Chapter 13 - KINETIC THEORY

A box of 1.00m3 is filled with nitrogen at 1.50 atm at 300K. The
box has a hole of an area 0.010 mm2. How much time is required
for the pressure to reduce by 0.10 atm, if the pressure outside is
1 atm.

Ten small planes are flying at a speed of 150 km/h in total
darkness in an air space that is 20 × 20 × 1.5 km3 in volume. You
are in one of the planes, flying at random within this space with
no way of knowing where the other planes are. On the average
about how long a time will elapse between near collision with
your plane. Assume for this rough computation that a saftey
region around the plane can be approximated by a sphere of
radius 10m.

Explain why
(a) there is no atmosphere on moon.
(b) there is fall in temperature with altitude.

An insulated container containing monoatomic gas of molar mass
m is moving with a velocity o v . If the container is suddenly
stopped, find the change in temperature

Calculate the number of degrees of freedom of molecules of
hydrogen in 1 cc of hydrogen gas at NTP.

A ballon has 5.0 g mole of helium at 7°C. Calculate
(a) the number of atoms of helium in the balloon,
(b) the total internal energy of the system.

When air is pumped into a cycle tyre the volume and pressure of
the air in the tyre both are increased. What about Boyle’s law in
this case?

We have 0.5 g of hydrogen gas in a cubic chamber of size 3cm
kept at NTP. The gas in the chamber is compressed keeping the
temperature constant till a final pressure of 100 atm. Is one
justified in assuming the ideal gas law, in the final state?

A gas mixture consists of molecules of types A, B and C with
masses A B C m >m >m . Rank the three types of molecules in
decreasing order of (a) average K.E., (b) rms speeds.

The container shown in Fig. 13.6 has two chambers, separated
by a partition, of volumes V1 = 2.0 litre and V2= 3.0 litre. The
chambers contain 1 2 μ = 4.0andμ = 5.0 moles of a gas at
pressures p1 = 1.00 atm and p2 = 2.00 atm. Calculate the pressure
after the partition is removed and the mixture attains equilibrium.

A gas mixture consists of 2.0 moles of oxygen and 4.0 moles of
neon at temperature T. Neglecting all vibrational modes, calculate
the total internal energy of the system. (Oxygen has two rotational
modes.)

The molecules of a given mass of a gas have root mean square
speeds of 100 m s−1 at 27°C and 1.00 atmospheric pressure.
What will be the root mean square speeds of the molecules of the
gas at 127°C and 2.0 atmospheric pressure?

The volume of a given mass of a gas at 27°C, 1 atm is 100 cc.
What will be its volume at 327°C?

Calculate the number of atoms in 39.4 g gold. Molar mass of
gold is 197g mole–1.

The earth with out its atmosphere would be inhospitably cold. Explain Why?

If Nine particles have speeds of 5, 8, 12, 12, 12, 14, 14, 17 and 20 m/s. find :®
1) the average speed
2) the Most Probable speed
3) the r.m.s. Speed of the particles?

Show that constant – temperature bulk modulus K of an ideal gas is the pressure
P of the gas?

At what temperature is the root mean square speed of an atom in an argon gas

cylinder equal to the r.m.s speed of a helium gas atom at- 20⁰C? Given Atomic

Mass of Ar = 39.9 and of He = 4.0?

Derive the Boyle’s law using kinetic theory of gases?

If a certain mass of gas is heated first in a small vessel of volume V₁ and then in a

large vessel of volume V₂. Draw the P – T graph for two cases?

What happens when an electric fan is switched on in a closed room?

What is Mean free path?

Find out the ratio between most probable velocity, average velocity and root
Mean square Velocity of gas molecules?

Given Samples of 1 cm³ of Hydrogen and 1 cm³ of oxygen, both at N. T. P. which

sample has a larger number of molecules?

What is the relation between pressure and kinetic energy of gas?

State Graham’s law of diffusion and derive it?

A tank of volume 0.3m3 contains 2 moles of Helium gas at 20⁰C. Assuming the

helium behave as an ideal gas;

1) Find the total internal energy of the system.

2) Determine the r. m. s. Speed of the atoms.

Determine the volume of 1 mole of any gas at s. T. P., assuming it behaves like an
ideal gas?

Calculate the degree of freedom for monatomic, diatomic and triatomic gas?

Two perfect gases at absolute temperature T₁ and T₂ are mixed. There is no loss

of energy. Find the temperature of the mixture if the masses of molecules are m1

and m2 and number of molecules is n₁ and n₂?

At very low pressure and high temperature, the real gas behaves like ideal gas.
Why?

If a vessel contains 1 mole of O₂ gas (molar mass 32) at temperature T. The

pressure of the gas is P. What is the pressure if an identical vessel contains 1

mole of He at a temperature 2 T?

Why the temperature less than absolute zero is not possible?

On what factors, does the average kinetic energy of gas molecules depend?

State the law of equi-partition of energy?

What are the assumptions of kinetic theory of gas?

Derive Avogadro’s law?

Four molecules of gas have speeds 2, 4, 6, 8, km/s. respectively.
Calculate 1) Average speed
2) Root Mean square speed?

Air pressure in a car tyre increases during driving? Why?

Show that average kinetic energy of translation per molecule of gas is directly
proportional to the absolute temperature of gas?

State Charles’s law? If air is filled in a vessel at 600c. To what temperature should it be heated in order that1/3rdof air may escape out of vessel?

What is an ideal perfect gas?

. Given below are densities of some solids and liquids. Give rough estimates of the size of their atoms:

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 = n₁ exp [-mg (h₂ – h₁)/ k_BT]
Where n₂, n₁ refer to number density at heights h₂ and h₁ respectively. Use this relation to derive the equation for sedimentation equilibrium of a suspension in a liquid column:
n₂ = n₁ exp [-mg N_A(ρ - P′) (h₂ –h₁)/ (ρRT)]
Where ρ is the density of the suspended particle, and ρ’ that of surrounding medium. [N_A is Avogadro’s number, and R the universal gas constant.] [Hint: Use Archimedes principle to find the apparent weight of the suspended particle.]

From a certain apparatus, the diffusion rate of hydrogen has an average value of 28.7 cm³ s^–1. The diffusion of another gas under the same conditions is measured to have an average rate of 7.2 cm³ s^–1. Identify the gas.
[Hint: Use Graham’s law of diffusion: R₁/R₂ = (M₂/M₁)^1/2, where R₁, R₂ are diffusion rates of gases 1 and 2, and M₁ and M₂ their respective molecular masses. The law is a simple consequence of kinetic theory.]

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?

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 Å. Compare the collision time with the time the molecule moves freely between two successive collisions (Molecular mass of N₂ = 28.0 u).

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). 

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 v_rms the largest?

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).

Estimate the total number of air molecules (inclusive of oxygen, nitrogen, water vapour and other constituents) in a room of capacity 25.0 m³ at a temperature of 27 °C and 1 atm pressure.

 An air bubble of volume 1.0 cm³ rises from the bottom of a lake 40 m deep at a temperature of 12 °C. To what volume does it grow when it reaches the surface, which is at a temperature of 35 °C?

An oxygen cylinder of volume 30 litres has an initial gauge pressure of 15 atm and a temperature of 27 °C. After some oxygen is withdrawn from the cylinder, the gauge pressure drops to 11 atm and its temperature drops to 17 °C. Estimate the mass of oxygen taken out of the cylinder (R = 8.31 J mol^–1 K^–1, molecular mass of O₂ = 32μ).

 Figure 13.8 shows plot of PV/T versus P for 1.00×10^–3 kg of oxygen gas at two different temperatures.

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.

 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 3Å.