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Physics Smart Booklet 1 13.Kinetic Theory of Gases Physics Smart Booklet Theory + NCERT MCQs + Topic Wise Practice MCQs + NEET PYQs
Physics Smart Booklet 2
Physics Smart Booklet 3 Kinetic Theory of Gases Ideal gas equation Many of the properties of gases are common at low pressures and high temperatures. The pressure, volume and the temperature (in kelvin) of such a gas obey the equation. pV = nRT where n is the amount of gas in number of moles and R is universal gas constant having a value 8.314 J/mol-K. The above equation is called ideal gas equation and a gas obeying this equation is called ideal gas. Kinetic theory of gases Any sample of a gas is made of molecules which is the smallest unit having all the chemical properties of the sample. The detailed behaviour of large number of molecules of gas results in its observed behaviour. The properties of a gas may be investigated from the point of view of kinetic theory based on the laws of classical mechanics. Postulates of kinetic theory gases a. All gases are made up of identical particles called molecules. b. The size of a molecule is much smaller than the average separation between the molecules. c. The molecules of a gas are in a state of continuous random motion in all the directions. d. The molecules collide with each other and also with the walls of the container. The molecules travel in straight line between collisions. e. The collisions are perfectly elastic and time spent during collisions is very small compared to the time of their random motion. f. The molecules exert no force on each other or on the walls of the container except during collisions. g. The molecules obey Newton’s laws of motion. The expression for the pressure of an ideal gas During the collisions, the molecules exert forces on the walls of the container. This is the origin of the pressure of a gas. Based on kinetic theory of gases it can be shown that 2 avg 1 pV Nm(v ) 3 = ... (1) where p = pressure of the gas V = volume of the gas N = total number of molecules in the sample m = mass of a single molecule (v2 )avg = average of the speeds squared of the molecules. It is also called mean square speed. Equation (1) can be written as 2 avg 2 1 pV N m(v ) 3 2   =     ... (2) 2 avg 1 m(v ) 2 gives average translational kinetic energy of single molecule. 2 avg 1 N m(v ) 2       gives total translational kinetic energy of all the molecules due to random motion. It is denoted by Ktr.  tr 2 pV K 3 = ... (3) From ideal gas equation pV = nRT ... (4) Comparing (3) and (4) tr 3 3 K nRT pV 2 2 = = ... (5) Thus, average translational kinetic energy of molecules of a gas is proportional to its absolute temperature.
Physics Smart Booklet 4 Average kinetic energy per mole is given by tr 2 avg K 3 1 RT M(v ) n 2 2 = = ... (6) where M = molecular weight The average translational kinetic energy of a molecule is given by 2 avg 1 3nRT m(v ) 2 2N = Since R = NAK ; nNA = N 2 avg 1 3 m(v ) kT 2 2 = ... (7) RMS speed The square root of the mean square speed is called root mean square speed and is denoted by vrms.  2 rms avg v (v ) = Thus average translational kinetic energy of a molecule is given by 2 rms 1 3 mv kT 2 2 = or rms 3kT v m = ... (8) The average translational kinetic energy per mole given by 2 rms 1 3 Mv RT 2 2 =  rms 3RT v M = ... (9) The average speed vavg is some what smaller than the rms speed. It can be shown that avg 8kT 8RT v m M = =   ... (10) where M = Molecular weight Most probable speed It is defined as the speed which is possessed by maximum fraction of total number of molecules of the gas. It is given by p 2kT 2RT v m M = =  When mixture of two gases A and B are in thermal equilibrium, the average kinetic energy of all molecules are equal. If v1 and v2 be the rms speeds of the molecules A and B respectively, 2 2 1 1 2 2 1 1 m v m v 2 2 = ...(11)  Thus, the heavier molecules move with smaller rms speed and the lighter molecules move with larger rms speed. Mean free path Mean free path is the average distance travelled by a molecule between collisions. It is given by 2 2 V 1 1 2 d N 2 d n V  = =   where d is the diameter of each molecule and V N n V = is the number of molecules per unit volume.