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WELCOME

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TOPIC: KINETIC THEORY OF GASES

PREPARED BY; MAHESH TRIPATHI PGT (PHYSICS) JAWAHAR NAVODAYA VIDAYALYA GANDHINAGAR

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Molecular nature of matter

We know that molecules which are made up of one or more atoms constitute matter.
In solids these atoms and molecules are rigidly fixed and space between them is very less of the order of few angstrom and hence they cannot move.
In liquids these atoms and molecules can more enabling liquids to flow.
In gases atoms are free to travel without colliding for large distances such that if gases were not enclosed in an enclosure they would disappear

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Molecular nature of MATTER

Atomic Hypothesis: All things are made of atoms little particles that move around in perpetual motion, attracting each other when they are a little distance apart,
but repelling upon being squeezed into one another.
Newton and several others tried to explain the behaviour of gases by considering that gases are made up of tiny atomic particles.
kinetic theory also relates measurable properties of gases such as viscosity, conduction and diffusion with molecular parameters, yielding estimates of molecular sizes and masses

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Important Characteristics of Gases

1) Gases are highly compressible

An external force compresses the gas sample and decreases its

volume, removing the external force allows the gas volume to

increase.

2) Gases are thermally expandable

When a gas sample is heated, its volume increases, and when it is

cooled its volume decreases.

3) Gases have high viscosity

Gases flow much easier than liquids or solids.

4) Most Gases have low densities

Gas densities are on the order of grams per liter whereas liquids

and solids are grams per cubic cm, 1000 times greater.

5) Gases are infinitely miscible

Gases mix in any proportion such as in air, a mixture of many gases.

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ASSUMPTIONS OF KINETIC THEORY

1.Every gas consists of extremely small particles known as

molecules. The molecules of a given gas are all identical

but are different from those of another gas.

2.The molecules of a gas are identical spherical,

rigid and perfectly elastic point masses.

3.Their molecular size is negligible in comparison

to intermolecular distance

4.The speed of gas molecules lies

between zero and infinity.

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5.The distance covered by the molecules between two successive collisions is known as Free-path and mean of all free path is known as mean free path.��6.The number of collisions per unit volume� in a gas remains constant.��7.No attractive or repulsive force acts between �gas molecules.�� 8.Gravitational to extremely attraction among �the molecules is ineffective due to small Masses� and very high speed of molecules

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Gas laws

Assuming permanent gases to be ideal, through experiments, it was established that gases irrespective of their nature obey the following laws.

Boyle’s Law:
Charle’s Law
Gay Lussac’s Law:
Avogadro Law

Click on the names to know more about them

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Dalton law of partial pressure

Dalton law of partial pressure: Total pressure exerted by a mixture of nonreacting

gases occupying a given volume is equal to the sum of partial pressures

which gas would exert if it alone occupied the same volume at given temp.

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Ideal Gas Law

The equality for the four variables involved in Boyle’s Law, Charles’ Law, Gay-Lussac’s Law and Avogadro’s law can be written

PV = nRT

R = ideal gas constant

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PV = nRT

R is known as the universal gas constant

Using STP conditions

P V

R = PV = (1.00 atm)(22.4 L) nT (1mol) (273K)

n T

= 0.0821 L-atm

mol-K

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Avogadro Law�

Avogadro stated that equal volume of all the gases under similar conditions of temperature and

pressure contain equal number molecules. This statement is called Avogadro’s hypothesis.

According Avogadro’s law

(i) Avogadro’s number The number of molecules present in 1g mole of a gas is defined as

Avogadro’s number.

NA = 6.023 X 1023 per gram mole

(ii) At STP or NTP (T = 273 K and p = 1 atm 22.4 L of each gas has 6.023 x 1023 molecules.

(iii) One mole of any gas at STP occupies 22.4 L of volume

BACKGas laws

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Charles’ Law�V ∝ T ⇒ V / T = constant�For a given gas, V1/T1 = V2/T2�At constant pressure the volume (V) of a given mass of a gas increases or decreases by�1/273.15 of its volume at 0°C for each 1°C rise or fall in temperature.�Volume of the gas at t°Ce�Vt = V0 (1 + t/273.15)�where V0 is the volume of gas at 0°C.

Linear Relation Between Temperature and Pressure

P

T (K)

0

100

200

300

P – T Diagram

isochors

V₁

V₂

V₃

V₁<V₂<V₃

BACKGas laws

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Gay Lussacs’ or Regnault’s Law�

At constant volume the pressure p of a given mass of gas is directly proportional to its absolute

temperature T,

i.e. ,

p ∝ T ⇒ V/T = constant

For a given gas,

P₁/T₁ = P₂/T₂

At constant volume (V) the pressure p of a given mass of a gas increases or decreases by

1/273.15 of its pressure at 0°C for each l°C rise or fall in temperature.

Volume of the gas at t°C, pt = p0 (1 + t/273.15)

where P0 is the pressure of gas at 0°C.

BACKGas laws

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�At constant temperature the volume (V) of given mass of a gas is inversely�proportional to its pressure (p), i.e.,V ∝ 1/p ⇒ pV = constant� For a given geas, P₁V₁ = P₂V₂

Hyperbolic Relation Between Pressure and Volume

p

V

p – V Diagram

isotherms

T₁

T₂

T₃

T₃>T₂>T₁

(courtesy F. Remer)

Boyle’s law

BACKGas laws

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IDEAL GAS EQUATION

To combine Boyle’s law and charles’s law in to a single equation i.e.� PV/T = Constant �If n moles is the mass of gas then we write PV = nRT �where, n is number of moles of gas R=N_AK_B is the universal constant known as gas constant and T is the absolute temperature.

A gas satisfying above equation at all values of preserves and temperatures is said to be an ideal gas�now no of moles of gas� n = m/M = N/N_A�where m - mass of gas
containing N molecules� M - molar mass� N_A – Avagadro’s number.�From this P = ρRT/M� ρ - mass density of gas.

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Kinetic Theory

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Kinetic Molecular Theory

Postulates

Evidence

1. Gases are tiny molecules in mostly empty space.	The compressibility of gases.
2. There are no attractive forces between molecules.	Gases do not clump.
3. The molecules move in constant, rapid, random, straight-line motion.	Gases mix rapidly.
4. The molecules collide classically with container walls and one another.	Gases exert pressure that does not diminish over time.
5. The average kinetic energy of the molecules is proportional to the Kelvin temperature of the sample.	Charles’ Law

The kinetic molecular theory of gases explains the laws that describe the behavior of gases and it was developed during the nineteenth century by Boltzmann, Clausius, and Maxwell

Kinetic molecular theory of gases provides a molecular explanation for the observations that led to the development of the ideal gas law

The kinetic molecular theory of gases is based on the following postulates:

1. A gas is composed of a large number of particles called molecules (whether monatomic or polyatomic) that are in constant random motion.

2. Because the distance between gas molecules is much greater than the size of the molecules, the volume of the molecules is negligible.

3. Intermolecular interactions, whether repulsive or attractive, are so weak that they are also negligible.

4. Gas molecules collide with one another and with the walls of the container, but collisions are perfectly elastic; they do not change the average kinetic energy of the molecules.

5. The average kinetic energy of the molecules of any gas depends on only the temperature, and at a given temperature, all gaseous molecules have exactly the same average kinetic energy.

Postulates 1 and 4 state that molecules are in constant motion and collide frequently with the walls of their container and are an explanation for pressure

1. Anything that increases the frequency with which the molecules strike the walls or increases the momentum of the gas molecules increases the pressure.

2. Anything that decreases that frequency or the momentum of the molecules decreases the pressure.

• Postulates 2 and 3 state that all gaseous particles behave identically, regardless of the chemical nature of their component molecules — this is the essence of the ideal gas law.

• Postulate 2 explains how to compress a gas — simply decrease the distance between the gas molecules.

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Kinetic Molecular Theory (KMT)

1. …are so small that they are assumed to have zero volume

…are in constant, straight-line motion

…experience elastic collisions in which no energy is lost

…have no attractive or repulsive forces toward each other

…have an average kinetic energy (KE) that is proportional

to the absolute temp. of gas (i.e., Kelvin temp.)

AS TEMP. , KE

explains why gases behave as they do

deals ^w/“ideal” gas particles…

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Elastic vs. Inelastic Collisions

8

3

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Kinetic Molecular Theory

Particles in an ideal gas…

have no volume.
have elastic collisions.
are in constant, random, straight-line motion.
don’t attract or repel each other.
have an avg. KE directly related to Kelvin temperature.

Courtesy Christy Johannesson www.nisd.net/communicationsarts/pages/chem

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PRESSURE OF AN IDEAL GAS

The change in momentum of a molecule is mv_1x - (-mv_1x) = 2 mv_1x
The number of molecules hitting the wall in time ∆t is ½ Av_1x ∆t n where n is no.of molecules per unit volume
The total momentum transferred to the wall by these molecules in time ∆t is Q= 2mv_1x ( ½ Av_1x ∆t n )
The force on the wall is the rate of momentum transfer and pressure is force per unit area P=Q/A ∆t =nm v_1x²

Σv_1x²=Σv_1y²=Σv_1z²� = 1/3Σ((v_1x)² + (v_1y)² +( v_1z)² )� = 1/3 Σv₁²�

pressure becomes

P = (1/3)ρv_mq² �

or PV = (1/3) Nmv_mq²

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Real Gases

Particles in a REAL gas…

have their own volume
attract each other

Gas behavior is most ideal…

at low pressures
at high temperatures
in nonpolar atoms/molecules

Courtesy Christy Johannesson www.nisd.net/communicationsarts/pages/chem

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A gas particle is shown colliding elastically with the right wall of the container and rebounding from it.

Kinetic interpretation of temperature

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Degree of freedom

No. of coordinate or independent quantities required to describe completely the position and configuration of a dynamical system is known as no. of degree of freedom of system.

represent by f and expressed as

f = 3N-K

N = No. of particle in a system

K = No. of independent relation between the particle

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DEGREE OF FREEDOM

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Law of equipartition of energy

It states that for a dynamic system in thermal equilibrium , the energy is distributed equally amongst all the degree of freedom and the energy associated with each molecule per degree of freedom is 1/2KBT

Where , KB = 1.38*10^-23 J/K is Boltzmann constant

T = Absolute temperature of system on the kelvin scale.

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SPECIFIC HEAT CAPACITY

Specific heat capacity of gases having different degree of freedom .It can be calculated by applying the law of equipartition of energy of gases .

1. Monoatomic gases

2. Diatomic gases

3. Polyatomic gases

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MONOATOMIC GASES

It have only three translational degrees of freedom . Thus , the average energy of a molecule at temperature T is 3/2 kBT .
The total energy of a mole of such a gas is

U = 3/2kBT*NA = 3/2 RT

The molar specific heat at constant volume ,CV ,is

CV( monoatomic gas) = dU/dT = 3/2 RT

For an ideal gas

CP – CV = R

Where CP is molar specific heat at constant pressure.

Thus ,

CP = 5/2R

The ratio of specific heat ỵ = CP / CV = 5/3

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Diatomic gas

2.For a diatomic gas with no vibrational mode f=5, so

3.For a diatomic gas with vibrational mode f=7, so

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Polyatomic gas

5. If ‘f’ is degree of freedom then for a gas of polyatomic molecules energy

associated with 1 mole of gas

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Polyatomic gases

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the mean free path

Average distance a molecule
can travel without colliding is called the mean
free path. The mean free path, in gases, is of
the order of thousands of angstroms. The atoms
are much freer in gases and can travel long
distances without colliding. If they are not
enclosed, gases disperse away. In solids and
liquids the closeness makes the interatomic force
important. The force has a long range attraction
and a short range repulsion. The atoms attract
when they are at a few angstroms but repel when
they come closer.

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Mean free path

In time Δt, it sweeps a

volume πd²<v> Δt wherein any other molecule

will collide with it (see Fig. 13.7). If n is the

number of molecules per unit volume,

the

molecule suffers nπd² <v> Δt collisions in

time

Δt. Thus the rate of collisions is nπd² <v>

or the

time between two successive collisions is

on the

average,

τ = 1/(nπ <v> d² )

The average distance between two successive

collisions, called the mean free path l, is :

l = <v> τ = 1/(nπd²)

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MEAN FREE PATH

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Specific heat capacity of solids

A solid of N atoms, each vibrating about its mean

position. An oscillation in one dimension has

average energy of 2 × ½ k_BT = k_BT . In three

dimensions, the average energy is 3 k_BT. For a

mole of solid, N = N_A, and the total

energy is

U = 3 k_BT × N_A = 3 RT

Now at constant pressure ΔQ = ΔU + PΔV

= ΔU, since for a solid ΔV is negligible. Hence,

=

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Specific heat capacity of water

We treat water like a solid. For each atom average

energy is 3K_BT. Water molecule has three atoms,

two hydrogen and one oxygen. So it has

U = 3 × 3 K_BT × N_A = 9 RT

and C = ΔQ/ ΔT =Δ U / ΔT = 9R .

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THANKS