Title Med-RM_Phy_SP-2_Ch-13-Kinetic Theory.pdf ✅

Chapter Contents Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 Introduction Molecular Theory of Matter Behaviour of Gases Kinetic Theory of an Ideal Gas Law of Equipartition of Energy Degrees of Freedom Mean Free Path Introduction Gases have no shape and size and can be contained in vessels of any shape and size. They have negligible force of molecular interaction. Many scientists like Boyle and Newton tried to explain the behaviour of gases. But the real theory was developed in the nineteenth century by Maxwell and Boltzmann. This theory is Kinetic theory, which explains the behaviour of gases. It is consistent with gas laws and Avogadro Hypothesis. It gives the interpretation of pressure and temperature of gases. In this chapter we shall study some of the features of kinetic theory. MOLECULAR THEORY OF MATTER John Dalton, about 200 years ago, proposed the atomic theory. According to this theory (a) The smallest constituents of an element are atoms. (b) Atoms of one element are identical but differ from those of other elements. (c) A small number of atoms of each element combine to form a molecule of a compound. From many observations, in recent times we now know that molecules (made up of more than one atoms) constitute matter. We are able to measure their dimensions with the help of electron microscope and scanning tunnelling microscope. The size of an atom is about an angstrom (10–10 m). In solids the atom is tightly packed (about 2 Å). In liquids the atoms are not as rigidly fixed as in solids and can move around, this enables a liquid to flow. In gases the interatomic distances are very large therefore, the mean free path (the average distance a molecule can travel without colliding) is very large i.e., of the order of thousands of angstroms. Hence, if the gases are not enclosed, they disperse away. Chapter 13 Kinetic Theory
188 Kinetic Theory NEET Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 BEHAVIOUR OF GASES In gases molecules are far away from each other, and due to this the interatomic forces between the molecules is negligible except, when two molecules collide. Hence, the properties of gases are easier to understand than those of solids and liquids. Avogadro’s Hypothesis Equal volume of all the gases under similar condition of temperature and pressure contain equal number of molecules. The number of molecules in 22.4 litres of any gas at STP are 6.02 × 1023. This is known as Avogadro number and is denoted by NA. The mass of 22.4 litres of any gas at S.T.P. (standard temperature 273 K and pressure 1 atm) is equal to its molecular weight which is equal to one mole. The perfect gas equation can be written as PV = RT ...(i)  is number of moles and R = NAkB which is universal gas constant. Temperature T is absolute temperature R = 8.314 J mol–1 K–1 0   A M N M N ...(ii) where M is the mass of the gas containing N molecules, M0 is the molar mass and NA the Avogadro’s number. Using equation (ii), equation (i) can be written as    A NRT PV N ∵        B A R k N  PV = NkBT   B B N P k T nk T V where n is the number density of molecules i.e., number of molecules per unit volume, kB is Boltzmann’s constant. Its value in SI unit is 1.38 × 10–23 JK–1. We can also write equation (i) as 0 0     M P RT RT RT V MV M         ∵ M V If a gas satisfy equation (i) at all pressures and temperature then, it is known as ideal gas. It is a theoretical model of a gas practically no gas is truly ideal. The figure given below shows the departure of real gases from ideal gas behaviour. A straight line parallel to x-axis shows an ideal gas. Curves of other gases approaches ideal gas behaviour at low pressure and high temperatures. T1 T2 T3 TTT 123 > > Ideal gas 1 0 200 400 600 800 pV T  J mol K–1 –1 P (atm) Fig.: Real gases approach ideal gas behaviour at low pressures and high temperatures.
NEET Kinetic Theory 189 Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 Now, the question arises why real gases approaches ideal behaviour at low pressure and high temperatures. Let us try to seek for the answer to this question. At low pressures and high temperatures the molecules of the gas are far apart and the interaction between the molecules becomes negligible, without interactions the gas behaves like an ideal one. Boyle’s law At constant temperature, pressure of given mass of a gas varies inversely to its volume. 1 P  V PV = constant PV P V 11 2 2  Graphs of Boyle’s law m = const. T = const. m = const. T = const. m = const. P P PV T = const. V 1/V P or V (a) (b) (c) Fig.: (a) Showing variation of P with V; (b) Showing variation of P with 1/V; (c) Showing variation of PV with P or V Charles’ law At constant pressure, volume of given mass of a gas is directly proportional to absolute temperature. The graph between V and T are straight lines as shown in the graph given below : V  T at constant P 1.2 1.0 0.8 0.6 0.4 0.2 0 100 200 300 400 500 T V PPP 123 > > P1 P2 P3 V = KT V K T  1 2 1 2 V V T T  Fig.: Experimental T-V curves (solid lines) for CO2 at three pressures compared with Charles’ law (dotted lines). Gay Lussac’s law or Pressure law At constant volume, pressure of given mass of a gas is directly proportional to absolute temperature P  T (at constant volume) 1 2 1 2 P P T T 
190 Kinetic Theory NEET Aakash Educational Services Limited - Regd. Office : Aakash Tower, 8, Pusa Road, New Delhi-110005 Ph. 011-47623456 Dalton’s law of partial pressure Let us consider a mixture of non-interacting ideal gases and 1, 2, 3...... be the number of moles of gases respectively, in a vessel of volume V at temperature T and pressure P. Gas equation becomes PV = (1 + 2 + 3 + ...)RT 123 RT RT RT P VVV        P = P1 + P2 + P3 + ...... Thus, the total pressure of a mixture of ideal gases is the sum of partial pressures. This is Dalton’s law of partial pressures. Example 1 : The pressure of a gas is increased 2 times. What should be the change in its volume so that the temperature and number of moles remain constant? Solution : Applying Boyle’s law P1V1 = P2V2 ...(i) P2 = 2P1 Substituting the values in equation (i) 1 2 2  V V Hence, the volume reduced to half. Example 2 : The pressure of a given mass of a gas filled in a vessel of volume V at constant temperature is reduced to 1 3 rd of its initial value. Calculate the percentage change in its volume. Solution : Applying Boyle’s law P1V1 = P2V2 ∵ 2 1 1 3 P P  Substituting the values in equation, we get 11 12 1 3 PV PV  then, V2 = 3V1 Change in volume V = V2 – V1 = 2V1 Percentage change 1 1 2 100 100   V V V V = 200%