# DC Pandey Solutions: Thermometry, Thermal Expansion & Kinetic Theory of Gases - 2 Notes | Study DC Pandey Solutions for JEE Physics - JEE

## JEE: DC Pandey Solutions: Thermometry, Thermal Expansion & Kinetic Theory of Gases - 2 Notes | Study DC Pandey Solutions for JEE Physics - JEE

The document DC Pandey Solutions: Thermometry, Thermal Expansion & Kinetic Theory of Gases - 2 Notes | Study DC Pandey Solutions for JEE Physics - JEE is a part of the JEE Course DC Pandey Solutions for JEE Physics.
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``` Page 1

Introductory Exercise 17.2
Q 1.  From the graph for an ideal gas state whether m
1
or m
2
is greater?

Q 2.  A vessel is filled with an ideal gas at a pressure of 20 atm and is at a temperature of 27°C. One-
half of the mass is removed from the vessel and the temperature of the remaining gas is increased
to 87°C. At this temperature find the pressure of the gas.
Q 3.  A vessel contains a mixture of 7 g of nitrogen and 11 g of carbondioxide at temperature T = 290K.
If pressure of the mixture is 1 atm (=1.01 × 10
5
N/m
2
), calculate its density (R = 8.31 J/mol-K).
Q 4.  An electric bulb of volume 250 cm
3
was sealed off during manufacture at a pressure of 10
-3
mm of
mercury at 27°C. Compute the number of air molecules contained in the bulb. Given that R = 8.31
J/mol -K and N
A
= 6.02 × 10
23
per mol.
Q 5.  State whether P
1
> P
2
or P
2
> P
1
for given mass of a gas?

Q 6.  For a given mass of a gas what is the shape of P versus
1
V
graph at constant temperature?
Solutions
1.

At constant volume T -P graph is a straight line of slope,
Slope of m
1
is less. Hence m
1
is greater.
2.

3.   n
1
= number of moles of nitrogen n
2
= number of moles of CO
2
=

4.  PV = nRT
Page 2

Introductory Exercise 17.2
Q 1.  From the graph for an ideal gas state whether m
1
or m
2
is greater?

Q 2.  A vessel is filled with an ideal gas at a pressure of 20 atm and is at a temperature of 27°C. One-
half of the mass is removed from the vessel and the temperature of the remaining gas is increased
to 87°C. At this temperature find the pressure of the gas.
Q 3.  A vessel contains a mixture of 7 g of nitrogen and 11 g of carbondioxide at temperature T = 290K.
If pressure of the mixture is 1 atm (=1.01 × 10
5
N/m
2
), calculate its density (R = 8.31 J/mol-K).
Q 4.  An electric bulb of volume 250 cm
3
was sealed off during manufacture at a pressure of 10
-3
mm of
mercury at 27°C. Compute the number of air molecules contained in the bulb. Given that R = 8.31
J/mol -K and N
A
= 6.02 × 10
23
per mol.
Q 5.  State whether P
1
> P
2
or P
2
> P
1
for given mass of a gas?

Q 6.  For a given mass of a gas what is the shape of P versus
1
V
graph at constant temperature?
Solutions
1.

At constant volume T -P graph is a straight line of slope,
Slope of m
1
is less. Hence m
1
is greater.
2.

3.   n
1
= number of moles of nitrogen n
2
= number of moles of CO
2
=

4.  PV = nRT

= 1.33 × 10
-8

?  Number of molecular = (n)
N
A
= (1.33 × 10
-8
) (6.02 × 10
23
) = 8 × 10
15

5.
For given mass, V-T graph is a straight line passing through origin having

Slope of P
2
is more. Hence P
2
< P
1
.
6.
i.e., P versus
1
V
graph is a straight line passing through origin of slope nRT.

Introductory Exercise 17.3
Q 1.  The average speed of all the molecules in a gas at a given instant is not zero, whereas the average
velocity of all the molecules is zero. Explain why?
Q 2.  A sample of helium gas is at a temperature of 300 K and a pressure of 0.5 atm. What is the
average kinetic energy of a molecule of a gas?
Q 3.  A sample of helium and neon gases has a temperature of 300 K and pressure of 1.0 atm. The molar
mass of helium is 4.0 g/mol and that of neon is 20.2 g/mol.
(a) Find the rms speed of the helium atoms and of the neon atoms.
(b) What is the average kinetic energy per atom of each gas?
Q 4.  At what temperature will the particles in a sample of helium gas have an rms speed of 1.0 km/s?
Q 5.  At 0°C and 1.0 atm (=1.01 × 10
5
N/m
2
) pressure the densities of air, oxygen and nitrogen are
1.284 kg/m
3
, 1.429 kg/m
3
and 1.251 kg/m
3
respectively. Calculate the percentage of nitrogen in
the air from these data, assuming only these two gases to be present.
Q 6.  An air bubble of 20 cm
3
volume is at the bottom of a lake 40 meters deep where the temperature is
4°C. The bubble rises to the surface which is at a temperature of 20°C. Take the temperature to be
the same as that of the surrounding water and find its volume just before it reaches the surface.
Q 7.  If the water molecules in 1.0 g of water were distributed uniformly over the surface of earth, how
many such molecules would there be in 1.0 cm
2
of earth's surface?
Q 8.  For a certain gas the heat capacity at constant pressure is greater than that at constant volume by
29.1J/K.
(a) How many moles of the gas are there?
(b) If the gas is monoatomic, what are heat capacities at constant volume and pressure ?
Page 3

Introductory Exercise 17.2
Q 1.  From the graph for an ideal gas state whether m
1
or m
2
is greater?

Q 2.  A vessel is filled with an ideal gas at a pressure of 20 atm and is at a temperature of 27°C. One-
half of the mass is removed from the vessel and the temperature of the remaining gas is increased
to 87°C. At this temperature find the pressure of the gas.
Q 3.  A vessel contains a mixture of 7 g of nitrogen and 11 g of carbondioxide at temperature T = 290K.
If pressure of the mixture is 1 atm (=1.01 × 10
5
N/m
2
), calculate its density (R = 8.31 J/mol-K).
Q 4.  An electric bulb of volume 250 cm
3
was sealed off during manufacture at a pressure of 10
-3
mm of
mercury at 27°C. Compute the number of air molecules contained in the bulb. Given that R = 8.31
J/mol -K and N
A
= 6.02 × 10
23
per mol.
Q 5.  State whether P
1
> P
2
or P
2
> P
1
for given mass of a gas?

Q 6.  For a given mass of a gas what is the shape of P versus
1
V
graph at constant temperature?
Solutions
1.

At constant volume T -P graph is a straight line of slope,
Slope of m
1
is less. Hence m
1
is greater.
2.

3.   n
1
= number of moles of nitrogen n
2
= number of moles of CO
2
=

4.  PV = nRT

= 1.33 × 10
-8

?  Number of molecular = (n)
N
A
= (1.33 × 10
-8
) (6.02 × 10
23
) = 8 × 10
15

5.
For given mass, V-T graph is a straight line passing through origin having

Slope of P
2
is more. Hence P
2
< P
1
.
6.
i.e., P versus
1
V
graph is a straight line passing through origin of slope nRT.

Introductory Exercise 17.3
Q 1.  The average speed of all the molecules in a gas at a given instant is not zero, whereas the average
velocity of all the molecules is zero. Explain why?
Q 2.  A sample of helium gas is at a temperature of 300 K and a pressure of 0.5 atm. What is the
average kinetic energy of a molecule of a gas?
Q 3.  A sample of helium and neon gases has a temperature of 300 K and pressure of 1.0 atm. The molar
mass of helium is 4.0 g/mol and that of neon is 20.2 g/mol.
(a) Find the rms speed of the helium atoms and of the neon atoms.
(b) What is the average kinetic energy per atom of each gas?
Q 4.  At what temperature will the particles in a sample of helium gas have an rms speed of 1.0 km/s?
Q 5.  At 0°C and 1.0 atm (=1.01 × 10
5
N/m
2
) pressure the densities of air, oxygen and nitrogen are
1.284 kg/m
3
, 1.429 kg/m
3
and 1.251 kg/m
3
respectively. Calculate the percentage of nitrogen in
the air from these data, assuming only these two gases to be present.
Q 6.  An air bubble of 20 cm
3
volume is at the bottom of a lake 40 meters deep where the temperature is
4°C. The bubble rises to the surface which is at a temperature of 20°C. Take the temperature to be
the same as that of the surrounding water and find its volume just before it reaches the surface.
Q 7.  If the water molecules in 1.0 g of water were distributed uniformly over the surface of earth, how
many such molecules would there be in 1.0 cm
2
of earth's surface?
Q 8.  For a certain gas the heat capacity at constant pressure is greater than that at constant volume by
29.1J/K.
(a) How many moles of the gas are there?
(b) If the gas is monoatomic, what are heat capacities at constant volume and pressure ?
(c) If the gas molecules are diatomic which rotate but do not vibrate, what are heat capacities at
constant volume and at constant pressure.
Q 9.  The heat capacity at constant volume of a sample of a monoatomic gas is 35 J/K. Find :
(a) the number of moles  (b) the internal energy at 0°C
(c) the molor heat capacity at constant pressure.
Q 10.  For any distribution of speeds v
rms
? v
av
.

Is this statement true or false?
Solutions
2.  Helium gas is monoatomic. So its degree of freedom f = 3. Average kinetic energy of 1 molecule
of gas

= 6.21 × 10
-21
J
3.  (a)
For He gas :
= 1368 m/s
For Ne gas :
= 609 m/s
(b) Each gas is monotonic for which degree of freedom f = 3. Hence average kinetic energy of one
atom

= 6.21 × 10
-21
J
4.

5.  Let mass of nitrogen = (m) g. Then mass of oxygen = (100 - m) g. Number of moles of nitrogen,
and number of moles of oxygen

For air

Page 4

Introductory Exercise 17.2
Q 1.  From the graph for an ideal gas state whether m
1
or m
2
is greater?

Q 2.  A vessel is filled with an ideal gas at a pressure of 20 atm and is at a temperature of 27°C. One-
half of the mass is removed from the vessel and the temperature of the remaining gas is increased
to 87°C. At this temperature find the pressure of the gas.
Q 3.  A vessel contains a mixture of 7 g of nitrogen and 11 g of carbondioxide at temperature T = 290K.
If pressure of the mixture is 1 atm (=1.01 × 10
5
N/m
2
), calculate its density (R = 8.31 J/mol-K).
Q 4.  An electric bulb of volume 250 cm
3
was sealed off during manufacture at a pressure of 10
-3
mm of
mercury at 27°C. Compute the number of air molecules contained in the bulb. Given that R = 8.31
J/mol -K and N
A
= 6.02 × 10
23
per mol.
Q 5.  State whether P
1
> P
2
or P
2
> P
1
for given mass of a gas?

Q 6.  For a given mass of a gas what is the shape of P versus
1
V
graph at constant temperature?
Solutions
1.

At constant volume T -P graph is a straight line of slope,
Slope of m
1
is less. Hence m
1
is greater.
2.

3.   n
1
= number of moles of nitrogen n
2
= number of moles of CO
2
=

4.  PV = nRT

= 1.33 × 10
-8

?  Number of molecular = (n)
N
A
= (1.33 × 10
-8
) (6.02 × 10
23
) = 8 × 10
15

5.
For given mass, V-T graph is a straight line passing through origin having

Slope of P
2
is more. Hence P
2
< P
1
.
6.
i.e., P versus
1
V
graph is a straight line passing through origin of slope nRT.

Introductory Exercise 17.3
Q 1.  The average speed of all the molecules in a gas at a given instant is not zero, whereas the average
velocity of all the molecules is zero. Explain why?
Q 2.  A sample of helium gas is at a temperature of 300 K and a pressure of 0.5 atm. What is the
average kinetic energy of a molecule of a gas?
Q 3.  A sample of helium and neon gases has a temperature of 300 K and pressure of 1.0 atm. The molar
mass of helium is 4.0 g/mol and that of neon is 20.2 g/mol.
(a) Find the rms speed of the helium atoms and of the neon atoms.
(b) What is the average kinetic energy per atom of each gas?
Q 4.  At what temperature will the particles in a sample of helium gas have an rms speed of 1.0 km/s?
Q 5.  At 0°C and 1.0 atm (=1.01 × 10
5
N/m
2
) pressure the densities of air, oxygen and nitrogen are
1.284 kg/m
3
, 1.429 kg/m
3
and 1.251 kg/m
3
respectively. Calculate the percentage of nitrogen in
the air from these data, assuming only these two gases to be present.
Q 6.  An air bubble of 20 cm
3
volume is at the bottom of a lake 40 meters deep where the temperature is
4°C. The bubble rises to the surface which is at a temperature of 20°C. Take the temperature to be
the same as that of the surrounding water and find its volume just before it reaches the surface.
Q 7.  If the water molecules in 1.0 g of water were distributed uniformly over the surface of earth, how
many such molecules would there be in 1.0 cm
2
of earth's surface?
Q 8.  For a certain gas the heat capacity at constant pressure is greater than that at constant volume by
29.1J/K.
(a) How many moles of the gas are there?
(b) If the gas is monoatomic, what are heat capacities at constant volume and pressure ?
(c) If the gas molecules are diatomic which rotate but do not vibrate, what are heat capacities at
constant volume and at constant pressure.
Q 9.  The heat capacity at constant volume of a sample of a monoatomic gas is 35 J/K. Find :
(a) the number of moles  (b) the internal energy at 0°C
(c) the molor heat capacity at constant pressure.
Q 10.  For any distribution of speeds v
rms
? v
av
.

Is this statement true or false?
Solutions
2.  Helium gas is monoatomic. So its degree of freedom f = 3. Average kinetic energy of 1 molecule
of gas

= 6.21 × 10
-21
J
3.  (a)
For He gas :
= 1368 m/s
For Ne gas :
= 609 m/s
(b) Each gas is monotonic for which degree of freedom f = 3. Hence average kinetic energy of one
atom

= 6.21 × 10
-21
J
4.

5.  Let mass of nitrogen = (m) g. Then mass of oxygen = (100 - m) g. Number of moles of nitrogen,
and number of moles of oxygen

For air

Solving this equation we get,
m = 76.5 g
This is also percentage of N
2
by mass on air as total mass we have taken is 100 g.
6.  n
1
= n
2

7.   Number of gram moles
A
= 6.02 × 10
23
/g-mol
?  Total number of molecules,
N =nN
A
= 3.34 × 10
22

?  Number of molecules per cm
2

8.  (a) Molar heat capacity and heat capacity
C
p
= nC
P

Similarly   C
V
= nC
V

Now   C
p
- C
V
= n (C
P
- C
V
) - nR

(b)
= (3.5) (2.5) (8.31)= 72.75 J/K
C
V
= nC
V
= n = (3.5) (1.5) (8.31)= 43.65 J/K
(c)
= (3.5) (3.5) (8.31) = 101.8 J/K
= (3.5) (2.5) (8.31)= 72.75 J/K
9.  (a) As discussed in the above problem.
C
V
= nC
V

(b) Internal energy
Page 5

Introductory Exercise 17.2
Q 1.  From the graph for an ideal gas state whether m
1
or m
2
is greater?

Q 2.  A vessel is filled with an ideal gas at a pressure of 20 atm and is at a temperature of 27°C. One-
half of the mass is removed from the vessel and the temperature of the remaining gas is increased
to 87°C. At this temperature find the pressure of the gas.
Q 3.  A vessel contains a mixture of 7 g of nitrogen and 11 g of carbondioxide at temperature T = 290K.
If pressure of the mixture is 1 atm (=1.01 × 10
5
N/m
2
), calculate its density (R = 8.31 J/mol-K).
Q 4.  An electric bulb of volume 250 cm
3
was sealed off during manufacture at a pressure of 10
-3
mm of
mercury at 27°C. Compute the number of air molecules contained in the bulb. Given that R = 8.31
J/mol -K and N
A
= 6.02 × 10
23
per mol.
Q 5.  State whether P
1
> P
2
or P
2
> P
1
for given mass of a gas?

Q 6.  For a given mass of a gas what is the shape of P versus
1
V
graph at constant temperature?
Solutions
1.

At constant volume T -P graph is a straight line of slope,
Slope of m
1
is less. Hence m
1
is greater.
2.

3.   n
1
= number of moles of nitrogen n
2
= number of moles of CO
2
=

4.  PV = nRT

= 1.33 × 10
-8

?  Number of molecular = (n)
N
A
= (1.33 × 10
-8
) (6.02 × 10
23
) = 8 × 10
15

5.
For given mass, V-T graph is a straight line passing through origin having

Slope of P
2
is more. Hence P
2
< P
1
.
6.
i.e., P versus
1
V
graph is a straight line passing through origin of slope nRT.

Introductory Exercise 17.3
Q 1.  The average speed of all the molecules in a gas at a given instant is not zero, whereas the average
velocity of all the molecules is zero. Explain why?
Q 2.  A sample of helium gas is at a temperature of 300 K and a pressure of 0.5 atm. What is the
average kinetic energy of a molecule of a gas?
Q 3.  A sample of helium and neon gases has a temperature of 300 K and pressure of 1.0 atm. The molar
mass of helium is 4.0 g/mol and that of neon is 20.2 g/mol.
(a) Find the rms speed of the helium atoms and of the neon atoms.
(b) What is the average kinetic energy per atom of each gas?
Q 4.  At what temperature will the particles in a sample of helium gas have an rms speed of 1.0 km/s?
Q 5.  At 0°C and 1.0 atm (=1.01 × 10
5
N/m
2
) pressure the densities of air, oxygen and nitrogen are
1.284 kg/m
3
, 1.429 kg/m
3
and 1.251 kg/m
3
respectively. Calculate the percentage of nitrogen in
the air from these data, assuming only these two gases to be present.
Q 6.  An air bubble of 20 cm
3
volume is at the bottom of a lake 40 meters deep where the temperature is
4°C. The bubble rises to the surface which is at a temperature of 20°C. Take the temperature to be
the same as that of the surrounding water and find its volume just before it reaches the surface.
Q 7.  If the water molecules in 1.0 g of water were distributed uniformly over the surface of earth, how
many such molecules would there be in 1.0 cm
2
of earth's surface?
Q 8.  For a certain gas the heat capacity at constant pressure is greater than that at constant volume by
29.1J/K.
(a) How many moles of the gas are there?
(b) If the gas is monoatomic, what are heat capacities at constant volume and pressure ?
(c) If the gas molecules are diatomic which rotate but do not vibrate, what are heat capacities at
constant volume and at constant pressure.
Q 9.  The heat capacity at constant volume of a sample of a monoatomic gas is 35 J/K. Find :
(a) the number of moles  (b) the internal energy at 0°C
(c) the molor heat capacity at constant pressure.
Q 10.  For any distribution of speeds v
rms
? v
av
.

Is this statement true or false?
Solutions
2.  Helium gas is monoatomic. So its degree of freedom f = 3. Average kinetic energy of 1 molecule
of gas

= 6.21 × 10
-21
J
3.  (a)
For He gas :
= 1368 m/s
For Ne gas :
= 609 m/s
(b) Each gas is monotonic for which degree of freedom f = 3. Hence average kinetic energy of one
atom

= 6.21 × 10
-21
J
4.

5.  Let mass of nitrogen = (m) g. Then mass of oxygen = (100 - m) g. Number of moles of nitrogen,
and number of moles of oxygen

For air

Solving this equation we get,
m = 76.5 g
This is also percentage of N
2
by mass on air as total mass we have taken is 100 g.
6.  n
1
= n
2

7.   Number of gram moles
A
= 6.02 × 10
23
/g-mol
?  Total number of molecules,
N =nN
A
= 3.34 × 10
22

?  Number of molecules per cm
2

8.  (a) Molar heat capacity and heat capacity
C
p
= nC
P

Similarly   C
V
= nC
V

Now   C
p
- C
V
= n (C
P
- C
V
) - nR

(b)
= (3.5) (2.5) (8.31)= 72.75 J/K
C
V
= nC
V
= n = (3.5) (1.5) (8.31)= 43.65 J/K
(c)
= (3.5) (3.5) (8.31) = 101.8 J/K
= (3.5) (2.5) (8.31)= 72.75 J/K
9.  (a) As discussed in the above problem.
C
V
= nC
V

(b) Internal energy

f = degree of freedom = 3

(c)
10.  Suppose n
1
molecules have v
1
velocity and n
2
molecules have v
2
velocity. Then

Now, v
rms
? V
av

because v
1
and v
2
may be in opposite direction also.
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