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1. Chemistry Teacher
KVS

2. Introduction to Gaseous State
some points to remember
 Gases do not have a fix volume.
 They assume the volume of the container in which
they are enclosed.
 The molecules tend have no packing.
 Gases have very low relative density.
 The attractive forces between individual molecules
of a gas is negligible.
 The molecules tend have very high Kinetic &
Thermal Energy.
 Gases have very high diffusion rate.
 Gases are “highly compressible”.

3. THE GAS LAWS
1. Boyle’s law
2. Charles law
3. Gay Lussac’s law
4. Avogadro’s law
5. Combined Gas law
6. Dalton’s law
7..Kinetic Molecular Theory
8. Behavior of Real Gases
9. Liquefaction of Gases

4. 1.BOYLE’S Law (p-V relationship)
This law was proposed by Robert Boyle in 1662.
At constant temperature, the pressure of a fixed amount of gas
varies inversely with its volume.
Mathematically-
p=k1*1/V , at constant T & n. .......(1.1)
where k1 is the proportionality constant.
The value of k1 depends upon the amount of gas, its temperature and the unit
in which p and V are expressed.
From equation (1),
pV=k1 .……(1.2)

5. Pressure – volume variation graphs
.
Fig1.1 p versus V graph Fig1.2 p versus V graph at different temperature
Each curve corresponds to a different constant temperature and are known as
ISOTHERM.

6. Fig1.3 p versus 1/V graph.

7. Finding out pressure (or volume) of a gas at same temperature
under different pressure (or volume)
Let V1 be the volume of a given mass of a gas at pressure p1 and at a given
temperature T. When the pressure is changed to p2 at the same temperature,
the volume changes to V2.
Then according to Boyle’s law,
p1V1=p2V2=constant (k) ……..(1.3)
or p1/p2=V1/V2 ……..(1.4)
The equation (1.4) can used to any of the four quantity if know any three of
them.

8. 2.CHARLES’ LAW (V-T RELATIONSHIP)
 At constant pressure, the volume of a fixed amount of
gas is directly proportional to its “absolute
temperature.”
or
 The volume of a given mass of a gas, at constant
pressure, increases or decreases by 1/273.15 times of he
volume of gas at 0°C, for each one degree rise or fall in
absolute temperature respectively.
V=k2T ………(2.1)
V/T=k2 ..…….(2.2)
where k2 is the proportionality constant.

9. Let Vo be the volume of a given mass of a gas at 0°C and Vt is its volume at any temperature
t°C, then the volume, Vt may be written in terms of Charles’ law(at constant pressure) as:
For 1 degree rise in temperature, volume increases=Vo*1/273.15
For t degree rise in temperature, volume increases=Vo*t/273.15
Therefore, volume at t°C, Vt = initial volume + increase in volume
= Vo + Voxt/273.15
= Vo[1+t/273.15] ..……….(2.3)
= Vo[(273.15+t)/273.15]
= VoxT(K)/To {T(K)=t + 273.15} …………(2.4)
where T(K) is the absolute temperature of the gas and To{=273.15(K)} corresponds
to 0°C on absolute temperature scale.
Vt = Vo*T/To
Vt/Vo= T/To
Vt/T = Vo/To
Vt/T = k2
**NOTE:- At t=-273.15°C,the volume the gas of tends to become zero, which is physically
unviable. This is because all the gases convert into liquid state before reaching this temperature
& hence such an absurd answer is received.

10. VOLUME VERSUS TEMPERATURE GRAPH
Fig2.1 V versus T graph Fig2.2V versus T graph at different pressure
Each line of the volume vs temperature graph is called ISOBAR.

11. 3.GAY LUSSAC’S Law (p-T RELATIONSHIP)
This law was put forward by Gay Lussac.
At constant volume, the pressure of a fixed amount of gas
is directly proportional to its absolute temperature.
Mathematically-
p=k3T ……….(3.1)
p/T=k3 ……….(3.2)
p1/t1=p2/t2 ………….(3.3)

12. Each line in the pressure versus absolute
temperature graph is called ISOCHORE.
Fig3.1 p versus T graph

13. This law was derived by Amedeo Avogadro.
 At a constant pressure and temperature, the volume of a gas is
directly proportional to its amount.
Mathematically-
V=k4n ……….(4.1)
where k4 is proportionality constant and n is the number of moles of the gas.
n=6.022*10²³ and is known as the Avogadro’s constant.
Thus,
V=k4*m/M ………..(4.2)
where m is the mass of the gas and M is the molar mass of the gas.

14. 5.The COMBINED GAS Law
Combining the three gas laws,
V=k1/p ……… Boyle’s law(1)
V=k2T ……... Charles’ law(2)
V=k4n ……… Avogadro’s law(3)
We arrive at the combined gas law which can be written as:
V=knT/p ……….(5.1)
Or pV=knT=kRT ……….(5.2)
pV=kRT ………..(5.3)
where R is the Universal Gas constant.
pV=nRT is also called the ideal gas equation.

15. 5.1.The nature of the Universal Gas constant ‘R’
The unit of the Universal gas constant can be found out from the ideal gas
equation-
pV= nRT
R=pV/nT
since p is Force/unit Area
R=(F/A * V)/n*T ……….(5.1.1)
where F-force, A-area, V-volume, n-moles and T-absolute temperature.
since A=(length)² and V=(length)³.
R=F*L/n*T ………..(5.1.2)
since F*L=Work done,
R=Work/n*T ………..(5.1.3)
Thus, R represents work done per degree per mol or energy per degree per mol.

16. 5.2.The numerical values of R
I. When pressure is expressed in atmosphere and volume in litres, the value of
R is in atmosphere-litre per mole.
At S.T.P., the pressure is 1 atm, volume is 22.4 L and the temperature is 273.15 K.
Therefore,
R=1 atm*22.4 L/(1 mole*273.15 K)
R=0.0827 l atm /(K mol) ……..(5.2.1)
II. When pressure is expressed in Pascal and volume in m³, then R is expressed
as Joule per degree per mol.
R=(100000Pa*22.7m³)/(1 mol*273.15K)
=8.314 Nm/(K mol)
R=8.314 J/(K mol) ……..(5.2.2)

17. III. When the pressure is expressed in bar and volume in dm³, then R is
expressed in bar dm³ per mol per K.
R=(1 bar*22.7 dm³)/(1 mol*K)
R=0.0837 bar dm³/(mol*K) ………..(5.2.3)
NOTE:- For pressure-volume calculations, R must be taken in the same units as
those for pressure and volume.

18. This law was formulated by John Dalton in 1801.
 The total pressure exerted by a mixture of two or more non-reacting
gases in a definite volume is equal to the sum of the partial
pressure of the individual gas.
Therefore, Dalton’s law can be stated as-
ptotal=p1+p2 ………..(6.1)
the same can be extended to ‘n’ number of gases
ptotal=p1+p2+p3+……+pn .………(6.2)
Partial pressure:- The partial pressure of each gas means the pressure
which that gas would exert if the present alone in the container at the
same and constant temperature as that of the mixture

19. Let us take a mixture of two gases(non-reacting) in a container of volume V at
temperature T.
Then ,
p1=n1RT/V
p2=n2RT/V
p3=n3RT/V
where n1,n2 and n3 are the number of moles of the three gases and p1, p2 and p3
are the partial pressure of the gases.
According to Dalton's law-
ptotal=p1+p2+p3
=n1RT/V+n2RT/V+n3RT/V
=(n1+n2+n3)RT/V ………..(6.3)
Dividing equation p1 the whole equation by ptotal, we get
P1/ptotal=[n1/(n1+n2+n3)]RTV/RTV
=n1/(n1+n2+n3)
=n1/n
=x1
Where n=n1+n2+n3
Similarly for the other two gases,
p2=x2*ptotal , p3=x3*ptotal
Generally, pi=xi*ptotal ………….(6.4)

20. 7.The KINETIC MOLECULAR Theory of
GASES
Assumptions of the Kinetic Molecular theory of Gases-
• All the gases are composed of tiny particles called molecules.
• There is negligible intermolecular force of attraction between the
molecules of gases.
• The molecules of a gas are always in constant random motion.
• The molecules of a gas do not have any gravitational force.
• The pressure of the gas is due to the collision of gas molecules on the
walls of the container.
• The volume of the gas molecules is negligible in comparison to the
total volume of the gas.
• The volume of the gas is directly proportional to the absolute
temperature of the gas. (V=kT)

21. According to the Kinetic Molecular theory of gases:
The Kinetic Gas equation is….
pV=mNurms²/3 ……..(7.1)
where urms is called the root mean square velocity of the molecules.
Average translational kinetic energy of a molecular temperature T is
=murms²/2
The total energy of the whole gas containing N molecules is given by-
Ek=mNurms²/2 …….(7.2)
And the Kinetic Gas equation is
pV=mNurms²/2
Therefore,
pV=2mNurms²/(2*3)
pV=2Ek/3 ………(7.3)
Comparing with the ideal gas equation
pV=RT for one mole of gas
Therefore,
RT=2Ek/3 or Ek=3RT/2 ……….(7.4)

22. Similarly, Kinetic Energy for n moles of the gas
Ek=3nRT/2 ………(7.5)
The average kinetic energy per molecule can obtained by both sides of
the equation by Avogadro number (6.022*10²³) of molecules, NA
The average kinetic energy for one molecule=3RT/2*NA
=3kbT/2 ………(7.6)
where kb=R/NA and is called the Boltzmann constant.
Verification of the gas laws by Kinetic Molecular theory of gases.
 Charles’ Law
K.E.=mu²/2
pV=mNu²/3
pV=mu²/2 (N=1)
therefore, pV=2K.E./3
At constant p,
V=2kt/3 (K.E.=kT)
or V=kT (2k/3=constant)

23.  Boyle’s Law
pV=2kT/3
At constant temperature,
pV=k (2kT/3=constant)
or p=k/V , which is the Boyle’s Law.

24. 8.Behavior of
Real gases:
Deviation from IDEAL GAS Behaviour
The gases which obey the ideal gas equation at all temperatures and pressures is
called on ideal gas or perfect gas.
The gases which do not the ideal gas equation at all temperatures and pressures is
called a real gas or non-ideal gas.
Causes for deviation from ideal behavior:-
 Deviation in pressure values
They show deviation because molecules tend to interact with each other. At high
pressures molecules of gases are very close to each. Molecular interactions start
to operating. At high pressures, molecules do not strike the walls of the
container wit full impact because of attraction from other molecules.
This reduces the total pressure of the gas on the walls of the container. Thus, they
exert less pressure than an ideal gas.
pideal=preal + an²/V³ ……….(8.1)
where n is the no. of moles of the gas and V is the volume of the gas and a is a
measure of the intermolecular attractive forces within the gas and is
independent of the temperature and pressure.

25.  Deviation in the value of volume
At high pressures, the volume of the individual gas become significant.
Repulsive interactions are short range interactions and are significant when
molecules are almost in contact.
The repulsive force causes the molecules to behave as small but impenetrable
spheres. As such the volume occupied by the molecules gets reduced. Now they
tend to move in volume V-nb where nb is the total volume occupied by
molecules themselves.
Vreal= Videal – nb .………..(8.2)
Here b is a constant.
Now the ideal gas equation along with the corrections becomes:
[p + an²/V²](V-nb)=nRT ……….(8.3)
This equation is known as van der Waals equation.

26. 8.1.Deviation:-Compressibility factor (Z)
The deviation from ideal behavior can also be expressed in terms of the
compressibility factor of the gas Z, which is the ratio of product pV and nRT.
Mathematically,
Z = pV/nRT …………(8.1.1)
For ideal gas Z=1 at all temperatures and pressures because pV=nRT.
It can be seen that for gases which
deviate from ideality , value of Z
deviates from unity.

27. At very low pressure all gases shown in the graph have Z≈1 and behave as ideal gas. At high
pressure all the gases have Z>1. These are more difficult to compress. At intermediate
pressures, most gases have Z<1.
Thus gases show ideal behavior when the volume occupied is so large that the volume of
the molecules can be neglected in comparison to it.
Some points to remember:-
 At low pressures all the gases show ideal behavior.
1
 At high pressures gases have Z> and is called negative deviation.
1
 At intermediate pressures gases have Z< and is called positive deviation.
Def:- The temperature at which a real gas obeys ideal gas law over an appreciable range
of pressure is called Boyle temperature or Boyle point.
The pressure till which a gas shows ideal gas law, is dependent on the nature of the gas and
its temperature.
The Boyle point of a gas is depends upon the nature of the gas. Above Boyle point
real gases show positive deviation fro ideality and Z values are greater the unity.
Below Boyle point real gases first show decrease in Z value with increase in
pressure, which reaches a minimum value. On further increase in pressure the
value of Z increases continuously.

28. We can also see further from the following derivation
Z=pVreal/n*RT
If the gas shows ideal behavior then
videal=nRT/p
Putting this value in (8.1.2) we have
Z=Vreal/Videal
From (8.1.3) we can see that the ratio of the actual volume of a gas to the molar
volume of it, if it were an ideal gas at that temperature and pressure.
Exception:- We have seen that the behavior of the H2 and He is always increasing.
This is due to the fact that ‘a’ for Hydrogen and He is very small indicating that forces of attraction in
these gases are very weak. Therefore, ‘a’/V²is negligible at all pressures so that Z is always greater
than unity.

29. Difference between a Real gas and an Ideal gas:-
IDEAL GAS REAL GAS
1. Ideal gas obeys all gas laws under all 1. Real gas obeys gas laws only under low
conditions of temperature and pressure. pressure and high temperature.
2. In ideal gas, the volume occupied the 2. In real gas, the volume occupied by the
molecules is negligible as compared to molecules is significant in comparison
the volume occupied by the gas. to the total volume occupied by the gas.
3. The force of attraction among the 3. The force of attraction among
molecules of gas are negligible. the molecules are significant at all
temperatures and pressures.
4. It obeys the ideal gas equation: 4. It obeys the van der Waals equation:
pV=nRT [p + an²/V²](V – nb) = nRT

30. According to the Kinetic Molecular theory of gases, the forces of attraction
between the molecules are negligible. When the temperature is lowered, the
K.E. of the molecules decreases. As a result the slow molecules tend to come
nearer to one other. At a sufficiently low temperature, some of the molecules
cannot resist the force of attraction and they come closer and ultimately the
gas changes its state into liquid. They can also be brought closer by increasing
the pressure as the volume decreases.
Thus the gases can be liquefied by increasing the pressure or decreasing
the temperatures or by both.
Thomas Andrews showed that at high temperatures isotherms look like that of
an ideal gas and the gas cannot be liquefied by even at very high pressures. But
as the temperature is lowered, the shape of the of the curves changes
considerably(Fig9.1). He used CO2 as the gas to show this property of the
gases.
From the graph(Fig 9.1) it can be seen that CO2 remains gas up to 73 atm
pressure at 30.98°C. This is the highest temperature at liquid CO2 is observed.
Above this temperature it remains as a gas.

32. Fig9.3 graph showing thermal equilibrium b/w Fig9.4Graph showing the compressibility
gas and liquid. factor of a real gas.

33. Definitions:-
1. Critical temperature:- It is the highest temperature at which a gas converts to a liquid.
2. Critical Volume:- The volume of one mole of gas at critical temperature is called
critical volume.
3. Critical pressure:- the pressure at the critical temperature is called critical pressure.
4. Critical constants:- The critical temperature, pressure and volume are called critical
constants.
Explaining the graph-
The steep line in the graph(fig 9.1 & 9.2) represents the isotherms of liquid.
a. Increasing pressure:-Further increase in pressure just compresses the liquid
CO2 and the curve represents the compressibility of the liquid. Even a slight
compression results in rise in pressure indicating low compressibility of the
liquid.
Below 30.98°C, the behavior of the gas on compression is quite different. At
21.5°C, CO2 remains as a gas only upto point B. at point B, liquid of a
particular volume appears. Further compression does not change the
pressure.
b. Condensation :-Liquid and gaseous CO2 co-exist and further application of
pressure causes condensation of more gas until the point C is reached. At
point c all the gas has been condensed.

34. c. Phase diagram:-
Below 30.98°C, the length of each curve increases. As is clear from the diagram,
the central horizontal line (FG) at 13.1 is larger then that at 21.5°C (BC).
Alternatively, we observe that the horizontal portion becomes smaller as the
temperature increases. At critical point A, we can say that it represents the
gaseous state.
Point C:- It represents the completion of the condensation process.
Point D:- it represents the liquid state.
The area under the point D represents the co-existence of liquid and gaseous
CO2 on equilibrium.
Points to understand:-
• There is always a certain temperature above which a gas cannot be
liquefied, no matter how much pressure is applied.
• The gas can only be liquefied below this temperature.
• Every gas has a critical pressure which is dependent on its temperature.

35. Substance Tc (K) Pc (bar) Vc (dm ³/mol)
H₂ 33.2 12.97 0.0650
He 5.3 2.29 0.0577
N₂ 126.0 33.9 0.0900
O₂ 154.3 50.4 0.0744
CO₂ 304.10 73.9 0.0956
H₂O 647.1 220.6 0.0450
NH₃ 405.5 113.0 0.0723

36. BE THOROUGH WITH THE CHAPTER