linde's method and Claude's method.
(i) Linde's method : This process is based upon Joule-Thomson
effect which states that “When a gas is allowed to expand adiabatically from
a region of high pressure to a region of extremely low pressure, it is
accompained by cooling.”
(ii)Claude's method : This process is based upon the principle that
when a gas expands adiabatically against an external pressure (as a piston in
an engine), it does some external work. Since work is done by the molecules
at the cost of their kinetic energy, the temperature of the gas falls causing
cooling.
(iii) By adiabatic demagnetisation.
(4) Uses of liquefied gases : Liquefied and gases compressed under a high
pressure are of great importance in industries.
(i) Liquid ammonia and liquid sulphur dioxide are used as
refrigerants.
(ii)Liquid carbon dioxide finds use in soda fountains.
(iii) Liquid chlorine is used for bleaching and disinfectant purposes.
(iv) Liquid air is an important source of oxygen in rockets and jet-
propelled planes and bombs.
(v) Compressed oxygen is used for welding purposes.
(vi) Compressed helium is used in airships.
(5) Joule-Thomson effect : When a real gas is allowed to expand
adiabatically through a porous plug or a fine hole into a region of low
pressure, it is accompanied by cooling (except for hydrogen and helium
which get warmed up).
Cooling takes place because some work is done to overcome the
intermolecular forces of attraction. As a result, the internal energy decreases
and so does the temperature.
Ideal gases do not show any cooling or heating because there are no
intermolecular forces of attraction i.e., they do not show Joule-Thomson
effect.
During Joule-Thomson effect, enthalpy of the system remains
constant.
Joule-Thomson coefficient. HPT )/(  .
For cooling, ve (because dT and dP will be ve )
For heating ve (because ), vedPvedT  .
For no heating or cooling 0 (because )0dT .
(6) Inversion temperature : It is the temperature at which gas shows
neither cooling effect nor heating effect i.e., Joule-Thomson coefficient
0 . Below this temperature, it shows cooling effect and above this
temperature, it shows heating effect.
Any gas like HeH ,2 etc, whose inversion temperature is low
would show heating effect at room temperature. However, if these gases are
just cooled below inversion temperature and then subjected to Joule-
Thomson effect, they will also undergo cooling.
 If the number of molecules present in 1 c.c. of the gas or vapour
at S.T.P., then that is called loschmidt number. Its value is 2.687
 1019
per c.c.
 CO2
> SO2
> SO3
> PCl3
is order of rate of diffusion.
 Vapour density is independent of temperature and has no unit
while absolute density is dependent of temperature and has unit
of gm–1
 The isotherms of CO2
were first studied by Andrews.
 1 Cal = 4.2 Joule, 1 Kcal = 4200 Joule
 The gas which has least mean free path has maximum value of a, is
easily liquefied and has maximum value of Tb
.
 Tc
< Tb
< Ti
 For critical constants, compression factor Z is < 1.
 Boyle’s law and Avogadro’s law are applicable under limiting
condition. This limiting condition is P  0.
 Tc
= 0.296 Tb
; Ti
= 6.75 Tc

234 Gaseous State
 Mean free path increases if H2
is replaced by He.

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