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Chapter 13, Kinetic Theory, , Introduction, What is Kinetic Theory?, , « Kinetic theory explains the behaviour of gases based on the idea that the gas consists of rapidly moving, atoms or molecules., , e Insolids the molecules are very tightly packed as inter molecular space is not present In liquids inter, molecular spaces are more as compared to solids and in gases the molecules are very loosely packed as, intermolecular spaces are very large., , * The random movement of molecules in a gas is explained by kinetic theory of gases., « Wewill also see that why kinetic theory is accepted as a success theory., « Kinetic theory explains the following:a) Molecular interpretation of pressure and temperature can be explained., b) Itis consistent with gas laws and Avogadro’s hypothesis., c) Correctly explains specific heat capacities of many gases., 13.2 Molecular nature of matter, , « Atomic hypothesis was given by many scientists. According to which everything in this universe is made, up of atoms., , « Atoms are little particles that move around in a perpetual order attracting each other when they are little, distance apart. But if they are forced very close to each other then they repel., , * Dalton’s atomic theory is also referred as the molecular theory of matter. This theory proves that matter, is made up of molecules which in turn are made up of atoms., , * According to Gay Lussac’s law when gases combine chemically to yield another gas, their volumes are in, ratios of small integers., , « Avogadro’s law states that the equal volumes of all gases at equal temperature and pressure have the, same number of molecules., , * Conclusion: - All these laws proved the molecular nature of gases., * Dalton’s molecular theory forms the basis of Kinetic theory., Why was Dalton’s theory a success?, , « Matter is made up of molecules, which in turn are made up of atoms., « Atomic structure can be viewed by an electron microscope., , Solids, Liquids, Gases in terms of molecular structure, , , , Basis of difference, , Solids, , Liquids, , Gases, , , , Inter Atomic, Distance(distance, between molecules)., , Molecules are very, tightly packed. Inter, atomic distance is, minimum., , Molecules are not so, tightly packed. Inter, atomic distance is more, as compared to solids., , Molecules are loosely, packed .Free to move., Inter atomic distance is, maximum., , , , Mean Free Path is the, average distance a, molecule can travel, without colliding., , , , , , No mean free path., , , , Less mean free path., , , , There is mean free path, followed by the, molecules.

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13.3 Behaviour of Gases, , °, , Gases at low pressures and high temperatures much above that at which they liquify (or solidify), approximately satisfy a relation between their pressure, temperature and volume:, , This is the universal relation which is satisfied by all gases., , where P, V, T are pressure, volume and temperature respectively, and K is the constant for a given, volume of gas. It varies with volume of gas., , K=Nke where, N=number of molecules and ke = Boltzmann Constant and its value never change., From equation (i) PV= Nkg, , Therefore PV/NT = constant=(ks) (Same for all gases)., , Consider there are 2 gases :- (P1,V1,T1) and (P2, V2,T2) where P, V and T are pressure, volume and, temperature resp., , Therefore P1,Vi/(NiT1) = P2V2/(N2T2), , Conclusion: - This relation is satisfied by all gases at low pressure and high temperature.

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Perfect Gas Equation, , o Perfect gas equation is given by PV = uRT,, Where P,V are pressure, volume, T =absolute temperature, 1: = number of moles and R =universal gas, constant, R= kgNa where, ks = Boltzmann constant and Na = Avogadro’s Number, , © This equation tells about the behaviour of gas at a particular situation., o If agas satisfies this equation then the gas is known as Perfect gas or an ideal gas., , Different Forms of Perfect Gas Equation, 1. PV=URT ow. oe (i), , Where p (no. of moles) = N/Na where N=no of molecules and Na = Avogadro number(no of molecules in 1, mole of gas).Orp. = M/Mo. where M=mass of sample of gas and Mo = molar mass., , PV = (N/Na)RT (putting p=N/Na in equation(i)), , By simplifying PV = NksT, , PV=NkeT => P = (N/V) keT => P=nkeT, , Where, n(number density) =N/V. where, N=number of molecules and V=volume.

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13.4 Kinetic Theory of an Ideal Gas, Basis of Kinetic Theory: , A, , woe WN, , Molecules of gas are in incessant random motion, colliding against one another and with the walls of the, container., , All collisions are elastic., , Total Kinetic energy is conserved., , Total momentum is conserved., , In case of an elastic collision total Kinetic energy and momentum before collision is equal to the total, Kinetic energy and momentum after collision., , What does Kinetic Theory tells?, , i,, , At ordinary temperature and pressure the molecular size is very small as compared to inter molecular, distance between them., , In case of gas, molecules are very far from each other and the size of molecules is small as compared to, the distance between them., , As a result, interaction between them is negligible. As there is no interaction between the molecules, there, will be no force between the molecules., , As a result molecules are moving freely as per newton’s first law of motion., , The molecules should move along straight line but when they come closer they experience the, intermolecular forces and as a result their velocities change., , This phenomenon is known as collision. These collisions are elastic., , Molecules moving randomly Molecules colliding with | Molecules change their direction, , each other after colliding

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13.4.1 Pressure of an ideal gas based on Kinetic theory, Assumptions:, o Consider a cube shape container filled with an ideal gas. We will consider only one molecule; the, molecule hits the walls of the container and bounces back., Let the velocity of the molecule when it is moving be (v,, Vy, Vz)., When the molecule bounces back, the velocity will be (-vx, Vy, v,)., Change in momentum = P; — P; where P; = final momentum and P, = initial momentum), Py — Pi = —mv, — mv, = —2mv,, This change in momentum is imparted to the wall due to the collision., Momentum imparted to the wall in collision by one molecule = 2mv,., But there are as many molecules, we have to calculate total momentum imparted to the wall by, all of them., © To calculate the number of molecules that hit the wall:, Area of wall= A, +. intime At, all molecules within a distance of Av, At can hit the wall., but on an average half of molecules move towards the wall and half away from the wall., , + (5) Av,Ac will hit the wall., , oooo0oo0o0o0, , © +. total momentum imparted to the wall =2mv, x zn Av, At = AnvzAtm, © Force exerted on the wall= rate of change of momentum=Anv2m, oO Pressure on the wall P = - = nmv2, o «+ P = nmvp;% is true for group of molecules moving with velocity vy, Note:, i. All the molecules inside the gas will not have the same value of velocity. All will have different, velocities, ii. The above equation therefore, is valid for pressure due to the group of molecules with speed v, in, the x-direction and n stands for the number density of that group of molecules., ¢ Therefore total pressure due to all such groups will be obtained by summing over the, contribution due to the molecule P = nmv2, © Where, v? is the average of v2., ©. Since the gas is isotropic the molecules move randomly which means the velocity of all the molecules can, , be in any direction. Therefore, , , , |, Hl, , vg = vy =, , )=3", , 1, 3, , 1, Therefore, Pressure, P = gnmv?, where v2= average squared speed., , Zz, , , , , , (vu, 4), , 4), , Cu.yy, , , , Ye, , , , , , x, , , , Elastic collision of a gas molecule with, the wall of the container.