Ncert Solutions of Physics Part-2 for Chapter 5 : Kinetic Theory

Q1 :  
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 3Ã….
Answer :
Diameter of an oxygen molecule, d= 3Ã…
Radius, r  = 1.5 Ã… = 1.5 × 10–8cm
Actual volume occupied by 1 mole of oxygen gas at STP = 22400 cm3
Molecular volume of oxygen gas,
Where, N is Avogadro's number = 6.023 × 1023 molecules/mole
Ratio of the molecular volume to the actual volume of oxygen 
= 3.8 × 10–4
Q2 :  
Molar volume is the volume occupied by 1 mol of any (ideal) gas at standard temperature and pressure (STP: 1 atmospheric pressure, 0 °C). Show that it is 22.4 litres.


Answer :
The ideal gas equation relating pressure (P), volume (V), and absolute temperature (T) is given as:
PV= nRT
Where,
Ris the universal gas constant = 8.314 J mol–1K–1
n= Number of moles = 1
T= Standard temperature = 273 K
P= Standard pressure = 1 atm = 1.013 × 105Nm–2
= 0.0224 m3
= 22.4 litres
Hence, the molar volume of a gas at STP is 22.4 litres.
Q3 :  
Figure 13.8 shows plot of PV/T versus Pfor 1.00 x 10-3kg of oxygen gas at two different temperatures.
(a) What does the dotted plot signify?
(b) Which is true: T1 > T2or T1< T2?
(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 x 10-3kg of hydrogen, would we get the same value of PV/Tat 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 H2 = 2.02 u, of O2= 32.0 u, R = 8.31 J mo1-1K-1.)


Answer :
(a) The dotted plot in the graph signifies the ideal behaviour of the gas, i.e., the ratio is equal.μR (μ 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 temperatureT1 is closer to the dotted plot than the curve of the gas at temperature T2. A real gas approaches the behaviour of an ideal gas when its temperature increases. Therefore, T1 > T2 is true for the given plot.
(c) The value of the ratio PV/T, where the two curves meet, is ÃŽÂ¼R. This is because the ideal gas equation is given as:
PV = ÃŽÂ¼RT
Where,
P is the pressure
T is the temperature
V is the volume
μ is the number of moles
R is the universal constant
Molecular mass of oxygen = 32.0 g
Mass of oxygen = 1 × 10–3 kg = 1 g
R = 8.314 J mole–1 K–1
= 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 ofPV/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:
R = 8.314 J mole–1 K–1
Molecular mass (M) of H2 = 2.02 u
m = Mass of H2
= 6.3 × 10–2 g = 6.3 × 10–5 kg
Hence, 6.3 × 10–5 kg of H2 will yield the same value of PV/T.
Q4 :  17 °C. Estimate the mass of oxygen taken out of the cylinder (

Answer :
Volumeof oxygen, V1= 30 litres = 30 × 10–3m3
Gauge pressure, P1= 15 atm = 15 × 1.013 × 105Pa
Temperature, T1= 27°C = 300 K
Universal gas constant, R= 8.314 J mole–1K–1
Let the initial number of moles of oxygen gas in the cylinder be n1.
The gas equation is given as:
P1V1= n1RT1
 = 18.276
But, 
Where,
m1= Initial mass of oxygen
M= Molecular mass of oxygen = 32 g
m1= n1M = 18.276 × 32 = 584.84 g
Aftersome oxygen is withdrawn from the cylinder, the pressure and temperature reduces.
Volume, V2= 30 litres = 30 × 10–3m3
Gauge pressure, P2= 11 atm = 11 × 1.013 × 105Pa
Temperature, T2= 17°C = 290 K
Let n2be the number of moles of oxygen left in the cylinder.
The gas equation is given as:
P2V2= n2RT2
 = 13.86
But, 
Where,
m2is the mass of oxygen remaining in the cylinder
m2= n2M = 13.86 × 32 = 453.1 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
= m1– m2
= 584.84 g – 453.1 g
= 131.74 g
= 0.131 kg
Therefore, 0.131 kg of oxygen is taken out of the cylinder.
Q5 :  An air bubble of volume 1.0 cm

Answer :
Volume of the air bubble, V1 = 1.0 cm3 = 1.0 × 10–6 m3
Bubble rises to height, d = 40 m
Temperature at a depth of 40 m, T1 = 12°C = 285 K
Temperature at the surface of the lake, T2 = 35°C = 308 K
The pressure on the surface of the lake:
P2 = 1 atm = 1 ×1.013 × 105 Pa
The pressure at the depth of 40 m:
P1 = 1 atm + dÃÂg
Where,
àis the density of water = 103 kg/m3
g is the acceleration due to gravity = 9.8 m/s2
P1 = 1.013 × 105 + 40 × 103 × 9.8 = 493300 Pa
We have: 
Where, V2 is the volume of the air bubble when it reaches the surface
= 5.263 × 10–6 m3 or 5.263 cm3
Therefore, when the air bubble reaches the surface, its volume becomes 5.263 cm3.
Q6 :  Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity 25.0 m

Answer :
Volume of the room, V= 25.0 m3
Temperature of the room, T= 27°C = 300 K
Pressure in the room, P= 1 atm = 1 × 1.013 × 105Pa
The ideal gas equation relating pressure (P), Volume (V), and absolute temperature (T) can be written as:
PV = kBNT
Where,
KBis Boltzmann constant = 1.38 × 10–23m2 kg s–2K–1
Nis the number of air molecules in the room
 = 6.11 × 1026molecules
Therefore, the total number of air molecules in the given room is 6.11 × 1026.
Q7 :  
Estimate the average thermal energy of a helium atom at (i) room temperature (27 °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).


Answer :
(i) At room temperature, T= 27°C = 300 K
Average thermal energy 
Where k is Boltzmann constant = 1.38 × 10–23m2 kg s–2K–1
= 6.21 × 10–21J
Hence, the average thermal energy of a helium atom at room temperature (27°C) is 6.21 × 10–21J.
(ii) On the surface of the sun, T= 6000 K
Average thermal energy 
= 1.241 × 10–19J
Hence, the average thermal energy of a helium atom on the surface of the sun is 1.241 × 10–19J.
(iii) At temperature, T= 107K
Average thermal energy 
= 2.07 × 10–16J
Hence, the average thermal energy of a helium atom at the core of a star is 2.07 × 10–16J.
Q8 :  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). Do the vessels contain equal number of respective molecules? Is the root mean square speed of molecules the same in the three cases? If not, in which case is

Answer :
Yes.All contain the same number of the respective molecules.
No. The root mean square speed of neon is the largest.
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 law, the three vessels will contain an equal number of the respective molecules. This number is equal to Avogadro's number, N= 6.023 × 1023.
The root mean square speed (vrms) of a gas of mass m, and temperature T, is given by the relation:
Where, k is Boltzmann constant
For the given gases, kand T are constants.
Hence vrmsdepends only on the mass of the atoms, i.e.,
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.
Q9 :  
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 °C? (atomic mass of Ar = 39.9 u, of He = 4.0 u).


Answer :
Temperatureof the helium atom, THe= –20°C= 253 K
Atomic mass of argon, MAr = 39.9 u
Atomic mass of helium, MHe= 4.0 u
Let, (vrms)Ar be the rms speed of argon.
Let (vrms)He be the rms speed of helium.
The rms speed of argon is given by:
 … (i)
Where,
Ris the universal gas constant
TAris temperature of argon gas
The rms speed of helium is given by:
 … (ii)
It is given that:
(vrms)Ar = (vrms)He
= 2523.675 = 2.52 × 103K
Therefore, the temperature of the argon atom is 2.52 × 103K.
Q10 :  Estimate the mean free path and collision frequency of a nitrogen molecule in a cylinder containing nitrogen at 2.0 atm and temperature 17 °C. Take the radius of a nitrogen molecule to be roughly 1.0 Ô¦. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of N

Answer :
Mean free path = 1.11 × 10–7m
Collision frequency = 4.58 × 109s–1
Successive collision time Ã¢â€°Ë†500 × (Collision time)
Pressure inside the cylinder containing nitrogen, P= 2.0 atm = 2.026 × 105Pa
Temperature inside the cylinder, T= 17°C =290 K
Radius of a nitrogen molecule, r= 1.0 Ã… = 1 × 1010m
Diameter, d = 2 × 1 × 1010= 2 × 1010m
Molecular mass of nitrogen, M= 28.0 g = 28 × 10–3kg
The root mean square speed of nitrogen is given by the relation:
Where,
Ris the universal gas constant = 8.314 J mole–1K–1
 = 508.26 m/s
The mean free path (l) is given by the relation:
Where,
kis the Boltzmann constant = 1.38 × 10–23kg m2s–2K–1
= 1.11 × 10–7m
Collision frequency 
= 4.58 × 109s–1
Collision time is given as:
 = 3.93 × 10–13s
Time taken between successive collisions:
 = 2.18 × 10–10s
∴ 
Hence, the time taken between successive collisions is 500 times the time taken for a collision.
Q11 :  
A metre long narrow bore held horizontally (and closed at one end) contains a 76 cm long mercury thread, which traps a 15 cm column of air. What happens if the tube is held vertically with the open end at the bottom?


Answer :
Length of the narrow bore, L= 1 m = 100 cm
Length of the mercury thread, l= 76 cm
Length of the air column between mercury and the closed end, la= 15 cm
Since the boreis held vertically in air with the open end at the bottom, the mercury length that occupies the air space is: 100 – (76 + 15) = 9 cm
Hence, the total length of the air column = 15 + 9 = 24 cm
Let h cm of mercury flow out as a result of atmospheric pressure.
Length of the air column in the bore= 24 + hcm
And, length of the mercury column = 76 – hcm
Initial pressure, P1= 76 cm of mercury
Initial volume, V1= 15 cm3
Final pressure, P2= 76 – (76 – h) = h cm of mercury
Final volume, V2= (24 + h) cm3
Temperature remains constant throughout the process.
P1V1= P2V2
76 × 15 = h (24 + h)
h2+ 24h â€“ 1140 = 0
= 23.8 cm or –47.8 cm
Height cannot be negative. Hence, 23.8 cm of mercurywill flow out from the boreand 52.2 cm of mercury will remain in it. The length of the air column will be 24 + 23.8 = 47.8 cm.
Q12 :  
From a certain apparatus, the diffusion rate of hydrogen has an average value of 28.7 cm3s-1. The diffusion of another gas under the same conditions is measured to have an average rate of 7.2 cm3s-1. Identify the gas.
[Hint:Use Graham's law of diffusion: R1/R2= (M2/M1)1/2, where R1, R2are diffusion rates of gases 1 and 2, and M1and M2their respective molecular masses. The law is a simple consequence of kinetic theory.]


Answer :
Rate of diffusion of hydrogen, R1 = 28.7 cm3s–1
Rate of diffusion of another gas, R2= 7.2 cm3s–1
According to Graham's Law of diffusion, we have:
Where,
M1is the molecular mass of hydrogen = 2.020 g
M2is the molecular mass of the unknown gas
= 32.09 g
32 g is the molecular mass of oxygen. Hence, the unknown gas is oxygen.
Q13 :  
A gas in equilibrium has uniform density and pressure throughout its volume. This is strictly true only if there are no external influences. A gas column under gravity, for example, does not have uniform density (and pressure). As you might expect, its density decreases with height. The precise dependence is given by the so-called law of atmospheres
n2= n1exp [-mg (h2 - h1)/ kBT]
Where n2, n1refer to number density at heights h2and h1respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:
n2= n1exp [-mg NA(à- P”²) (h2-h1)/ (ÃÂRT)]
Where ÃÂ is the density of the suspended particle, and ÃÂ' that of surrounding medium. [NAis Avogadro's number, and R the universal gas constant.] [Hint:Use Archimedes principle to find the apparent


Answer :
According to the law of atmospheres, we have:
n2= n1exp [-mg (h2 â€“ h1)/ kBT] … (i)
Where,
n1 is thenumber density at heighth1, and n2is the number density at height h2
mg is the weight of the particle suspended in the gas column
Density of the medium = ÃÂ'
Density of the suspended particle = ÃÂ
Mass of one suspended particle = m'
Mass of the medium displaced = m
Volume of a suspended particle = V
According to Archimedes'principle for a particle suspended in a liquid column, the effective weight of the suspended particle is given as:
Weight of the medium displaced – Weight of the suspended particle
= mg– m'g
Gas constant, R = kBN
 … (iii)
Substituting equation (ii) in place of mg in equation (i) and then using equation (iii), we get:
n2= n1exp [-mg (h2 â€“ h1)/ kBT]
= n1exp [-(h2 â€“ h1)]
= n1exp [-(h2 â€“ h1)]
Q14 :  
Given below are densities of some solids and liquids. Give rough estimates of the size of their atoms:
Substance
Atomic Mass (u)
Density (103 Kg m-3)
Carbon (diamond)
Gold
Nitrogen (liquid)
Lithium
Fluorine (liquid)
12.01
197.00
14.01
6.94
19.00
2.22
19.32
1.00
0.53
1.14
[Hint:Assume the atoms to be ‘tightly packed' in a solid or liquid phase, and use the known value of Avogadro's number. You should, however, not take the actual numbers you obtain for various atomic sizes too literally. Because of the crudeness of the tight packing approximation, the results only indicate that atomic sizes are in the range of a few Å].

Answer :
Substance
Radius (Ã…)
Carbon (diamond)
Gold
Nitrogen (liquid)
Lithium
Fluorine (liquid)
1.29
1.59
1.77
1.73
1.88
Atomic mass of a substance = M
Density of the substance = ÃÂ
Avogadro's number = N= 6.023 × 1023
Volume of each atom
Volume of Nnumber of molecules N … (i)
Volume of one mole of a substance = … (ii)
N =
For carbon:
M= 12.01 × 10–3kg,
à= 2.22 × 103kg m–3
= 1.29 Ã…
Hence, the radius of a carbon atom is 1.29 Ã….
For gold:
M= 197.00 × 10–3kg
à= 19.32 × 103kg m–3
= 1.59 Ã…
Hence, the radius of a gold atom is 1.59 Ã….
For liquid nitrogen:
M= 14.01 × 10–3kg
à= 1.00 × 103kg m–3
= 1.77 Ã…
Hence, the radius of a liquid nitrogen atom is 1.77 Ã….
For lithium:
M= 6.94 × 10–3kg
à= 0.53 × 103kg m–3
= 1.73 Ã…
Hence, the radius of a lithium atom is 1.73 Ã….
For liquid fluorine:
M= 19.00 × 10–3kg
ÃÂ = 1.14 × 103kg

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