DC Pandey Solutions: First Law of Thermodynamics - 2 - Notes | Study Physics Class 11 - NEET

 Page 1


 
Introductory Exercise 18.2 
Q 1.  One mole of an ideal mono atomic gas is initially at 300 K. Find the final temperature if 200 J of 
heat are added as follows: 
  (a) at constant volume   (b) at constant pressure. 
Q 2.  Prove that work done by an ideal gas in an adiabatic process is 
i i f f
PV P V
W
1
?
?
??
, using the 
integration PdV
?
. 
Q 3.  When a gas expands along AB it does 500 J of work and absorbs 250 J of heat. When the gas 
expands along AC, it does 700 J of work and absorbs 300 J of heat. 
 
  (a) How much heat does the gas exchange along BC? 
(b) When the gas makes the transition from C to A along CDA, 800 J of work are done on it from 
C to D. How much heat does it exchange along CDA ? 
Q 4.  One mole of an ideal monoatomic gas occupies a volume of 1.0 × 10
-2
 m
3
 at a pressure of  
  2.0 × 10
5
 N/m
2
. 
  (a) What is the temperature of the gas? 
(b) The gas undergoes an adiabatic compression until its volume is decreased to 5.0 × 10
-3
 m
3
. 
What is the new gas temperature? 
(c) How much work is done on the gas during the compression?  
  (d) What is the change in the internal energy of the gas? 
Q 5.  A bullet of mass 10 g travelling horizontally at 200 m/s strikes and embeds in a pendulum bob of 
mass 2.0 kg. 
  (a) How much mechanical energy is dissipated in the collision? 
(b) Assuming that C
v
 for the bob plus bullet is 3 R, calculate the temperature increase of the 
system due to the collision. Take the molecular mass of the system to be 200 g/mol. 
Q 6.  An ideal gas is carried through a thermodynamic cycle consisting of two isobaric and two 
isothermal processes, as shown in figure. Show that the net work done in the entire cycle is given 
by the equation. 
 
Q 7.  Consider the cyclic process depicted in figure. If Q is negative for the process BC, and if. ?U is 
negative for the process CA, what are the signs of Q, W and ?U that are associated with each 
process? 
Page 2


 
Introductory Exercise 18.2 
Q 1.  One mole of an ideal mono atomic gas is initially at 300 K. Find the final temperature if 200 J of 
heat are added as follows: 
  (a) at constant volume   (b) at constant pressure. 
Q 2.  Prove that work done by an ideal gas in an adiabatic process is 
i i f f
PV P V
W
1
?
?
??
, using the 
integration PdV
?
. 
Q 3.  When a gas expands along AB it does 500 J of work and absorbs 250 J of heat. When the gas 
expands along AC, it does 700 J of work and absorbs 300 J of heat. 
 
  (a) How much heat does the gas exchange along BC? 
(b) When the gas makes the transition from C to A along CDA, 800 J of work are done on it from 
C to D. How much heat does it exchange along CDA ? 
Q 4.  One mole of an ideal monoatomic gas occupies a volume of 1.0 × 10
-2
 m
3
 at a pressure of  
  2.0 × 10
5
 N/m
2
. 
  (a) What is the temperature of the gas? 
(b) The gas undergoes an adiabatic compression until its volume is decreased to 5.0 × 10
-3
 m
3
. 
What is the new gas temperature? 
(c) How much work is done on the gas during the compression?  
  (d) What is the change in the internal energy of the gas? 
Q 5.  A bullet of mass 10 g travelling horizontally at 200 m/s strikes and embeds in a pendulum bob of 
mass 2.0 kg. 
  (a) How much mechanical energy is dissipated in the collision? 
(b) Assuming that C
v
 for the bob plus bullet is 3 R, calculate the temperature increase of the 
system due to the collision. Take the molecular mass of the system to be 200 g/mol. 
Q 6.  An ideal gas is carried through a thermodynamic cycle consisting of two isobaric and two 
isothermal processes, as shown in figure. Show that the net work done in the entire cycle is given 
by the equation. 
 
Q 7.  Consider the cyclic process depicted in figure. If Q is negative for the process BC, and if. ?U is 
negative for the process CA, what are the signs of Q, W and ?U that are associated with each 
process? 
 
Q 8.  An ideal gas is enclosed in a cylinder with a movable piston on top. The piston has mass of 8000 g 
and an area of 5.00 cm
2
 and is free to slide up and down, keeping the pressure of the gas constant. 
How much work is done as the temperature of 0.200 mol of the gas is raised from 200°C to 
300°C? 
Q 9.  A sample of ideal gas is expanded to twice its original volume of 1.00 m
3
 in a quasi-static process 
for which P = ?V
2
, with 
? 
= 5.00 atm/m
6
, as shown in figure. How much work is done by the 
expanding gas? 
 
 
Solutions 
1.  (a) 
    
  or   200 = 1.5 × 8.31(T
f
 -300) 
  Solving we get, T
f
 = 316 K 
  (b) 
    
  or   200 = 2.5 × 8.31(T
f
 -300) 
  Solving we get, T
f
 = 309.6 K 
2.  In adiabatic process, 
  PV
?
 = constant = K (say) 
   
    
    
Page 3


 
Introductory Exercise 18.2 
Q 1.  One mole of an ideal mono atomic gas is initially at 300 K. Find the final temperature if 200 J of 
heat are added as follows: 
  (a) at constant volume   (b) at constant pressure. 
Q 2.  Prove that work done by an ideal gas in an adiabatic process is 
i i f f
PV P V
W
1
?
?
??
, using the 
integration PdV
?
. 
Q 3.  When a gas expands along AB it does 500 J of work and absorbs 250 J of heat. When the gas 
expands along AC, it does 700 J of work and absorbs 300 J of heat. 
 
  (a) How much heat does the gas exchange along BC? 
(b) When the gas makes the transition from C to A along CDA, 800 J of work are done on it from 
C to D. How much heat does it exchange along CDA ? 
Q 4.  One mole of an ideal monoatomic gas occupies a volume of 1.0 × 10
-2
 m
3
 at a pressure of  
  2.0 × 10
5
 N/m
2
. 
  (a) What is the temperature of the gas? 
(b) The gas undergoes an adiabatic compression until its volume is decreased to 5.0 × 10
-3
 m
3
. 
What is the new gas temperature? 
(c) How much work is done on the gas during the compression?  
  (d) What is the change in the internal energy of the gas? 
Q 5.  A bullet of mass 10 g travelling horizontally at 200 m/s strikes and embeds in a pendulum bob of 
mass 2.0 kg. 
  (a) How much mechanical energy is dissipated in the collision? 
(b) Assuming that C
v
 for the bob plus bullet is 3 R, calculate the temperature increase of the 
system due to the collision. Take the molecular mass of the system to be 200 g/mol. 
Q 6.  An ideal gas is carried through a thermodynamic cycle consisting of two isobaric and two 
isothermal processes, as shown in figure. Show that the net work done in the entire cycle is given 
by the equation. 
 
Q 7.  Consider the cyclic process depicted in figure. If Q is negative for the process BC, and if. ?U is 
negative for the process CA, what are the signs of Q, W and ?U that are associated with each 
process? 
 
Q 8.  An ideal gas is enclosed in a cylinder with a movable piston on top. The piston has mass of 8000 g 
and an area of 5.00 cm
2
 and is free to slide up and down, keeping the pressure of the gas constant. 
How much work is done as the temperature of 0.200 mol of the gas is raised from 200°C to 
300°C? 
Q 9.  A sample of ideal gas is expanded to twice its original volume of 1.00 m
3
 in a quasi-static process 
for which P = ?V
2
, with 
? 
= 5.00 atm/m
6
, as shown in figure. How much work is done by the 
expanding gas? 
 
 
Solutions 
1.  (a) 
    
  or   200 = 1.5 × 8.31(T
f
 -300) 
  Solving we get, T
f
 = 316 K 
  (b) 
    
  or   200 = 2.5 × 8.31(T
f
 -300) 
  Solving we get, T
f
 = 309.6 K 
2.  In adiabatic process, 
  PV
?
 = constant = K (say) 
   
    
    
3.  (a) U
C
 - U
A
 along two paths should be same. 
    (Q
AB
 + Q
BC
) - (W
AB
 + W
BC
) = Q
AC 
- W
AC  
  ?  (250 + Q
BC
) - (500 + 0) = 300 - 700 Solving we get 
   Q
BC 
= -150 J 
  (b) Q
CDA
 = W
CD
 + W
DA
)+ (U
A 
- U
C
) 
   = (-800+ 0) - (U
C 
- U
A
)  
   = - 800 - (300 - 700) = - 400 J 
4.  (a) 
  (b) In adiabatic process  
   
    
  (c) Work done on the gas = ?U 
    
   = 1.5 × 8.31(383-240.7) ? 1770 J 
  (d) ?U ? 1770 J 
5.  (a) From conservation of linear momentum 
   0.01 × 200 = (2+0.01) v 
   V ? 1 m/s 
  Mechanical energy dissipated in collision = K
i 
- K
f
 
    
   ? 199 J 
  (b) 
  Using, 
   Q = nC
V
?T 
   
   = 0.80°C 
6.  Along the process CD apply, 
   PV - constant and find V
C
  
  Similarly along path AB, again apply, 
  PV = constant and find Y
B  
  
Now, W
nett
 = W
AB
 + W
BC
 + W
CD
 + W
DA
 
    
Page 4


 
Introductory Exercise 18.2 
Q 1.  One mole of an ideal mono atomic gas is initially at 300 K. Find the final temperature if 200 J of 
heat are added as follows: 
  (a) at constant volume   (b) at constant pressure. 
Q 2.  Prove that work done by an ideal gas in an adiabatic process is 
i i f f
PV P V
W
1
?
?
??
, using the 
integration PdV
?
. 
Q 3.  When a gas expands along AB it does 500 J of work and absorbs 250 J of heat. When the gas 
expands along AC, it does 700 J of work and absorbs 300 J of heat. 
 
  (a) How much heat does the gas exchange along BC? 
(b) When the gas makes the transition from C to A along CDA, 800 J of work are done on it from 
C to D. How much heat does it exchange along CDA ? 
Q 4.  One mole of an ideal monoatomic gas occupies a volume of 1.0 × 10
-2
 m
3
 at a pressure of  
  2.0 × 10
5
 N/m
2
. 
  (a) What is the temperature of the gas? 
(b) The gas undergoes an adiabatic compression until its volume is decreased to 5.0 × 10
-3
 m
3
. 
What is the new gas temperature? 
(c) How much work is done on the gas during the compression?  
  (d) What is the change in the internal energy of the gas? 
Q 5.  A bullet of mass 10 g travelling horizontally at 200 m/s strikes and embeds in a pendulum bob of 
mass 2.0 kg. 
  (a) How much mechanical energy is dissipated in the collision? 
(b) Assuming that C
v
 for the bob plus bullet is 3 R, calculate the temperature increase of the 
system due to the collision. Take the molecular mass of the system to be 200 g/mol. 
Q 6.  An ideal gas is carried through a thermodynamic cycle consisting of two isobaric and two 
isothermal processes, as shown in figure. Show that the net work done in the entire cycle is given 
by the equation. 
 
Q 7.  Consider the cyclic process depicted in figure. If Q is negative for the process BC, and if. ?U is 
negative for the process CA, what are the signs of Q, W and ?U that are associated with each 
process? 
 
Q 8.  An ideal gas is enclosed in a cylinder with a movable piston on top. The piston has mass of 8000 g 
and an area of 5.00 cm
2
 and is free to slide up and down, keeping the pressure of the gas constant. 
How much work is done as the temperature of 0.200 mol of the gas is raised from 200°C to 
300°C? 
Q 9.  A sample of ideal gas is expanded to twice its original volume of 1.00 m
3
 in a quasi-static process 
for which P = ?V
2
, with 
? 
= 5.00 atm/m
6
, as shown in figure. How much work is done by the 
expanding gas? 
 
 
Solutions 
1.  (a) 
    
  or   200 = 1.5 × 8.31(T
f
 -300) 
  Solving we get, T
f
 = 316 K 
  (b) 
    
  or   200 = 2.5 × 8.31(T
f
 -300) 
  Solving we get, T
f
 = 309.6 K 
2.  In adiabatic process, 
  PV
?
 = constant = K (say) 
   
    
    
3.  (a) U
C
 - U
A
 along two paths should be same. 
    (Q
AB
 + Q
BC
) - (W
AB
 + W
BC
) = Q
AC 
- W
AC  
  ?  (250 + Q
BC
) - (500 + 0) = 300 - 700 Solving we get 
   Q
BC 
= -150 J 
  (b) Q
CDA
 = W
CD
 + W
DA
)+ (U
A 
- U
C
) 
   = (-800+ 0) - (U
C 
- U
A
)  
   = - 800 - (300 - 700) = - 400 J 
4.  (a) 
  (b) In adiabatic process  
   
    
  (c) Work done on the gas = ?U 
    
   = 1.5 × 8.31(383-240.7) ? 1770 J 
  (d) ?U ? 1770 J 
5.  (a) From conservation of linear momentum 
   0.01 × 200 = (2+0.01) v 
   V ? 1 m/s 
  Mechanical energy dissipated in collision = K
i 
- K
f
 
    
   ? 199 J 
  (b) 
  Using, 
   Q = nC
V
?T 
   
   = 0.80°C 
6.  Along the process CD apply, 
   PV - constant and find V
C
  
  Similarly along path AB, again apply, 
  PV = constant and find Y
B  
  
Now, W
nett
 = W
AB
 + W
BC
 + W
CD
 + W
DA
 
    
  Further,   nRT
A
=P
A
V
A
 
  and   nRT
C
 = P
C
V
C
 
7.  AB 
  V is increasing, so W = + ve. Product of PV and therefore T and therefore U is increasing. 
  So, ?U
 
is also +ve 
   Q = W + ?V  
  ?   Q is also positive.  
  EC  
     V = constant 
  ?   W = 0 
   Q = ?U = - ve  
  CA 
   V is decreasing.  
  ?  W =-ve 
  ?U
 
is also negative.  
  ?   Q = W + ?U is also negative. 
8.  PV = nRT 
  ? P ?V = nR ?T 
  Work done under constant pressure is, 
   W = P ?V = nR ?T 
   = (0.2)(8.31)(100)= 166.2 J 
9.  
    
  Solving we get, W = 1.18 × 10
6
 J 
   = 1.18 MJ  
 
Introductory Exercise 18.3 
Q 1.  Three moles of an ideal gas being initially at a temperature T
i
 = 273 K were isothermally 
expanded 5 times its initial volume and then isochorically heated so that the pressure in the final 
state become equal to that in the initial state. The total heat supplied in the process is 80 kJ. Find 
P
V
C
C
??
??
??
??
 of the gas. 
Q 2.  As a result of the isobaric heating by ?T = 72 K, one mole of a certain ideal gas obtains an amount 
of heat Q =1.6 kJ. Find the work performed by the gas, the increment of its internal energy and ?.
 
 
Q 3.  A gas undergoes the cycle shown in figure. The cycle is repeated 100 times per minute. Determine 
the power generated. 
Page 5


 
Introductory Exercise 18.2 
Q 1.  One mole of an ideal mono atomic gas is initially at 300 K. Find the final temperature if 200 J of 
heat are added as follows: 
  (a) at constant volume   (b) at constant pressure. 
Q 2.  Prove that work done by an ideal gas in an adiabatic process is 
i i f f
PV P V
W
1
?
?
??
, using the 
integration PdV
?
. 
Q 3.  When a gas expands along AB it does 500 J of work and absorbs 250 J of heat. When the gas 
expands along AC, it does 700 J of work and absorbs 300 J of heat. 
 
  (a) How much heat does the gas exchange along BC? 
(b) When the gas makes the transition from C to A along CDA, 800 J of work are done on it from 
C to D. How much heat does it exchange along CDA ? 
Q 4.  One mole of an ideal monoatomic gas occupies a volume of 1.0 × 10
-2
 m
3
 at a pressure of  
  2.0 × 10
5
 N/m
2
. 
  (a) What is the temperature of the gas? 
(b) The gas undergoes an adiabatic compression until its volume is decreased to 5.0 × 10
-3
 m
3
. 
What is the new gas temperature? 
(c) How much work is done on the gas during the compression?  
  (d) What is the change in the internal energy of the gas? 
Q 5.  A bullet of mass 10 g travelling horizontally at 200 m/s strikes and embeds in a pendulum bob of 
mass 2.0 kg. 
  (a) How much mechanical energy is dissipated in the collision? 
(b) Assuming that C
v
 for the bob plus bullet is 3 R, calculate the temperature increase of the 
system due to the collision. Take the molecular mass of the system to be 200 g/mol. 
Q 6.  An ideal gas is carried through a thermodynamic cycle consisting of two isobaric and two 
isothermal processes, as shown in figure. Show that the net work done in the entire cycle is given 
by the equation. 
 
Q 7.  Consider the cyclic process depicted in figure. If Q is negative for the process BC, and if. ?U is 
negative for the process CA, what are the signs of Q, W and ?U that are associated with each 
process? 
 
Q 8.  An ideal gas is enclosed in a cylinder with a movable piston on top. The piston has mass of 8000 g 
and an area of 5.00 cm
2
 and is free to slide up and down, keeping the pressure of the gas constant. 
How much work is done as the temperature of 0.200 mol of the gas is raised from 200°C to 
300°C? 
Q 9.  A sample of ideal gas is expanded to twice its original volume of 1.00 m
3
 in a quasi-static process 
for which P = ?V
2
, with 
? 
= 5.00 atm/m
6
, as shown in figure. How much work is done by the 
expanding gas? 
 
 
Solutions 
1.  (a) 
    
  or   200 = 1.5 × 8.31(T
f
 -300) 
  Solving we get, T
f
 = 316 K 
  (b) 
    
  or   200 = 2.5 × 8.31(T
f
 -300) 
  Solving we get, T
f
 = 309.6 K 
2.  In adiabatic process, 
  PV
?
 = constant = K (say) 
   
    
    
3.  (a) U
C
 - U
A
 along two paths should be same. 
    (Q
AB
 + Q
BC
) - (W
AB
 + W
BC
) = Q
AC 
- W
AC  
  ?  (250 + Q
BC
) - (500 + 0) = 300 - 700 Solving we get 
   Q
BC 
= -150 J 
  (b) Q
CDA
 = W
CD
 + W
DA
)+ (U
A 
- U
C
) 
   = (-800+ 0) - (U
C 
- U
A
)  
   = - 800 - (300 - 700) = - 400 J 
4.  (a) 
  (b) In adiabatic process  
   
    
  (c) Work done on the gas = ?U 
    
   = 1.5 × 8.31(383-240.7) ? 1770 J 
  (d) ?U ? 1770 J 
5.  (a) From conservation of linear momentum 
   0.01 × 200 = (2+0.01) v 
   V ? 1 m/s 
  Mechanical energy dissipated in collision = K
i 
- K
f
 
    
   ? 199 J 
  (b) 
  Using, 
   Q = nC
V
?T 
   
   = 0.80°C 
6.  Along the process CD apply, 
   PV - constant and find V
C
  
  Similarly along path AB, again apply, 
  PV = constant and find Y
B  
  
Now, W
nett
 = W
AB
 + W
BC
 + W
CD
 + W
DA
 
    
  Further,   nRT
A
=P
A
V
A
 
  and   nRT
C
 = P
C
V
C
 
7.  AB 
  V is increasing, so W = + ve. Product of PV and therefore T and therefore U is increasing. 
  So, ?U
 
is also +ve 
   Q = W + ?V  
  ?   Q is also positive.  
  EC  
     V = constant 
  ?   W = 0 
   Q = ?U = - ve  
  CA 
   V is decreasing.  
  ?  W =-ve 
  ?U
 
is also negative.  
  ?   Q = W + ?U is also negative. 
8.  PV = nRT 
  ? P ?V = nR ?T 
  Work done under constant pressure is, 
   W = P ?V = nR ?T 
   = (0.2)(8.31)(100)= 166.2 J 
9.  
    
  Solving we get, W = 1.18 × 10
6
 J 
   = 1.18 MJ  
 
Introductory Exercise 18.3 
Q 1.  Three moles of an ideal gas being initially at a temperature T
i
 = 273 K were isothermally 
expanded 5 times its initial volume and then isochorically heated so that the pressure in the final 
state become equal to that in the initial state. The total heat supplied in the process is 80 kJ. Find 
P
V
C
C
??
??
??
??
 of the gas. 
Q 2.  As a result of the isobaric heating by ?T = 72 K, one mole of a certain ideal gas obtains an amount 
of heat Q =1.6 kJ. Find the work performed by the gas, the increment of its internal energy and ?.
 
 
Q 3.  A gas undergoes the cycle shown in figure. The cycle is repeated 100 times per minute. Determine 
the power generated. 
 
Solutions 
1.  First process 
   T = constant 
   
  V is made 5 times. Therefore P will become 
1
th
5
. 
  Second process 
   V = constant 
   
  Pressure again becomes first times. So, temperature will also become 5 times. 
   Q
1
 + Q
2
 = 80 × 10
3
 
   
   = 80 × l0
3 
Solving we get, ?
 
= 1.4 
2.  W = nR ?T (in isobaric process) 
   = (1) (8.31)(72) = 600 J = 0.6 kJ 
    
    
3.  Work done per cycle = area under the cycle 
    
   = 1010 J 
  Total work done per second 
    
   = 1683 J/s = 1.68 W