# Question-1

**Distinguish between average speed and r.m.s. speed. If three molecules have speeds u**

_{1},u_{2},u_{3, }what will be their average speed and r.m.s speed?**Solution:**

Average speed is the arithmetic mean of the speeds of the molecules

Average speed=

r.m.s. speed is the root mean square speed and is defined as the square root of the mean of the squares of different speeds of the individual molecules.

r.m.s. speed=.

# Question-2

**Helium gas is filled in a closed vessel whose coefficient of thermal expansion is negligible. Find the average kinetic energy of helium atom when it is heated from 300K to 600K.**

**Solution:**

The average kinetic energy becomes double because average K. E. per molecule of gas is directly proportional to temperature in Kelvin.

# Question-3

**There are N molecules of gas in a box, if the number of molecules is increased to 2N, what will be the effect on the pressure of the gas and on the root-mean-square speed of the molecules?**

**Solution:**

P =

i.e., P ∝ n,

Therefore pressure will be double when number of molecules n become 2n.

As kinetic energy of the gas =mnc^{2}, K. E. of the gas also becomes double when n changes to 2n.

Thus r.m.s. velocity of molecules of a gas only depends on the temperature.

# Question-4

**Define degrees of freedom.**

**Solution:**

The total number of independent quantities or coordinates which must be known to completely specify the position and configuration of a dynamic system is called the degrees of freedom of that system.

The total number of independent ways in which the particles of a system can acquire energy are called the degrees of freedom of that system.

Degree of freedom for various atomic systems of gas are as follows:

f = 3 for monoatomic gas molecule at all temperatures

f = 5 for diatomic gas molecule, if T lies between 250K and 500K

f = 3 for diatomic gas molecule, if T < 70K

f = 7 for diatomic gas molecule, if T>5000K

f = 6 for non-linear triatomic gas molecule

f = 7 for linear triatomic gas molecule

# Question-5

**Under what conditions do real gases obey the gas equation, PV = RT more strictly? Explain.**

**Solution:**

The basic properties of the molecules of an ideal gas are (i) zero size of the molecules and hence zero volume of the molecules and (ii) no mutual intermolecular forces between them.

At low pressure, the volume of the given gas becomes large. Therefore, the volume of the molecules becomes negligible in comparison to the volume of the gas. At high temperature, the molecules have large K.E. and so the effect of the intermolecular force on the motion of the molecules becomes negligible. Hence at low pressure and high temperature the real gases behaves as ideal gas and gas equation is obeyed.

# Question-6

**A gas is filled in a cylinder fitted with a piston at a definite temperature and pressure. Explain on the basis of kinetic theory why the pressure of gas decreases on pulling the piston out.**

**Solution:**

On pulling the piston out, the volume of the cylinder for the given gas increases. Due to this, the molecules of gas get more space to move about. As a result, of which less molecules will collide with the wall of the cylinder per second and hence less momentum is transferred to the wall per second. In addition to this, now these collisions take place on the larger area of the walls. Due to both these reasons, the pressure decreases.

# Question-7

**An air bubble of volume 1.0cm**

^{3}rises from the bottom of a lake 40m deep at a temperature of 12^{0}C. To what volume does it grow when it reaches the surface, which is at a temperature of 35^{0}C?**Solution:**

When the air bubble is at 40m depth, then

V

_{1}= 1.0 cm

^{3 }= 1.0 ×10

^{-6 }m

^{3 }

T

_{1 }= 12

^{0}C = 12 + 273 = 285 K

P

_{1 }= 1atm + h

_{1}ρg

= 1.01 × 105 + 40 × 10

^{3}×9.8

= 493000 Pa.

# Question-8

**The velocities of three molecules are, 3ν , 4ν , and 5ν . Calculate their root-mean-square velocity.**

**Solution:**

# Question-9

**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 16 times heavier than hydrogen?**

**Solution:**

The ratio of kinetic energies of oxygen molecule and hydrogen molecule when oxygen is 16 times heavier than hydrogen is 1, because the kinetic energy per molecule of the gas depends only on the temperature.

# Question-10

**Absolute temperature of a gas is increased four times its original value. What will be the change is r.m.s velocity of its molecules?**

**Solution:**

v

_{rms}∝ vT

Therefore, when absolute temperature of the gas is made four times its original value, its r.m.s velocity will become double. Hence the change in r.m.s velocity will be equal to its initial value.

# Question-11

**Two perfect gases at absolute temperatures T**

_{1}and T_{2 }are mixed. There is no loss of energy. Find the temperature of the mixture if the masses of the molecule are m_{1 }and m_{2 }and the number of the molecule in the gases are n_{1 }and n_{2 }respectively?**Solution:**

According to kinetic theory, the average K.E. per molecule of gas = 3/2 KT.

The average K. E. of all the molecules of two gases before it has been mixed together is given by,

The average K. E. of both the gases after it has been mixed is given by

Where T is the temperature of mixture. Since there is no loss of energy,

=

T =

# Question-12

**Helium gas is filled in a closed vessel whose coefficient of thermal expansion is negligible. When it is heated from 300 K to 600 K, find the average kinetic energy of helium atom?**

**Solution:**

When helium is heated from 300 K to 600 K the average kinetic energy of the helium gas becomes double, because average K.E. per molecule of gas is directly proportional to temperature in Kelvin.

# Question-13

**The root mean square (r.m.s) speed of oxygen molecule (v**

_{2}) at a temperature T is v. If the temperature is doubled and oxygen gas dissociates into atomic oxygen, what is the speed of atomic oxygen?**Solution:**

Let C and C′ be the root mean square velocity of oxygen at temperature T and T’ respectively

Therefore, C = = v ---(1)

C′ =

= 2 = 2v

Hence the velocity of atomic oxygen will becomes double.

# Question-14

**The pressure of a given mass of a gas is halved at constant temperature. What will be the volume of the gas in comparison to its initial volume. Explain on the basis of kinetic theory?**

**Solution:**

We know that P =

or V =

i.e., V ∝

1/3 MC

_{2}= a constant when temperature is constant

Therefore P becomes P/2 and V becomes double.

# Question-15

**Two gases A and B each at temperature T, pressure P and volume V are mixed. If the mixture be at the same temperature T and its volume also be C, then what should be its pressure? Explain.**

**Solution:**

Since P =

But C

^{2}∝ T

Therefore, if V and T are constant, then P ∝ M.

As M becomes 2 M, P becomes 2 P.

# Question-16

**Two gases of the same volume are filled with the same gas at the same temperature. If the pressure of the gas in these vessels be in the ration 1 : 2, then state :**

(i) the ratio of the r.m.s speed of the molecules

(ii) the ratio of the number of molecules.

(i) the ratio of the r.m.s speed of the molecules

(ii) the ratio of the number of molecules.

**Solution:**

Since P =

Therefore P ∝ nc

^{2}when m and v are constants. But c ∝ vT. Because the temperature of tow vessels is the same, the r.m.s velocity of gas molecules in tow vessels will be the same. When temperature is same for two vessels then P ∝ n.

i.e,.

# Question-17

**The volume of vessel A is twice the volume of another vessel B and both of them are filled with the same gas. If the gas in A is at twice the temperature and twice the pressure in comparison to the gas in B, what is the ratio of gas molecules in A and B.**

**Solution:**

V

_{A}= 2 V

_{ B }

T_{A }= 2 T_{ B }

and P_{A }= 2T_{B }

and

_{}