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P11 – Thermodynamics and Kinetic Theory of Gases
1
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P11. Thermodynamics and Kinetic Theory
of Gases
TABLE OF CONTENTS
P11. Thermodynamics and Kinetic Theory of Gases
1
P11.1 First Law of Thermodynamics…………………………………………………………….……………………………………….. 2
P11.2 Thermodynamic Processes………………………………………………………..……….……………………………………… 10
P11.3 Heat Engine, Refrigerator and Second Law of Thermodynamics………………………..………………………… 16
P11.4 Kinetic Theory of Gases……………………………………………………...……………………………………………………… 21
P11.5 Polytropic Process and problems of thermodynamic processes [ADVANCED] …………..………………… 27
Additional Practice Problems…………………………………………………..……………………………………………………………32
Answer Key…………………………………………………………………………………………………………….……………………………36
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P11 – Thermodynamics and Kinetic Theory of Gases
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P11.2 Thermodynamic Processes
LEARNING OBJECTIVES
1.
2.
3.
4.
5.
Understand Quasi-static, Reversible and Irreversible process
Define isothermal, adiabatic, isobaric and isochoric process
Understand the relation between 𝐶 and 𝐶 and their relation with 𝛾
Write the expression of the work done in different kinds of processes
Understand cyclic processes with the help of PV, VT and TP diagrams and calculate heat and work in the
whole cyclic process
PRE-READING
Category
Book Name (Edition)
Chapter
Section
REQUIRED
NCERT XI
12
12.8, 12.12
ADDITIONAL
Concept of Physics, H. C. Verma
26
26.5
27
27.5 to 27.7
PRE-READING EXERCISE
Q1.
Q2.
Identify the Isothermal process out of the processes given below.
A)
B)
C)
D)
If an Ideal gas expands at constant temperature then which of the following is true?
A) 𝑊 = 0
B) ∆(𝑃𝑉) = 0
C) ∆𝑃 = 0
D) 𝑄 = 0
Q3.
A gas is compressed or expanded at constant pressure then which of the following is true?
C) ∆𝑃 = 0
D) 𝑄 = 0
Q4.
A) ∆𝑈 = 0
B) ∆(𝑃𝑉) = 0
What is the characteristics of adiabatic process?
A) ∆𝑈 = 0
C) ∆𝑃 = 0
D) 𝑄 = 0
B) ∆(𝑃𝑉) = 0
P11.2
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Q5.
If an insulated piston is compressed such that volume is decreased in reversible manner then 𝑄 = 0 and
∆𝑈 = 0. (True/False)
Q6.
Match the following
Column I
Column II
A) ∆𝑈 = 0
P) Adiabatic
B) 𝑄 = 0
Q) Isothermal
C) ∆𝑃 = 0
R) Isochoric
D) 𝑊 = 0
S) Isobaric
SELF ASSESSMENT
Total Qs.
Correctly solved
Attempted
Not Attempted
Individually
6
After GD
IN-CLASS EXERCISE
LEVEL 1
Q1.
For a gas of 𝛾 = 5/3 and 800 𝑐𝑐 of this gas is suddenly compressed to 100 𝑐𝑐 adiabatically. If the initial
pressure is P. Find the final pressure (in terms of P)?
Q2.
When an ideal gas at a pressure P, temperature T and volume V is isothermally compressed to 𝑉/𝑛, its
pressure becomes 𝑃 . If the gas is compressed adiabatically to 𝑉/𝑛, its pressure becomes 𝑃 . The ratio 𝑃 /𝑃
is (𝛾 = 𝐶 /𝐶 )
A) 1
Q3.
B) 𝑛
C) 𝑛
D) 𝑛(
)
A gasoline engine takes in 0.1 mole of air at 20°𝐶 and 1 atm, and compresses it adiabatically to 1/8𝑡ℎ of its
original volume. Find the final temperature and pressure. Assume (𝛾 = 5/3). Also find the work done and
change in internal energy. (Take 𝑅 = 25/3 )
LEVEL 2
Q4.
Q5.
[Compulsory PSV] If the temperature of 2 moles of helium gas is raised from 0℃ 𝑡𝑜 100℃ then what is the
heat supplied? (Give your answer in terms of R. Given for Helium 𝛾 = )
A) In an Isochoric process
B) In an Isobaric process
C) What is the work done by the gas in part (B)?
[Compulsory PSV] In a cylinder, 2.0 moles of an ideal gas initially at 1.0 × 10 Pa and 300 K expands until
its volume doubles. Compute the work done if the expansion is : (Take 𝑙𝑜𝑔2 = 0.3, log 10 = 2.3 , 𝑅 =
,
= 0.6)
A) Isothermal
Q6.
B) adiabatic (𝛾 = )
C) isobaric
[Compulsory PSV] A mass of 8 grams of oxygen at the pressure of one atmosphere and at temperature 27℃
is enclosed in a cylinder fitted with a frictionless piston. The following operations are performed in the
order given:
I. The gas is heated at constant pressure to 127℃
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II. Then it is compressed isothermally to its initial volume and
III. Finally it is cooled to its initial temperature at constant volume.
Calculate the following:
A)
B)
C)
D)
What is the heat absorbed by the gas during operation (a)
How much work is done by the gas in state (a)
What is the work done on the gas in stage (b)
How much heat is extracted from the gas in stage (c)
(𝛾 for oxygen= 1.4, Molecular weight of oxygen=32, 1 atm= 10 𝑃𝑎, Take 𝑅 =
, = 1.2, = 0.8, 𝑙𝑜𝑔2 =
0.30, 𝑙𝑜𝑔3 = 0.48, log 10 = 2.3)
Q7.
[PSV] One mole of an ideal gas, initially at a pressure and temperature of 10 𝑁/𝑚 and 300 K respectively
expand isothermally until its volume is doubled and then adiabatically until its volume is again doubled. Find
the final pressure and temperature of the gas. Find the total work done during the isothermal and adiabatic
processes. Also draw the 𝑃 − 𝑉 diagram for the process. (Given 𝛾 = 1.4, (0.5) . = 0.38, (0.5) . = 0.75, 𝑙𝑜𝑔2 =
0.3, 𝑙𝑜𝑔3 = 0.48, log 10 = 2.3 )
Q8.
[PSV] 6 grams of hydrogen gas at a temperature of 300 K was isothermally expanded to ten times its initial
volume and then isochorically heated so that the pressure in the final state becomes equal to that in the initial
state. Find the total amount of heat absorbed by the gas during the entire process. For hydrogen gas (𝐶 =
, 𝐶 = , Take log 10 = 2.3)
LEVEL 3
Q9.
[PSV] One mole of an ideal gas whose pressure changes with volume as 𝑃 = 𝛼𝑉, where 𝛼 is a constant, is
expanded so that its volume increase to 𝜂𝑉 , where 𝑉 is the initial volume of the gas. Find the change in internal
energy and molar heat capacity (C) for the process. ( Take 𝛾 = ratio of molar heat capacities )
SELF ASSESSMENT
Total Qs.
Correctly solved
Attempted
Not Attempted
Individually
6
After GD
NOTE: Compulsory PSV’s are not to be counted for self-assessment
HOMEWORK
LEVEL 1
Q1.
The ratio
= 𝛾 for a gas and its molecular weight is M. What is its specific heat capacity at constant
pressure?
A)
Q2.
B)
C)
(
)
D)
(
)
A gas is compressed to one fourth of its original volume in an adiabatic process. What will be its final
pressure if its initial pressure is P:
A) P/4
C) Less than 4P
B) 4P
D) more than 4P
P11.2
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P11 – Thermodynamics and Kinetic Theory of Gases
Q3.
One mole of an ideal gas undergoes a cyclic change as shown in the figure. the process AB is isothermal. The
pressure and volume at point C is 1.013 × 10 𝑃𝑎 𝑎𝑛𝑑 22.4L respectively. The temperature of gas at C is :
A) 0°𝐶
Q4.
13
B) 273°𝐶
C) 0K
D) 27°𝐶
A monatomic gas 𝛾 = is suddenly compressed to
its volume adiabatically. The pressure of the gas
will change to 𝑥𝑃 (where 𝑃 =initial pressure). What is 𝑥?
B) 8
A)
D) 32
C)
Q5.
An ideal gas undergoes an isothermal expansion at 0°𝐶 from 0.010 𝑚 to 0.200 𝑚 . If the quantity of gas is
0.05 moles, compute the work done (upto 3 decimal places), the change in internal energy (upto 3 decimal
places) and the heat added (upto 3 decimal places). (Take 𝑅 = , log 2 = 0.3)
Q6.
A compressor takes in 0.1 mole of air at 27°𝐶 and 0.125 atm., and compresses it adiabatically to 1/32𝑛𝑑 of
the original volume. Find the final temperature and pressure. Assume air to be diatomic (𝛾 = 7/5). Also
find the work done (in 𝑘𝐽) and change in internal energy (in 𝑘𝐽).
*Q7. Two gases have the same initial pressure, volume and temperature. They expand to the same final volume,
one adiabatically and the other isothermally.
A)
B)
C)
D)
Q8.
The final temperature is greater for the isothermal process
The final pressure is greater for the isothermal process
The work done by the gas is greater for the isothermal process
All of the above options are incorrect
In the cyclic process shown in the V-P diagram is represented in the form of an ellipse
whose area is given by 𝜋 × 𝑎 × 𝑏 (where a is half of maximum possible diameter of the
ellipse and b is half of minimum possible diameter of the ellipse) such that the radius is
maximum possible radius of the curve). Find the magnitude of the work done.
A) 𝜋
C)
Q9.
B) 𝜋
D) 𝜋(𝑃 𝑉 − 𝑃 𝑉 )
(𝑃 − 𝑃 )(𝑉 − 𝑉 )
An ideal gas has a specific heat capacity at constant pressure C = 𝑅. The gas is kept in a closed vessel of
volume 0.0083 𝑚 , at a temperature of 300K and a pressure of 1.6 × 10 𝑁/𝑚 . An amount of 2.49 × 10 𝐽
of heat energy is supplied to the gas. Calculate the final temperature and pressure of the gas.
LEVEL 2
Q10. Two moles of helium gas (γ = 5/3) are initially at temperature 27°C and occupy a volume of 20L. The gas
is first expanded at constant pressure until the volume is doubled. Then it undergoes an adiabatic change
until the temperature returns to its initial value. (Take
√
= 0.707,
= 2.83)
A) Sketch the process on a P-V diagram
B) What are the final volume (in L) and pressure (in kPa) of the gas? (Upto one decimal place)
C) What is the work done (in kJ) by the gas? (Upto two decimal places)
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Q11. Two identical containers A and B with frictionless pistons contain the same ideal gas at the same
temperature and the same volume V. The mass of the gas in A is 𝑚 and that in B is 𝑚 . The gas in each
cylinder is now allowed to expand isothermally to the same final volume 2V. The change in the pressure in
A and B are found to be ∆𝑝 and 1.5 ∆𝑝 respectively. Then find the relation between 𝑚 and 𝑚 .
Q12. The piston cylinder arrangement shown contains a diatomic gas at temperature 300
K. the cross-sectional area of The piston cylinder arrangement shown contains a
diatomic gas at temperature 300 K and the cross-sectional area of the cylinder
is1 𝑚 . Initially the height of the piston above the base of the cylinder in 1 m. the
temperature is now raised to 400K at constant pressure. Find the new height of the
piston above the base of the cylinder. If the piston is now brought back to its original
height without any heat loss. Find the new equilibrium temperature of the gas, you
can leave the answer in fraction. (Take
.
= 1.122)
Q13. 2 moles of an ideal gas are carried round a cycle as shown. If the
process 𝑏 → 𝑐 is isothermal and 𝐶 = 3 𝑐𝑎𝑙/𝑚𝑜𝑙/𝐾.
(Take log 2 = 0.3 , 1𝑐𝑎𝑙 = 4.2𝐽) . During the processes 𝑎 → 𝑏, 𝑏 → 𝑐
and 𝑐 → 𝑎, determine:
A) work done (in J upto 2 decimal places)
B) change in internal energy (in cal)
C) heat supplied to the system (in J upto 2 decimal places)
Q14. An ideal monoatomic gas is confined in a cylinder by a spring –loaded piston of cross-section 8.0 × 10 𝑚 .
Initially the gas is at 300K and occupies a volume of 2.4 × 10 𝑚 and the spring is in its relaxed
(unstretched, uncompressed) state. The gas is heated by a small electric heater until the piston moves out
slowly by 0.1𝑚.
Calculate the final temperature of the gas and the heat supplied (in joules) by the heater. (The force
constant of the spring is 8000 𝑁/𝑚, and the atmospheric
pressure is 1 × 10 𝑁𝑚 . The cylinder and the piston
are thermally insulated. The piston is massless and there
is no friction between the piston and the cylinder.
Neglect heat loss through the lead wires of the heater.
The heat capacity of the heater coil is negligible. Assume
the spring to be massless)
Q15. Figure shows three processes for an ideal gas. The temperature at “a” is 600K, pressure 16 atm and volume 1L.
The volume at "𝑏" is 4 L. Out of two processes, ab and ac, one is adiabatic and other is isothermal. The process
bc is isobaric. The ratio of molar heat capacities of the gas is 1.5
A) Which of the processes "𝑎𝑏" and "𝑎𝑐" is adiabatic ? Why ?
B) Compute the pressure of the gas at "𝑏" and "𝑐"
C) Compute the temperature at "𝑏" and "𝑐"
D) Compute the volume at "𝑐"
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LEVEL 3
Q16. An ideal gas, whose adiabatic exponent is equal to 𝛾, is expanded so that the amount of heat transferred to the
gas is equal to the decrease of internal energy. Find
A) The molar heat capacity of the gas in this process,
B) The equation of the process in the variables T, V;
C) The work performed by one mole of the gas when its volume increases 𝜂 times and also the initial
temperature of the gas is 𝑇 .
Q17. A piston can freely move inside a horizontal cylinder closed from both
ends. Initially, the piston separates the inside space of the cylinder into
two equal parts each of volume 𝑉 , in which an ideal gas is contained
under the same pressure 𝑃 and at the same temperature. What work has
to be performed in order to increase isothermally the volume of one part
of gas η times compared to that of the other by slowly moving the piston?
Q18. Three moles of an ideal gas being initially at a temperature 𝑇 = 273 𝐾 were isothermally expanded 𝜂 =
5.0 times its initial volume and then isochorically heated so that the pressure in the final state became
equal to that in the initial state. The total amount of heat transferred to the gas during the process
equals80 𝑘𝐽. Find the ratio 𝛾 = for this gas.
Q19. The rectangular box as shown in figure has a partition which can slide without friction along the length of
the box. Initially each of the two chambers of the box has one mole of a monoatomic ideal gas (𝛾 = 5/3) at a
pressure 𝑝 , volume 𝑉 and temperature 𝑇 . The chamber on the left is
slowly heated by an electric heater. The walls of the box and the partition
are thermally insulated. Heat loss through the lead wires of the heater is
negligible. The gas in the left chamber expands pushing the partition
until the final pressure in both chambers becomes243 𝑝 /32. Determine
(a) the final temperature of the gas in each chamber (in terms of 𝑇 ) (b)
the work done by the gas in the right chamber (in terms of 𝑇 )
Q20. Two moles of an ideal monoatomic ideal gas (𝛾 = 5/3) initially at pressure 𝑝 and volume 𝑉 undergoes an
adiabatic compression until its volume is 𝑉 . Then the gas is given heat Q at constant volume 𝑉 .
A) Sketch the complete process on a P-V diagram
B) Find (i)the total work done by the gas , (ii)the total change in internal energy and (iii)the final
temperature of the gas
(Give your answer in terms of 𝑝 , 𝑉 , 𝑉 , 𝑄 𝑎𝑛𝑑 𝑅)
Q21. One mole of a monoatomic ideal gas is taken through the cycle shown in the figure
𝐴 → 𝐵: Adiabatic expansion
𝐵 → 𝐶: Cooling at constant volume
𝐶 → 𝐷 : Adiabatic compression
𝐷 → 𝐴 : heating at constant volume
The pressure and temperature at A, B, etc., are denoted by 𝑝 , 𝑇 , 𝑝 , 𝑇 etc., respectively. Given that, 𝑇 =
1000𝐾, 𝑝 = (2⁄3)𝑝 𝑎𝑛𝑑 𝑝 = (1⁄3)𝑝 , calculate the following quantities
A) The work done by the gas in the process A→B
B) The heat lost by the gas in the process 𝐵 → 𝐶
C) The temperature 𝑇 .
(Given: (2⁄3) ⁄ = 0.85)
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