KINETIC THEORY OF GASES CLASS XI

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1 KINETIC THEORY OF GASES CLASS XI Obtain the dimensional formula for R used in the ideal gas equation, PV =RT R = PV/T Why do the gases at low temperature and high pressure, show large deviations from ideal behaviour? At low temperature and high pressure, the intermolecular attractions become appreciable.. Following fig. shows the variation of the product PV with respect to the pressure (P)of given masses of three gases, A,B,C. The temperature is kept constant. State with proper arguments which of these gases is ideal. Marks Gas C is ideal because PV is constant for it. That is gas C obeys Boyle s law at all pressures. 4 The ratio of vapour densities of two gases at the same temperature is 8:9. Compare the rms. velocity of their molecules? Ans :- : Oxygen and hydrogen are at the same temperature T. What is the ratio of kinetic energies of oxygen molecule and hydrogen molecule when oxygen is 6 times heavier than hydrogen? ANS: I:, K.E αt Kinetic energy of a gas molecule depends only on temperature. 6 Two vessels A and B of the same gas at the same temperature. The pressure of the gas in vessel B is twice the pressure of the gas in

2 A. What is the ratio of the number of molecules in B and A ANS: : 7 Two gases each at temperature T, Volume V and pressure P are mixed such that the temperature and volume of the mixtures are T and V respectively. What would be the pressure of the mixture? Ans:- P= Pressure gets doubled because mass m gets doubled 8 Give the relation between number of degree of freedom and independent relations among the particle. ANS: N =A-R 9 Define RMS speed of gas molecules. 0 State the law of equipartition of energy. What is degree of freedom? How many degrees of freedoms are there for a diatomic molecule? The velocities of three molecules are v, 4v,and v. Calculate their rms velocity. C = 0/ =4.08v The pressure of a gas at -7 0 c is atmosphere. Keeping the volume constant, to what temperature should the gas be heated so that the pressure becomes atmosphere? ANS : P /T =P /T T = (P / P ) T = -7 0 C 4 The absolute temperature of the gas is increased times. What will be the increase in root mean square velocity of the gas molecules? ANS: As C α T the r.m.s velocity becomes C Increase in rms velocity C C =0.7C A gas in a closed vessel is at the pressure Po. If the masses of all the molecules be made half and their speeds be made double, then find the resultant. 6 If pressure? the mass of each molecule of a gas is halved and speed is doubled, find the ratio of initial and final pressure. ANS: P/P =/ 7 Define the mean free path. Name any two factors on which the mean free path depends. Ans:- It is defined as the average distance travelled by the molecules between two successive collisions. λ = / Πd n 8 There are N molecules of a gas in a container. If the number of molecules is increased to N,what will be the pressure of the gas? ii)total energy of the gas? ANS: Pressure of the gas is doubled ii) total energy of the gas is

3 doubled 9 (a) Name at least two prominent phenomena which provide conclusive evidence of molecules motion. (b) What is the relative between pressure and K.E of gas molecules? (c) Define mean free path. 0 Prove that the average K.E of molecules of an ideal gas is directly proportional to the absolute temp. Of the gas. Using expression for pressure exerted by a gas, deduce Avogadro s law and graham s law of diffusion.. Explain the concept of absolute zero temp. On the basis of Kinetic theory of gases. (a) State the law of equi-partition of energy. (b) State the no. of degree of freedom possessed by a monoatomic molecules in space. Also give the expression for total energy possed by it at a given temp. hence give the total energy of the atom at 00k 4 Three vessel of equal capacity have gases at the same temp. and pressure. The first vessel contain neon (monoatomic) the second vessel contain chlorine (diatomic) and the third vessel contain uranium hexafluoride (polyatomic). Do the vessel contain equal no. of respective molecules? Is the root mean sq. speed of molecules same in three cases? If not in which case is Vrms the largest vrms Explain qualitatively how the extent of Brownian motion is affected by the (a) size of the Brownian particle (b) density of the medium (c) temp. of the medium (d) viscosity of the medium 6 A gas is filled in a cylinder fitted with a piston at a const. temp. Explain on the basis of kinetic theory. (a) The pressure of the gas increases by raising the temp. (b) in pulling the piston out, the pressure of the gas decreases 7 A gas is filled in a cylinder fitted with a piston at a const. temp. Explain on the basis of kinetic theory. (a) The pressure of the gas increases by raising the temp. (b) in pulling the piston out, the pressure of the gas decreases 8 A vessel is filled with a mixture of two different gases. However the no.

4 of molecules per unit volume of the two gases in the molecules is same. (a) will the mean K.E per molecules of both tha gases be equal. (b) will the root mean sq. velocities of the molecules be equal. (c) will the pressure be equal. Give reasons 9. Write the basic assumptions of kinetic theory of gases? On their basis derive an expression for the pressure exerted by an ideal gas. 0 Prove that the pressure exerted by a gas is P=/ ᵟ C, where ᵟ density and C is the root mean square velocity. is the What is meant by degrees of freedom? State the law of equi-partition of energy. Hence calculate the values of molar specific heats at constant volume and pressure for monatomic, diatomic and triatomic gases. Derive the following laws on the basis of kinetic theory of gases. (a) Boyle s law (b) Charles law (c) Gay Lussac s law. What is Dulong and Petit law? Illustrate it graphically also. (b) State four factors on which Brownian motion depends. 4 (a) Deduce Grahim s law of diffusion from kinetic theory of gases using expression for pressure. (b) State Dalton s law of partial pressures. Deduce it from the kinetic theory of gases. (a) Define absolute zero (b) Deduce the dimensional formula for R, using ideal gas equation PV=nRT. (b) Find the degree of freedom of monoatomic gas. 6 Derive the relation between the ratio of two specific heats of gas and degree of freedom. 7 (a) Using the expression for pressure exerted by a gas deduce Avogadro s law and Graham s law of diffusion. (b) Using the law of equi-partition of energy. Show that for ideal gas having f degrees of freedom. ᵧ=+/f NUMERICAL 8 Calculate the ratio of the neon free paths of the molecule of two gases having molecular diameter A 0 and A 0. Ans: - The gases may be considered under identical conditions of temperature, pressure and volume λ α /d Hence, λ /λ = d /d = / = 4:

5 9 Calculate the temperature at which rms velocity of SO is the same as that of Oxygen at STP. For O, Vrms = For SO, Vrms = As V 0 = V... T = 600t = = 7 0 C 40 Kinetic energy of oxygen molecule at 0 0 C is 9.4 x 0 - J. Calculate Avogadro number when R = 8.J mole - K - ANS: K.E of an oxygen molecule = / (R/N) T N= 6.0 x 0 4 Calculate the mean free path if the number of molecules per cm is x0 9, the diameter of molecule is x 0-8 cm ANS: λ = / Πd n =.876 x 0-7 m 4 You have the following group of particles,n represents the number of molecules with speed v n V m/s Calculate i)the average speed ii)the rms speed 4 44 A tank used for filling helium balloons has a volume of 0. m and contains.0 mol of helium gas at C. Assuming that the helium behaves like an ideal Gas i) What is the total translational K.E of the molecules of the gas? ii) What is the average K.E per molecules? ) K.E= / nrt n= R=8., T= 7±0=9,=/XX8.X9 K.E =7.X0 J ) The average K.E per molecule is / KT Find the rms speed of hydrogen molecules at room temperature

6 (00k). Mass of mole of hydrogen gas. V (rms)= RT/M= x8.x00/0.00=90m/s. 4 Find the temperature at which oxygen molecules would have the same rms speed as of hydrogen molecule at 00k. If T be the corresponding temperature RT/M = R (00)/M T=00[M /M] =00 [6x0.00/0.00]=4800k 46 4g of hydrogen is mixed with ½ liters of He at STP in a container of volume 0 L. If the final temperature is 00k.Find the pressure. Let pressure of hydrogen be P and pressure of helium be P 4g of hydrogen = moles of hydrogen. L of He at STP =/ mole of He PV=nRT P=nRT/V P=P +P = (n +n) RT/V = (+/)8.x00/0.0 = One mole of an ideal monoatomic gas is taken at a temperature of 00k. Its volume is doubled keeping its pressure constant. Find the change in internal energy. Pressure is constant, V proportional to T V/T=V/T T= (V/V) T T=T=600k Internal energy U=f/xnxRx T Force monoatomic gas= U=/xRx T=/xR (00) U=40R 49 Calculate the change in internal energy of moles of helium gas when its temperature is increased by 0k. U=/nRT /xx8.x=74.8 J 0 An air bubble of volume cm rises from the bottom of a lake 40m deep at a temperature of C. To what volume does it grow when it reaches the surface, which is at a temperature of C?

7 h=40m, T= C, or, 8k atm=.0x0 pa T= C, or, 08k V= cm =x0-6 m P=atm+hᵽg=4900pa P= atm When the air bubble reaches at the surface of the lake, then P= atm PV/T=PV/T V=PVT/TP = 49000xx0-6x08/8x.0x0 =.7x0-6m. Molar volume is the volume occupied by mole of any ideal gas at standard temperature and pressure. Show that it is.4l P= atm=.0x0 pa T=7K R=8. n= PV=nRT V=nRT/P= x8.x7/.0x0 =.4L A balloon of partially filled with helium has a volume of 0m at the earth s surface, where pressure is 76cm of Hg and temperature is 7 C.what will be the increase in volume of gas if balloon rises to a height, where pressure is 7.6 cm of Hg and temperature is -4 C? P=7.6 cm of Hg V=0m P=76cm of Hg T= 7-4=9k T=7+7=00k PV/T=PV/T V= PVT/TP=76x0x9/00x7.6=9m Increase in volume=v-v=9-0=89m HOTS AND VALUE BASED QUESTIONS An air bubble starts rising from the bottom of a lake. Its diameter is.6mm at the bottom and 4mm at the surface. The depth of the lake is 0 cm and the temperature at the surface is 40 C.What is the temperature at the bottom of the lake? Given atmospheric pressure =76 cm of Hg and g=980cm/s. D=.6mm so, r=0.08cm At the bottom of the lake volume of the bubble V=4/πr=4/π (0.08x0.08x0.08) cm pressure in the bubble P= atmospheric pressure +pressure due to a column of 0 cm of HO = 76x.6x980+0xx980 = (76x.6+0)980

8 At the surface of the lake, volume pf the bubble V=4/πr = 4/π (0.) x0.x0. Pressure on the bubble P= atmospheric pressure = 76x.6x980 T=7+40= k Now, PV/T=PV/T T=8.7 C T=8.7-7=0.7 C (a)calculate the value of ratio of specific heats ( ) for a (a) Diatomic gas and Poly atomic gas (b)explain why there is no atmosphere on moon. Uranium has two isotopes of masses and 8 units. If both are present in Uranium hexafluoride gas which would have the larger average speed? If atomic mass of fluorine is 9 units, estimate the percentage difference in speeds at any temperature. 4 Ravi has to attend an interview He was not well. He took the help of his friend Raghavan. On the way to office Ravi felt giddy. He vomited on his dress. Raghavan washed his shirt. He made Ravi to drink enough amount of water. In spite of doing.a foul smell was coming from the shirt. Then Raghavan purchased a scent bottle from the nearby cosmetics and applied on Ravi. Ravi attended the interview, performed well. Finally he was selected. a)what values do you find in Raghavan? b)the velocity of air molecules is nearly 00m/s.But the smell of scent spreads very slowly, why? Geeta a student of class xi has found the factors on which the rms speed of gas molecules depends and arrived at the expression Vrms = RT/M.She wants to know how the speed of gas molecules gets affected on the surface of moon and arrived at the conclusion that moon has no

9 atmosphere a)what values does Geetaposses? b)explain why moon has no atmosphere? An ideal gas or a perfect gas is that which strictly obeys the gas laws ( such as Boyle s law, Charles law etc.). Two essential characteristics of an ideal gas are : Size of molecule of an ideal gas is 0. There is no force of attraction or repulsion amongst the molecules of an ideal gas. In view of this, no real gas is perfect. Read the above passage and answer the following questions:. Under what condition do some real gases behave as nearly perfect gases?. What does this concept imply in day to day life? According to the law of equipartition of energy,for any dynamical system in thermal equilibrium,the total energy is distributed equally amongst all the degrees of freedom. The energy associated with each molecule per degree of freedom is ½ K B T. Read the above passage and answer the following question a)what is the internal energy of one gram mole of a polyatomic gas having n degrees of freedom? b) What values of life do you learn from this law?

(2) The volume of molecules is negligible in comparison to the volume of gas. (3) Molecules of a gas moves randomly in all direction.

9.1 Kinetic Theory of Gases : Assumption (1) The molecules of a gas are identical, spherical and perfectly elastic point masses. (2) The volume of molecules is negligible in comparison to the volume of