◊ STATES OF MATTER

Matter can be classified into three main divisions, namely, gases, liquids .and solids

Gases: have neither shape nor size but fill completely any space into which they are introduced. All gases mix completely and a gas•
exerts pressure equally in all directions. The volume of a gas Changes corresponding to changes in temperature and pressure.
Liquids: have a definite volume but no shape.
Solids: have both size and shape.

The Gaseous State

It is the most disordered state of matter. Characteristics of this state of matter are:

(i) In gases, the intermolecular forces are weakest.

(ii) Gases are highly compressible.

(iii) Gases exert pressure equally in all directions.

(iv) Gases have much lower density than the solids and liquids.

(v) The volume and the shape of gases are not fixed.

(vi) Gases mix evenly and completely in all proportions without any mechanical aid.

Measurable Properties of Gases

(i) Mass It is expressed in gram or kg.

(ii) Volume It is equal to the volume of the container and is expressed in terms of litre (L), millilitre (mL), cubic centimeter (cm3),
cubic meter (m3) or cubic decimeter (dm3).

1 L = 1000 mL = 1000 cm3 = 1 dm3

1 m3= 103 dm3 =106 cm3= 106 mL = 103 L

(iii) Pressure Gas pressure is measured with manometer and atmospheric pressure is measured by barometer.

1 atm = 76 cm of Hg = 760 mm of Hg = 760 torr

1 atm = 101.325 kPa = 101325 Pa = 101.325 Nm−2

= 1.01325 bar

1 bar =105 Pa = 0.987 atm

Measurement of pressure of gas

(a) Open end manometer, pgas = patm − h

(b) Closed end manometer, pgas = h

where h is difference in the mercury levels in the two columns of density (d) (of a gas).

(iv) Temperature It is measured in Celsius scale (°C) or in Kelvin scale (K). SI unit of temperature is kelvin (K).

T (K) = t° (C) + 273

Standard temperature and pressure (STP or NTP) mean 273.15 K (0°C) temperature and 1 bar (i.e., exactly 105 pascal) pressure.

At STP, molar volume of an ideal gas is 22.71098 L mol-1.

◊ Gas Laws
Boyle’s Law (1662)
The volume of a given mass of a gas is inversely proportional to its pressure at constant temperature.
 𝟏
V∝ or Vp = K
𝑷

K is a constant and its value depends on mass, temperature and nature of gas.

∴ p1V1 = p2V2

Graphical Representation of Boyle’s Law

𝟏
Graphs of p V or p or pV p at constant temperature are known as isotherms.
𝑽

Air is dense at the sea level

because it is compressed by the mass of air above it.

Charles’ Law (1787)

𝟏
The volume of the given mass of a gas increases or decreases by of its volume for each degree rise or fall of temperature
𝟐𝟕𝟑
respectively at constant pressure.
𝒕
Vt = V0 (1 + ) at constant p
𝟐𝟕𝟑

Or

The volume of a given mass of a gas is directly proportional to the absolute temperature at constant pressure.
𝑽 𝑽𝟏 𝑽𝟐
V ∝ T (at constant p), = constant or =
𝑻 𝑻𝟏 𝑻𝟐

Absolute zero is the theoretically possible temperature at which the volume of the gas becomes zero. It is equal to 0°C or 273.15
K.

Graphical Representation of Charles’ Law

A graph of V vs T at constant pressure is known as isobar.

Charles’ law explains that gases expand on heating, so hot air is less dense than cold air.

Gay Lussac’s Law (1802)

𝟏
The pressure of a given mass of gas increases or decreases by of its pressure for each degree rise or fall of temperature
𝟐𝟕𝟑
respectively at constant volume.
𝒕
pt = p0 (1 + ) at constant V and n
𝟐𝟕𝟑

The pressure of a given mass of a gas at constant volume is directly proportional to absolute temperature.
𝑷 𝑷𝟏 𝑷𝟐
p ∝ T or p = KT or = K at constant V and n or =
𝑻 𝑻𝟏 𝑻𝟐
Graphical Representation of Gay Lussac’s Law

A graph of pvs T at constant volume is known as isochore.

Avogadro’s Law
It states that equal volumes of all gases under the same conditions of temperature and pressure contain equal number of
molecules.

Mathematically

V∝n (at constant T and p)

𝑽 𝒎
Or =K [ n = number of moles, n = ]
𝒏 𝑴

Molar gas volume

The volume of one mole of a gas, i.e. 22.4 L at STP (0°C,1atm) is known as molar gas volume.

Ideal Gas Equation

𝟏
V∝ T and n constant (Boyle’s law)
𝑷

V∝T p and n constant (Charles’ law)

V∝n p and T constant (Avogadro’s law)

𝒏𝑻
⇒ V∝
𝑷

Or pV ∝ nT

Or pV = nRT.

This is known as ideal gas equation. R is known as universal gas constant. From the ideal gas equation, density,
𝐩𝐌
d= (where, M = molecular mass)
𝐑𝐓

Numerical Values of R
(i) R = 0.0821 L atm mol−1 K−1

(ii) R = 0.083 L bar mol−1 K−1

(iii) R = 8.314 J K−1 mol−1

(iv) R = 8.314×107 erg K−1 mol−1

(v)R=1.987 or 2 cal K−1 mol−1

Ideal gas The gas which obeys the equation pV = nRT at every temperature and pressure range strictly is known as ideal gas.

Real gases Since none of the gases present in universe strictly obey the equation pV = nRT, hence they are known as real or non-
ideal gases. Real gases behave ideally at low p and high T.
Graham’s Law of Diffusion
Under similar conditions of temperature and pressure, the rates of diffusion (or effusion) of different gases (under identical
conditions) are inversely proportional to the square root of their densities (or mole weights).

𝒓𝟏 𝒅𝟐 𝑴𝟐
Mathematically, =√ =√
𝒓𝟐 𝒅𝟏 𝑴𝟏

Since 1 mole of gas, i.e., M grams, occupies 22.4 dm3 at s.t.p them the density of d of a gas at s.t.p is
𝑴
d= g dm-3
𝟐𝟐.𝟒

Hence, density may be replaced by the mole weights of the gas.

The rate of effusion is the volume of gas escaping in unit time. The time taken for a given volume of gas to effuse is inversely
proportional to the rate of effusion. Hence
𝐓𝐢𝐦𝐞 𝐭𝐚𝐤𝐞𝐧 𝐟𝐨𝐫 𝟏 𝐯𝐨𝐥𝐮𝐦𝐞 𝐨𝐟 (𝐠𝐚𝐬 𝟏) 𝐭𝐨 𝐞𝐬𝐜𝐚𝐩𝐞 𝐫𝐚𝐭𝐞 𝐨𝐟 𝐞𝐟𝐟𝐮𝐬𝐢𝐨𝐧 𝐨𝐟 (𝐠𝐚𝐬 𝟐)
𝐓𝐢𝐦𝐞 𝐭𝐚𝐤𝐞𝐧 𝐟𝐨𝐫 𝟏 𝐯𝐨𝐥𝐮𝐦𝐞 𝐨𝐟 (𝐠𝐚𝐬 𝟐) 𝐭𝐨 𝐞𝐬𝐜𝐚𝐩𝐞
= 𝐫𝐚𝐭𝐞 𝐨𝐟 𝐞𝐟𝐟𝐮𝐬𝐢𝐨𝐧 𝐨𝐟 (𝐠𝐚𝐬 𝟏)

𝒅𝟐 𝑴𝟐
=√ =√
𝒅𝟏 𝑴𝟏

Effusiometer
This instrument enables the times required by different gases to escape,
under identical conditions, through a small hole in a metal plate to be
compared.

The apparatus is filled with the gas under test using the three-way tap, and
placed in a thermostatically controlled bath.

When a steady temperature has been obtained, the three-way tap T is

adjusted to allow the gas under test to escape through the fine hole in the
thin platinum disc at the upper end of the apparatus.

The time required for the liquid to rise from the lower to the upper mark is noted. The experiment is repeated using a different
gas at the same temperature. The liquid used in the thermostat bath should not, of course, react with or dissolve either gas.

In effusion, a gas is made to pass through a small orifice under pressure. Diffusion relates to the mixing of two gases at the same
pressure. In spite of the fact that gas molecules have high velocities, diffusion is a slow process. This is due to the fact that each
molecule is, at ordinary pressures, involved continually in colliding with other molecules, thus greatly restricting the rate of
forward movement. However, the rate of bulk motion of a gas does depend on molecular velocities (and also on the mean free
path of the molecules) which agrees with Graham's law, that the rate of diffusion, is inversely proportional to the square root of
the density of the gas. Since the rates of effusion and diffusion depend on gas densities, both methods are employed in the
separation of gaseous Isotopes.

MEAN FREE PATH

The average distance traversed by a gas molecule between successive collisions is termed its mean free path. This quantity is
inversely proportional to:

(i) the number of molecules per unit volume;

(ii) the square of the diameter of the molecule.

An estimation of molecular diameters can be made by measuring the rates of diffusion which are related to the mean free path.
At s.t.p. the mean free path of a molecule in a gas is of the order of 10-5 cm. At low pressures, the number of molecules per unit
,volume is much less, and at 10-5 atm the mean free path is about 1 cm.

Diffusion is the tendency of gases to distribute itself uniformly throughout the available space while effusion is the moveme nt of
gas through a small hole when it is subjected to pressure.
CALCULATIONS BASED ON THE GAS LAWS
Errors in these calculations usually arise from:

1.Using inconsistent units on opposite side of the equation

𝑷𝟏 𝑽 𝟏 𝑷𝟐 𝑽 𝟐
. =
𝑻𝟏 𝑻𝟐

(For example, if P1 is given in N m-2, P2 must be in the same units).

2.Not converting temperatures to the absolute scale.

3.Either omitting or using an incorrect value of n in the equations PV=nRT.

Dalton’s Law of Partial Pressure

At constant temperature, the total pressure exerted by a mixture of non-reacting gases is the sum of partial pressures of different
gases present in the mixture.

p = p 1 + p2 + p 3 + …

Partial pressure of a gas = mole fraction of the gas × total pressure.

If n1, n2 and n3 are moles of non-reacting gases filled in a vessel of volume V at temperature T, the total pressure, p is given by

P = (n1 + n2 + n3) RT/V

This is the equation of state of a gaseous mixture.

Pressure of a dry gas can be determined by Dalton’s law. When a gas is collected over water, its observed pressure is equal to the
sum of the pressure of dry gas and the pressure of water vapour (aqueous tension)

then pressure of moist gas = pressure of dry gas + aqueous tension.

Aqueous tension It is the pressure exerted by water vapours at a particular temperature. It depends upon temperature.

Kinetic Theory of Gases

Main assumptions of this theory are:

1. A gas consists of large number of small particles, called molecules.

2. Volume occupied by gas molecules is negligible as compared to the total volume of the gas.

3. There is continuous rapid random motion of gas molecules. The molecules collide with each other and with the walls of
container.

4. The molecules are perfect elastic bodies and there is no loss of kinetic energy during collisions.

5. There are no attractive forces between the gaseous molecules.

6. The pressure exerted by a gas is due to the bombardment of gas molecules against the walls of the container.

7. The different molecules possess different velocities and hence, different energies. The average KE is directly proportional to
absolute temperature.
𝟑
KE= RT
𝟐

𝟑
∴ Average kinetic energy per molecule= kT
𝟐

Here, k is Boltzmann constant, it is gas constant per molecule.

𝑹
K= = 1.38 x 10-23 J K−1 mol−1
𝑵𝑨
From the above postulates, the kinetic gas equation derived is
𝟏
pV = mnU2
𝟑

𝟑𝑹𝑻
where, U = root mean square velocity = √
𝑴

Velocities of Gas Molecules

The different velocities possessed by gas molecules are:

(i) Most probable velocity (α) It is the velocity possessed by

maximum fraction of gas molecules at a particular
temperature.

𝟐𝑹𝑻
α= √
𝑴

(ii) Average velocity (ῡ) This is the average of the different

velocities of all the molecules.

𝟖𝑹𝑻
ῡ=√ c
π𝑴

(iii) Root mean square velocity (Urms) It is the square root of the mean of the square of the different velocities of the molecules.

𝒏𝟏 𝑐 2 1 +𝒏𝟐 𝑐 2 2 +𝒏𝟑 𝑐 2 3 +⋯
Urms = √
𝒏𝟏 +𝒏𝟐 +𝒏𝟑 +⋯

Mathematically, U = √
𝟑𝑹𝑻
𝑴
=√
𝟑𝒑𝑽
𝑴
=√
𝟑𝒑
𝒅
[ √𝟑𝒑𝒅 = √𝟑𝑹𝑻
𝑴
]
α: ῡ: U = 1:1.128:1.224

Deviation from Ideal Behaviour

At high pressure and low temperature, the gases deviate considerably from the ideal behaviour. Deviation can be expressed in
terms of compressibility factor (Z), expressed as
𝒑𝑽
Z=
𝒏𝑹𝑻

In case of ideal gas, pV = nRT , Z = 1

In case of real gas pV ≠ nRT, Z ≠ 1

It can be seen easily that at constant temperature,

pV p plot for real gas is not a straight line.

Negative deviation (-) In such case, Z <1, gas is more compressible.

Positive deviation (+) In such case, Z >1, gas is less compressible.

The factors affecting the deviation are:

(i) Nature of the gas, In general, the most easily liquefiable and highly soluble gases show larger deviation.

(ii) Pressure The deviation is more at high pressure.CO2 and N2 show negative deviation at low pressure and positive deviation at
high pressure.

(iii) Temperature The deviation is more at low temperature. H and He always show positive deviations at 0°C.
Cause of deviation from the ideal behaviour It is due to two faulty assumptions of kinetic theory of gases, particularly
not valid at high pressure and low temperature.

1. Volume occupied by the gas molecules is negligible as compared to the total volume of the gas.

2. There are no attractive forces between the gas molecules.

van der Waals’ Equation

After volume and pressure correction, van der Waals’ obtained the following equation for n moles of a gas.
𝟐
(p + 𝒏𝑽𝟐𝒂 ) (V - nb) = nRT
(p + 𝑽𝒂𝟐 ) (V - b) = RT, (for one mole)

where, b = excluded volume or co-volume = 4 × actual volume of gas molecules

a = magnitude of attractive forces between gas molecules.

The greater the value of ‘a’, the greater the strength of van der Waals’ forces and greater is the ease with which a gas can be
liquefied.

Units for van der Waals’ constant

Pressure correction,
𝒏𝟐 𝒂 𝒑𝑽𝟐
p= or a = = atm L2 mol−2
𝑽𝟐 𝒏𝟐

Volume correction,
𝑽
V = nb or b = = L mol−1
𝒏

Limitation of van der Waals’ Equation

There is specific range of temperature and pressure, to apply the equation. It deviates at very high pressure and very low
temperature.

Liquefaction of Gases and Critical Points

The phenomenon of conversion of a gas into liquid is known as liquefaction. The liquefaction of a gas takes place when the
intermolecular forces of attraction becomes so high that it exist in the liquid. A gas can be liquefied by

(i) increasing pressure

(ii) decreasing temperature.

The critical points are as follows

(i) Critical temperature (TC) It may be defined as the temperature above which no gas can be liquefied. Critical temperature of
CO2 is 30.98°C.

Critical temperature (TC) of some gases are He (5.4), H2 (33.2), N2 (126.0), CO (134.4), O2 (154.3), CO2 (304.1),

NH4 (405.5),
𝟖𝒂
TC = 𝟐𝟕𝑹𝒃

(ii) Critical pressure (PC) At critical temperature, the pressure needed to liquefy a gas is known as critical pressure.
𝟖𝒂
PC =
𝟐𝟕𝒃𝟐

(iii) Critical volume (VC) The volume occupied by one mole of a gas at critical temperature and critical pressure is known as
critical volume.
 VC = 3b

(iv) Boyle’s temperature (Tb) Temperature at which a real gas exhibits ideal behaviour for considerable range of pressure is
called Boyle’s temperature.
𝒂
Tb =
𝒃𝑹