Kinetic Theory

Class 11 NCERT Physics

NCERT

1   Estimate the fraction of molecular volume to the actual volume occupied by oxygen gas at STP. Take the diameter of an oxygen molecule to be $3 Å $.

Solution :

Diameter of an oxygen molecule,$d=3 Å.$$\\$ Radius,$r=\dfrac{d}{2}=\dfrac{3}{2}=1.5 Å \\ 1.5 * 10^{-8} cm$$\\$ Actual volume occupied by $1$ mole of oxygen gas at STP =$ 22400 cm^3$$\\$ Molecular volume of oxygen gas,$V=\dfrac{4}{3}\pi r^3.N$$\\$ Where, $N$ is Avogadro’s number $\\$ $=6.023*10^{23}$ molecules/mole$\\$ $\therefore V=\dfrac{4}{3}*3.14*(1.5*10^{-8})^3*6.023*10^23\\ =8.51 cm^3$$\\$ Ratio of the molecular volume to the actual volume of oxygen$\\$ $=\dfrac{8.51}{22400}=3.8*10^{-4}$

2   Molar volume is the volume occupied by $1$ mole of any (ideal) gas at standard temperature and pressure (STP: $1$ atmospheric pressure, $0^o C$ ). Show that it is $22.4$ litres.

Solution :

The ideal gas equation relating pressure $(P)$, volume $(V)$, and absolute temperature $(T)$ is given as: $PV = nRT$ Where, $R$ is the universal gas constant =$8.314 J mol^{-1}K^{-1}$$\\$ $n=$ Number of moles $=1$$\\$ $T =$ Standard temperature$ = 273 K$$\\$ $P =$ Standard pressure =$1$ atm =$1.013 *10^5 Nm ^{-2}$$\\$ $\therefore V=\dfrac{nRT}{P}\\ =\dfrac{1*8.314*273}{1.013*10^5}\\ =0.0224 m^3\\ =22.4 \text{litres}$$\\$ Hence, the molar volume of a gas at STP is $22.4$ litres.

3   Figure shows plot of $PV /T$ versus $P$ for $1.00 * 10^{-3} kg$ of oxygen gas at two different temperatures.$\\$ (a)What does the dotted plot signify?$\\$ (b)Which is true: $T_1 > T_2$ or$T_1< T_2$ ?$\\$ (c)What is the value of $PV/T$ where the curves meet on the y-axis?$\\$ (d)If we obtained similar plots for $1.00 * 10^{-3} kg$ of hydrogen, would we get the same value of $PV/T$ at the point where the curves meet on the y-axis? If not, what mass of hydrogen yields the same value of $PV/T$ (for low pressure high temperature region of the plot)? (Molecular mass of $H_2 = 2.02 u ,$ of $O_2 = 32.0 u $ , $R=8.31 J mo1^{ -1} K^{-1} . )$

Solution :

(a)The dotted plot in the graph signifies the ideal behaviour of the gas, i. e. , the ratio $\dfrac{PV}{T}$ is equal. $\mu R$($\mu $ is the number of moles and $R$ is the universal gas constant )is a constant quality. It is not dependent on the pressure of the gas. $\\$ (b)The dotted plot in the given graph represents an ideal gas. The curve of the gas at temperature $T_1$ is closer to the dotted plot thanthe curve of the gas at temperature $T_2$ . A real gas approaches the behaviour of an ideal gas when its temperature increases. Therefore, $T_1 > T_2$ is true for the given plot.$\\$ (c ) The value of the ratio $PV/T,$ where the two curves meet, is $\mu R$ . This is because the ideal gas equation is given as:$\\$ $PV=\mu RT\\ \dfrac{PV}{T}=\mu R$$\\$

Where $\\$ $P$ is the pressure $\\$ $T$ is the temperature$\\$ $V$ is the volume$\\$ $\mu$ is the number of moles$\\$ $R$ is the universal constant$\\$ Molecular mass of oxygen $= 32.0 g$$\\$ Mass of oxygen $=1*10^{-3}kg =1g$$\\$ $R=8.314 J mole^{-1}K^{-1}\\ \therefore \dfrac{PV}{T}=\dfrac{1}{32}*8.314\\ =0.26 J K^{-1}$$\\$ Therefore, the value of the ratio $PV/T$, where the curves meet on the y-axis, is $0.26 J K^{-1}$$\\$ (d) If we obtain similar plots for $1.00 *10^{-3}$ kg of hydrogen, then we will not get the same value of $PV/T$ at the point where the curves meet the y-axis. This is because the molecular mass of hydrogen $(2.02 u)$ is different from that of oxygen $(32.0 u).$$\\$ We have:$\\$ $\therefore \dfrac{PV}{T}=0.26 JK^{-1}\\ R=8.314 J mole^{-1}K^{-1}$$\\$ Molecular mass (M) of $H_2=2.02 u$$\\$ $\dfrac{PV}{T}=\mu R $ at constant temperature $\\$ $\dfrac{PV}{T}=\mu R $ at constant temperature $\\$ where, $ \mu =\dfrac{m}{M}$$\\$ $m=\text{Mass of H_2}\\ m=\dfrac{PV}{T}* \dfrac{M}{R}\\ =\dfrac{0.26* 2.02}{8.31}\\ =6.3* 10^{-2}g\\ =6.3*10^{-5}k g$$\\$ Hence ,$6.3*10^{-5}kg $ of $H_2$ will yield the same value of $\dfrac{PV}{T}$

4   An oxygen cylinder of volume $30$ litres has an initial gauge pressure of $15$ atm and a temperature of $27^o C$ . After some oxygen is withdrawn from the cylinder, the gauge pressure drops to $11$ atm and its temperature drops to $17^o C$ . Estimate the mass of oxygen taken out of the cylinder ($ R =8.31 J mol ^{-1} K^{-1}$ , molecular mass of $O_2 = 32 u $).

Solution :

Volume of oxygen,$V_1=30$litres =$30*10^{-3}m^3$$\\$ Gauge pressure,$P_1=15 atm=15*1.013*10^5 Pa$$\\$ Temperature,$T_1=27^oC=300K$$\\$ Universal gas constant, $R=8.314 J mole^{-1}K^{-1}$$\\$ Let the initial number of moles of oxygen gas in the cylinder be $n_1.$$\\$ The gas equation is given as:$\\$ $P_1V_1=n_1RT_1\\ \therefore n_1=\dfrac{P_1V_1}{RT_1}\\ =\dfrac{15.195*10^5*30*10^{-3}}{(8.314)*300}\\ =18.276$ $\\$ But $ n_1=\dfrac{m_1}{M}$ $\\$ Where,$\\$ $m_1=$Initial mass of oxygen$\\$ M=Molecular mass of oxygen=$32 g$$\\$ $\therefore m_1=n_1M=18.276 *32 \\ =584.84 g$$\\$ After some oxygen is withdrawn from the cylinder, the pressure and temperature reduces.$\\$ Volume,$V_2=30 $ litres $=30*10^{-3}m^3$$\\$ Gauge pressure, $P_2 = 11 atm = 11 * 1.013 * 10^5 Pa$$\\$ Temperature, $T_2 = 17^o C = 290 K$$\\$ Let $n_2$ be the number of moles of oxygen left in the cylinder. The gas equation is given as:$\\$ $P_2V_2=n_2RT_2$

$\therefore n_2=\dfrac{P_2V_2}{RT_2}\\ =\dfrac{11.143*10^5*30*10^{-3}}{8.314*290}\\ =13.86$$\\$ But,$n_2=\dfrac{m_2}{M}$$\\$ Where, $\\$ $m_2$ is the mass of oxygen remaining in the cylinder$\\$ $\therefore m_2=n_2 M=13.86*32=443.52 g$$\\$ The mass of oxygen taken out of the cylinder is given by the relation:$\\$ Initial mass of oxygen in the cylinder - Final mass of oxygen in the cylinder$\\$ $=m_1-m_2\\ =584.84 g-443.52 g\\ =141.32 g\\ =0.141 kg$$\\$ Therefore, $0.141 kg$ of oxygen is taken out of the cylinder.

5   An air bubble of volume $1.0 cm^3$ rises from the bottom of a lake $40 m$ deep at a temperature of $12^o C$ . To what volume does it grow when it reaches the surface, which is at a temperature of $35^o C$ ?

Solution :

Volume of the air bubble, $V_1 = 1.0 cm^3 = 1.0* 10^{-6} m^3$$\\$ Bubble rises to height, $d = 40 \ m$$\\$ Temperature at a depth of $40 m, T_1 = 12^o C = 285 K$$\\$ Temperature at the surface of the lake, $T_2= 35^o C= 308 K$$\\$ The pressure on the surface of the lake:$\\$ $P_2=1 atm =1*1.013*10^5 Pa$$\\$ The pressure at the depth of $40 m:$$\\$ $P_1 = 1 atm +d \rho g$$\\$ Where,$\\$ $\rho$ is the density of water $= 10^3 kg / m^3$$\\$ $g$ is the acceleration due to gravity $=9.8 m/s^2$$\\$ $\therefore P_1=1.1013*10^5+40*10^3*9.8\\ =493300 Pa$$\\$ We have:$\dfrac{P_1V_1}{T_1}=\dfrac{P_2V_2}{T_2}$$\\$ Where, $V _2$ is the volume of the air bubble when it reaches the surface$\\$ $V_2=\dfrac{P_1V_1T_2}{T_1P_2}\\ =\dfrac{(493300)(1.0*10^{-6})308}{285*1.013*10^5}\\ =5.263*10^{-6}m^3 or 5.263 c m^3$$\\$ Therefore, when the air bubble reaches the surface, its volume becomes $5.263 cm^3 .$

6   Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity $25.0 m^3$ at a temperature of $27^oC$ and $1$ atm pressure.

Solution :

Volume of the room, $V = 25.0 m^3$$\\$ Temperature of the room, $T = 27 ^o C = 300 K$$\\$ Pressure in the room, $P = 1 atm = 1 * 1.013 * 10^ 5 Pa$$\\$ The ideal gas equation relating pressure (P), Volume (V), and absolute temperature (T) can be written as:$\\$ $PV = k_ B NT$$\\$ Where,$\\$ $K _B$ is Boltzmann constant = $1 .38 *10 ^{-23} m ^2 kg s^{-2} K^{-1 }N$ is the number of air molecules in the room$\\$ $N=\dfrac{PV}{k_BT}=\dfrac{1.013 * 10^5*25}{1.38*10^{-23}*300}=6.11*10^{26} \text{molecules}$ Therefore, the total number of air molecules in the given room is $ 6.11 *10^{26}$ .

7   Estimate the average thermal energy of a helium atom at$\\$ (i) room temperature $(27 ^o C )$ ,$\\$ (ii) the temperature on the surface of the Sun $(6000 K),$$\\$ (iii) the temperature of $10$ million Kelvin (the typical core temperature in the case of a star).

Solution :

(i) At room temperature, $T = 27^o C = 300 K$$\\$ Average thermal energy$=\dfrac{3}{2}kT$$\\$ Where $k $ is Boltzmann constant =$ 1.38 * 10^{-23} m^2 kg s^{-2}K^{-1}$$\\$ $\therefore \dfrac{3}{2}kT=\dfrac{3}{2}*1.38*10^{-38}*300\\ =6.21* 10^{-21}J$$\\$ Hence, the average thermal energy of a helium atom at room temperature $(27^oC)$ is $ 6.21 * 10^{-21}J.$$\\$ (ii) On the surface of the sun, $T = 6000 K$$\\$ $=\dfrac{3}{2}*1.38*10^{-38}*6000\\ =1.241*10^{-19}J$$\\$ Hence, the average thermal energy of a helium atom on the surface of the sun is$\\$ $1.241*10^{-19}J$$\\$ (iii) At temperature, $T = 10^7 K$$\\$ Average thermal energy $=\dfrac{3}{2}kT$$\\$ $=\dfrac{3}{2}*1.38*10^{-23}*10^7$$\\$ $=2.07 * 10^{-16} J$$\\$ Hence, the average thermal energy of a helium atom at the core of a star is $2.07 * 10^{-16} J$

8   Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains neon (monatomic), the second contains chlorine (diatomic), and the third contains uranium hexafluoride (polyatomic).$\\$ (a) Do the vessels contain equal number of respective molecules?$\\$ (b) Is the root mean square speed of molecules the same in the three cases? If not, in which case is $v _{rms}$ the largest?

Solution :

(a)Yes. All contain the same number of the respective molecules. Since the three vessels have the same capacity, they have the same volume. Hence, each gas has the same pressure, volume, and temperature. According to Avogadro’s la, the three vessels will contain an equal number of the respective molecules. This n umber is equal to Avogadro’s number, $N = 6.023 * 10^{ 23}$ .$\\$ (b)No. The root mean square speed of neon is the largest. The root mean square speed ($v _{rms} )$ of a gas of mass $m$, and temperature $T$, is given by the relation:$\\$ $v_{rms}=\sqrt{3kT}{m}$$\\$ Where,$ k$ is Boltzmann constant $\\$ For the given gases, $k$ and $T$ are constants.$\\$ Hence $ v_{ rms } $ depends only on the mass of the atoms, i.e.,$\\$ $ v_{rms} \infty \sqrt{\dfrac{1}{m}}$$\\$ Therefore, the root mean square speed of the molecules in the three cases is not the same. Among neon, chlorine, and uranium hexafluoride, the mass of neon is the smallest. Hence, neon has the largest root mean square speed among the given gases.

9   At what temperature is the root mean square speed of an atom in an argon gas cylinder equal to the $rms$ speed of a helium gas atom at $-20^oC $ ? (atomic mass of $Ar = 39.9 u,$ of He$ = 4.0 u$).

Solution :

Temperature of the helium atom, $T_{ He} = - 20^o C = 253 K$$\\$ Atomic mass of argon, $M_{ Ar }= 39.9 u$$\\$ Atomic mass of helium, $M_{ He }= 4.0 u$$\\$ Let , $( v_{ rms} )$ Ar be the $rms$ speed of argon.$\\$ Let $(v_{ rms} )$ He be the $rms$ speed of helium.$\\$ The $rms$ speed of argon is given by:$\\$ $(v_{rms})_Ar=\sqrt{3RT_{Ar}}{M_{Ar}}.....(i)$$\\$ Where,$\\$ R is the universal gas constant$\\$ $T_Ar$ is temperature of argon gas$\\$ The rms speed of helium is given by:$\\$ $(v_{rms})_{He}=\sqrt{\dfrac{3RT_{He}}{M_{He}}} .....(ii)$$\\$ It is given that:$\\$ $(v_{rms})_{Ar}=(v_{rms})_He\\ \sqrt{3RT_{Ar}{M_{Ar}}}=\sqrt{\dfrac{3RT_{He}}{M_{He}}}\\ \dfrac{T_{Ar}}{M_{Ar}}=\dfrac{T_{He}}{M_{He}}\\ T_{At}=\dfrac{T_{He}}{M_{He}}*M_{Ar}\\ =\dfrac{253}{4}*39.9\\ =2523.675=2.52*10^3 K$$\\$ Therefore, the temperature of the argon atom is $2.52*10^3 K$

10   Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at $2.0$ at $m$ and temperature $17^o C$ . Take the radius of a nitrogen molecule to be roughly $1.0 Å$ . Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of $N _2 = 28. 0 u$ ).

Solution :

Mean free path $=1.11 * 10^{-7 }m$$\\$ Collision frequency $=4.58 * 10^9 s^{-1}$$\\$ Successive collision time $\approx 500 ×$ (Collision time)$\\$ Pressure inside the cylinder containing nitrogen,$ P = 2.0 atm = 2.026 * 10^5 P a$$\\$ Temperature inside the cylinder, $T =17 ^o C = 290 K$$\\$ Radius of a nitrogen molecule, $r = 1.0 Å = 1 * 10^{ 10} m$$\\$ Diameter, $d = 2 *1 * 10 ^{10} =2 * 10^{ 10} m$$\\$ Molecular mass of nitrogen, $M = 28.0 g = 28 * 10 ^{-3 }kg$$\\$ The root mean square speed of nitrogen is given by the relation:$\\$ $v_{rms}=\sqrt{\dfrac{3RT}{M}}$$\\$ Where,$\\$ $R$ is the universal gas constant $=8.314 J mole^{-1}K^{-1}$$\\$ $\therefore v_{rms}=\sqrt{\dfrac{3*8.314 * 290}{28*10^{-3}}}\\ =508.26 m/s$$\\$ The mean free path (l) is given by the relation:$\\$ $l=\dfrac{kT}{\sqrt{2}*d^2* P}$$\\$ Where,$\\$ $k$ is the Boltzmann constant $= 1.38 * 10 ^{-23} kg m ^2 s ^{-2} K ^{-1}$$\\$ $\therefore l=\dfrac{1.38*10^{-23}*290}{\sqrt{2}*3.14*(2*10^{-10})^2*2.026*10^5}$$\\$ $=1.11*10^{-7}m$$\\$ Collision frequency $ =\dfrac{v_{rms}}{l}$$\\$ $=\dfrac{508.26}{1.11*10^{-7}}=4.58*10^9 s^{-1}$$\\$ Collision time is given as:$\\$ $T=\dfrac{d}{v_{rms}}$$\\$ $=\dfrac{2*10^{-10}}{508.26}=3.93*10^{-13}s$$\\$ Time taken between successive collisions:$\\$ $T'=\dfrac{l}{v_{rms}}$$\\$ $=\dfrac{1.11*10^{-7}m}{508.26m/s}=2.18*10^{-10}s\\ \therefore \dfrac{T'}{T}=\dfrac{2.18*10^{-10}}{3.93*10^{-13}}=500$$\\$ Hence, the time taken between successive collisions is $500$ times the time taken for a collision.