Kinetic Molecular Theory Of Gases

A gas which obeys the gas laws and the gas equation PV = nRT strictly at all temperatures and pressures is said to be an ideal gas. The molecules of ideal gases are assumed to be volume less points with no attractive forces between one another. But no real gas strictly obeys the gas equation at all temperatures and pressures. Deviations from ideal behaviour are observed particularly at high pressures or low temperatures. The deviation from ideal behaviour is expressed by introducing a factor Z known as compressibility factor in the ideal gas equation. Z may be expressed as Z = PV/nRT

• In case of ideal gas, PV = nRT ∴ Z = 1

• In case of real gas, PV ≠ nRt ∴ Z ≠ 1

Thus in case of real gases Z can be • 1 or • 1

(i) When Z • 1, it is a negative deviation. It shows that the gas is more compressible than expected from ideal behaviour.

(ii) When Z • 1, it is a positive deviation. It shows that the gas is less compressible than expected from ideal behaviour.

Causes of deviation from ideal behaviour

The causes of deviations from ideal behaviour may be due to the following two assumptions of kinetic theory of gases. There are

• The volume occupied by gas molecules is negligibly small as compared to the volume occupied by the gas.

• The forces of attraction between gas molecules are negligible.

The first assumption is valid only at low pressures and high temperature, when the volume occupied by the gas molecules is negligible as compared to the total volume of the gas. But at low temperature or at high pressure, the molecules being in compressible the volumes of molecules are no more negligible as compared to the total volume of the gas.

The second assumption is not valid when the pressure is high and temperature is low. But at high pressure or low temperature when the total volume of gas is small, the forces of attraction become appreciable and cannot be ignored.

Van Der Waal’s Equation

The general gas equation PV = nRT is valid for ideal gases only Van der Waal is 1873 modified the gas equation by introducing two correction terms, are for volume and the other for pressure to make the equation applicable to real gases as well.

Volume correction

Let the correction term be v

∴ Ideal volume vi = (V – v)

Now v ∝ n or v = nb

[n = no. of moles of real gas; b = constant of proportionality called Van der Waal’s constant]

∴ Vi = V – nb

b = 4 × volume of a single molecule.

Pressure Correction

Let the correction term be P

∴ Ideal pressure Pi = (P + p)

Now,

Where a is constant of proportionality called another Van der Waal’s constant.

Hence ideal pressure

Pi =

Here, n = Number of moles of real gas

V = Volume of the gas

a = A constant whose value depends upon the nature of the gas

Substituting the values of ideal volume and ideal pressure, the modified equation is obtained as

Vander Waals equation, different forms

• At very low pressures: V becomes so large that both b and a/V²  become negligible and the Vander Waals equation reduces to PV = RT. This shows why gases approach ideal behaviour at very low pressures.

• Hydrogen and Helium: These are two lightest gases known. Their molecules have very small masses. The attractive forces between such molecules will be extensively small. So a/V² is negligible even at ordinary temperatures. Thus PV • RT. Thus Vander Waals equation explains quantitatively the observed behaviour of real gases and so is an improvement over the ideal gas equation.

Vander Waals equation accounts for the behaviour of real gases. At low pressures, the gas equation can be written as,

Where Z is known as compressibility factor. Its value at low pressure is less than 1 and it decreases with increase of P. For a given value of Vm, Z has more value at higher temperature.

At high pressures, the gas equation can be written as

P (Vm – b) = RT

Z =

Here, the compressibility factor increases with increase of pressure at constant temperature and it decreases with increase of temperature at constant pressure. For the gases H2 and He, the above behaviour is observed even at low pressures, since for these gases, the value of ‘a’ is extremely small.

Some other important definitions

Mean Free Path (λ)

The average distance covered by a molecule between two successive collisions is called mean free path and is denoted by 

∴ where n = no. of molecules/cc. Again, if P & T denote the pressure and temperature of the gas, from kinetic theory

P × 10^–3 = n/N0 RT or n =

∴ l =

Thus mean free path is directly proportional to temperature and inversely to pressure.

Collision Frequency (z)

It is the number of collisions taking place per second

z = πσ²Nuav

where σ = collision diameter. It is the distance between the centers of two molecules without collision.

N = number of molecules per unit volume,

And uav = average velocity

Relative Humidity (RH)

At a given temperature it is given by equation

RH = partial pressure of water in air/vaour pressure of water

Boyle’s Temperature (Tb)

Temperature at which real gas obeys the gas laws over a wide range of pressure is called Boyle’s Temperature. Gases which are easily liquefied have a high Boyle’s temperature [Tb(O2)] = 46 K] whereas the gases which are difficult to liquefy have a low Boyle’s temperature [Tb(He) = 26K].

Boyle’s temperature Tb = a/Rb = 1/2Ti

where Ti is called Inversion Temperature and a, b are called van der Waals constant.

Critical Constants

• Critical Temperature (Tc): It (Tc) is the maximum temperature at which a gas can be liquefied i.e. the temperature above which a gas can’t exist as liquid.

Tc = 8a/27Rb

• Critical Pressure (Pc): It is the minimum pressure required to cause liquefaction at Tc

Pc = a/27b²

• Critical Volume: It is the volume occupied by one mol of a gas at Tc and Pc

Vc = 3b

Molar heat capacity of ideal gases: Specific heat c, of a substance is defined as the amount of heat required to raise the temperature of is defined as the amount of heat required to raise the temperature of 1 g of substance through 10C, the unit of specific heat is calorie g-1 K-1. (1 cal is defined as the amount of heat required to raise the temperature of 1 g of water through 10C)

Molar heat capacity C, is defined as the amount of heat required to raise the temperature of 1 mole of a gas trough 10C. Thus,

Molar heat capacity = Sp. Heat × molecular wt. Of the gas

For gases there are two values of molar heats, i.e., molar heat at constant pressure and molar heat at constant molar heat at constant volume respectively denoted by Cp and Cv. Cp is greater than Cv and Cp-R = 2 cal mol-1 K^-1.

From the ratio of Cp and Cv, we get the idea of atomicity of gas.

For monatomic gas Cp = 5 cal and Cv =3 cal

λ = 4/3 = 1.67 ( γ is poisson's ratio = Cp/Cv)

for diatomic gas Cp = 7 cal and Cv = 5 cal

γ = 7/5 = 1.40

For polyatomic gas Cp = 8 cal and Cv= cal

γ = 8/6 = 1.33

also Cp = Cp × m, C_v = C_v x m

where, Cp and Cv are specific heat and m, is molecular weight.

Gas Eudiometry

The relationship amongst gases, when they react with one another, is governed by two laws, namely Gay-Lussac law and Avogadro’s law.

Gaseous reactions for investigation purposes are studied in a closed graduated tube open at one end and the other closed end of which is provided with platinum terminals for the passage of electricity through the mixture of gases. Such a tube is known as Eudiometer tube and hence the name Eudiometry also used for Gas analysis.

During Gas analysis, the Eudiometer tube filled with mercury is inverted over a trough containing mercury. A known volume of the gas or gaseous mixture to be studied is next introduced, which displaces an equivalent amount of mercury. Next a known excess of oxygen is introduced and the electric spark is passed, whereby the combustible material gets oxidised. The volumes of carbon dioxide, water vapour or other gaseous products of combustion are next determined by absorbing them in suitable reagents. For example, the volume of CO2 is determined by absorption in KOH solution and that of excess of oxygen in an alkaline solution of pyrogallol. Water vapour produced during the reaction can be determined by noting contraction in volume caused due to cooling, as by cooling the steam formed during combustion forms liquid (water) which occupies a negligible volume as compared to the volumes of the gases considered. The excess of oxygen left after the combustion is also determined by difference if other gases formed during combustion have already been determined. From the data thus collected a number of useful conclusions regarding reactions amongst gases can be drawn.

• Volume-volume relationship amongst Gases or simple Gaseous reactions.

• Composition of Gaseous mixtures.

• Molecular formulae of Gases.

• Molecular formulae of Gaseous Hydrocarbons.

The various reagents used for absorbing different gases are

O₃ → turpentine oil

O₂→ alkaline pyrogallol

NO → FeSO₄ solution

CO_2,SO₂ →alkali solution (NaOH, KOH, Ca(OH)2, HOCH2CH₂NH₂, etc.)

NH3 → acid solution or CuSO₄ solution

Cl₂ →  water

Equation for combustion of hydrocarbons.

CxHy + (x + y/4)O₂ → xCO₂ + y/2 H₂O

General Assumptions: In all problems, it is assumed that the sparking occurs at room temperature. This implies that water formed would be in liquid state and that nitrogen gas is inert towards oxidation.

Liquid State

• The liquid molecules are relatively close together.

• The intermolecular forces of attraction in case of liquids are much larger than in gases.

• Unlike gases, liquids have a definite volume although no definite shape (similarity with gases).

• The molecules are in constant random motion.

• The average kinetic energy of molecules in a given sample is proportional to the absolute temperature.

• Guldberg’s rule: Normal boiling point (Tb) of the liquid is nearly two – third of its critical temperature (Tc) when both are expressed on the absolute scale.

Tb = 2/3 Tc

• Trouton’s rule : The molar heat of vapourisation of a liquid expressed in Joules divided by normal b.p. of the liquid on the absolute scale is approximately equal to 88.

Evaporation

Evaporation is the spontaneous change in which a liquid changes into vapours at the surface of liquid. Evaporation occurs at all temperatures. Evaporation increases with increase in surface area, increase in temperature and decrease in intermolecular attractive forces. In contrast to evaporation, boiling takes place at a definite temperature and it involves bubble formation below the surface. Evaporation produces cooling.

Vapour Pressure

Vapour pressure of a liquid at any temperature may be defined as the pressure exerted by the vapour present above the liquid in equilibrium with the liquid at that temperature. The magnitude of vapour pressure depends upon the nature of liquid and temperature.

• Non – polar or less polar liquids have fairly high vapour pressure due to weak forces of attraction (e.g.,CCl₄, CHCl₃ ether etc). Polar liquids (e.g. water, alcohols, etc.) have lower vapour pressure because of strong dipole – dipole interaction between their molecules.

• Vapour pressure of a liquid is constant at a given temperature. Further the vapour pressure of a liquid increases with increase in temperature. When the vapour pressure of the liquid is equal to the external pressure (normal pressure or 1 atm pressure) acting upon the surface of the liquid, the bubbles increase in size and escape freely; the temperature at which this happens is called the boiling point of the liquid. In case the external pressure is more than the atmospheric pressure, more heat will be required to make the vapour pressure equal to the external pressure and hence higher will be the boiling point. In case, the external pressure is low (as on the top of a mountain), the boiling point of the liquid decreases. This explains why a liquid boils at a lower temperature on the top of a mountain. (where pressure in low) than on the sea shore.

• Substances having high vapour pressure (e.g. petrol) evaporate more quickly than substances of low vapour pressure. (e.g. motor oil)