Thermodynamics
Thermo—heat; dynamics --- power or capacity
Thermodynamics means the work created by heat.
Thermodynamics assigns its own special meanings not only to the word heat but also to the terms system, boundary
and state.
System:
A thermodynamic system is simply that part of the universe with which we are concerned; everything else in the
universe constitutes the surroundings.
Boundary:
Each thermodynamic system is surrounded by a boundary separating the system from its environment
(surroundings). Boundaries regulate the interaction between the system and its surroundings. Boundaries can be
divided into several kinds: those that permit or forbid work to be done on or by the system and those that permit or
forbid heat to be absorbed or rejected by the system. For instance, a movable boundary allows mechanical work to
be done on or by the system while a rigid one does not.
Depending up on the type of boundary, the systems are divided into three:
(i)
Open System: A system which can exchange matter and energy with the surroundings is called an
open system. E.g. Air compressor: Air at low pressure enters and air at high pressure leaves the system
i.e. there is an exchange of matter and energy with the surroundings.
(ii)
Closed System: A system which can exchange only energy (and not matter) with the surroundings is
called a closed system. E.g. Gas enclosed in a cylinder expands when heated and drives the piston
outwards. The boundary of the system moves but the matter (here gas) in the system remains constant.
(iii)
Isolated System: A system which is thermally insulated and has no communication of heat or work
with the surroundings is called isolated system.
Equilibrium:
Classical thermodynamics is concerned with equilibrium states described by a small set of macroscopic variables
that change only when the system’s environment changes. Relatively sudden or violent interactions, such as those
caused by pouring or stirring a fluid, destroy equilibrium because they set in motion changes that persist even after
the interaction is complete.
Two systems are in mutual equilibrium if the state variables of neither change when the two interact. Two
systems are in mutual thermal equilibrium if the state variables of neither change when the two are placed in thermal
contact. When two initially isolated systems, not in mutual thermal equilibrium, are placed in thermal contact, they
eventually achieve mutual thermal equilibrium.
Dr. M Chaitanya Varma
Dept. of Physics, GIT
Equilibrium Process
Changes or processes that unfold very slowly—so slowly that the system always remains in, or arbitrarily close to,
equilibrium even as it passes from one state to another—are called equilibrium or quasistatic processes.
Quasistatic Process
. Quasistatic processes can be analyzed into indefinitely small, temporally ordered parts, each standing for an
equilibrium state, and can be represented by continuous lines on a state variable diagram. A finite unbalanced
force may cause the system to pass through non-equilibrium states. A quasistatic process is defined as the process in
which the deviation from thermodynamic equilibrium is infinitesimal and all the states through which the system
passes during a quasistatic process can be considered as equilibrium states.
A quasistatic process is an ideal concept which can never be satisfied rigorously in practice. However, in actual
practice, many processes closely approach a quasistatic process with no significant error.
Thermodynamic Equilibrium
Any state of homogeneous system in which any two of the three variables P, V and T remain constant, as long as
the external conditions remain unchanged is said to be in thermodynamic equilibrium.
A system in thermodynamic equilibrium must satisfy the following requirements strictly:
(i) Mechanical Equilibrium. For a system to be in mechanical equilibrium, there should be no macroscopic
movement within the system (i.e. no unbalanced forces acting) or of the system with respect to its surroundings.
(ii) Thermal Equilibrium. For a system to be in thermal equilibrium there should be no temperature difference
between the parts of the system or between the system and the surroundings.
(iii) Chemical Equilibrium. For a system to be in chemical equilibrium there should be no chemical reaction
within the system and also no movement of any chemical constituent from one part of the system to the other.
Zeroth Law of Thermodynamics
When a hot body A is brought in thermal contact with a cold body B, heat flows from A to B and after
some time the flow stops. The bodies are then said to be in thermal equilibrium with each other.
Statement: The zeroth law of thermodynamics states that if two bodies A and B are each separately in thermal
equilibrium with a third body C, then A and B are also in thermal equilibrium with each other.
Now if the insulating partition is removed and systems A and B are brought into thermal contact, we find
that there is no further change. This means that the systems A and B are also in thermal equilibrium with another.
This is known as zeroth law of thermodynamics.
Empirical Temperature:
The zeroth law of thermodynamics allows us to associate with each equilibrium state a state variable T, called the
temperature, chosen in accordance with the rule that any two systems in thermal equilibrium have the same
temperature and any two systems not in thermal equilibrium have different temperatures. So chosen, a system’s
Dr. M Chaitanya Varma
Dept. of Physics, GIT
temperature reveals all its relations of potential thermal equilibrium or disequilibrium with other systems whose
temperature is known.
While the zeroth law itself does not dictate a method of assigning temperatures to equilibrium states,
scientists naturally prefer the convenience of a standard method.
Work:
In thermodynamics, work performed by a system is the energy transferred to another system that is measured by the
external generalized mechanical constraints on the system. As such, thermodynamic work is a generalization of the
concept of mechanical work in mechanics. Thermodynamic work encompasses mechanical work plus many other
types of work, such as electrical or chemical. It does not include energy transferred between systems by heat, as heat
is modeled distinctly in thermodynamics. Therefore, all energy changes in a system not a result of heat transfer into
or out of the system is thermodynamic work.
Example
Imagine a system that consists of a sample of ammonia trapped in a piston and cylinder, as shown in the figure
below. Assume that the pressure of the gas pushing up on the piston just balances the weight of the piston, so that
the volume of the gas is constant. Now assume that the gas decomposes to form nitrogen and hydrogen, increasing
the number of gas particles in the container. If the temperature and pressure of the gas are held constant, this means
that the volume of the gas must increase.
2 NH3(g)
N2(g) + 3 H2(g)
The volume of the gas can increase by pushing the piston partway out of the cylinder. The amount of work done is
equal to the product of the force exerted on the piston times the distance the piston is moved.
w=Fxd
The pressure (P) the gas exerts on the piston is equal to the force (F) with which it pushes up on the piston divided
by the surface area (A) of the piston.
Dr. M Chaitanya Varma
Dept. of Physics, GIT
Thus, the force exerted by the gas is equal to the product of its pressure times the surface area of the piston.
F=PxA
Substituting this expression into the equation defining work gives the following result.
w = (P x A) x d
The product of the area of the piston times the distance the piston moves is equal to the change that occurs in the
volume of the system when the gas expands. By convention, the change in the volume is represented by the symbol
V.
V=Axd
The magnitude of the work done when a gas expands is therefore equal to the product of the pressure of the gas
times the change in the volume of the gas.
|w| = P V
Convention used:
When the external force acting on a thermodynamic system is in the same direction as the displacement of the
system, work is done ON the system; the work is regarded as POSITIVE. When the external force is opposite to the
displacement, work is done BY the system; the work is regarded as NEGATIVE.
Concept of Heat
The concepts of temperature and heat were often confused before Joseph Black (1728–1799) carefully distinguished
between the two in the late eighteenth century. While both are uniquely thermodynamic concepts, each plays its own
role within the subject. Temperature, for instance, is an intensive state variable—intensive because temperature does
not depend in a direct way upon the size of the system.
In contrast, heat is not a state variable at all but, rather, quantifies the interaction between a system and its
environment allowed by a diathermal boundary.
Heat is defined as energy in transit. As it is not possible to speak of work in a body, it is also not possible to speak
of heat in a body. Work is either done on a body or by a body. Similarly, heat can flow from a body or to a body. If a
body is at a constant temperature, it has both mechanical and thermal energies due to the molecular agitations and it
is not possible to separate them. So, in this case, we cannot talk of heat energy. It means, if the flow of heat stops, the
word heat cannot be used. It is only used when there is transfer of energy between two or more systems.
Dr. M Chaitanya Varma
Dept. of Physics, GIT
First law of thermodynamics
Since heat, Q, transferred to and work, W, done on a system may produce the same change of state, each must be
one part of a single quantity. Since work done on a purely mechanical system is known to increase its mechanical
energy, it is natural to identify the sum Q +W introduced into a thermodynamic system as an increase of its internal
energy, ΔU. For this reason one common verbal expression of the first law of thermodynamics is “energy is
conserved if heat is taken into account.
Algebraically the first law of thermodynamics states that
Q + W = ΔU.
Because heat, Q, work, W, and the change in internal energy, ΔU, are, in general, signed quantities, the above
equation go beyond recapitulating the bare facts of Joule’s experiments. The work may be done on or by the system
or not at all (W > 0, W < 0, or W = 0) and by any means (mechanical, electrical, or magnetic); heat may be
transferred to or rejected from the system or not at all (Q > 0, Q < 0, or Q = 0); internal energy may increase,
decrease, or remain the same (ΔU > 0, ΔU < 0, or ΔU = 0); and the work, W, and heat transfer, Q, may be
simultaneous or successive. Adiabatic (Q = 0), diathermal (Q ≠ 0), work- allowing (W ≠ 0), and work- prohibiting
(W = 0) boundaries help realize these possibilities. In each process the algebraic sum of heat transferred to and work
performed on the system Q + W changes the internal energy by ΔU.
Example:
Consider two systems A and B in thermal contact with one another and surrounded by adiabatic walls.
For the system A,
Q' = U2-Ux + W
.
where Q is the heat energy transferred, Ux is the initial internal energy, U2 is the final internal energy
and W is the work done.
.
Similarly for the system B,
Q'  U 2'  U1'  W '
Adding (i) and (ii)
Q  Q '  (U 2  U1 )  W  (U 2'  U1' )  W
Q  Q '  (U 2  U1 )  (U1  U1' )  (W  W ' )
The total change in the internal energy of the composite system
[(U 2  U1 )  (U1  U1' )]
The work done by the composite system = W + W
It means that the heat transferred by the composite system = Q + Q. But the composite system is surrounded
by adiabatic walls and the net heat transferred is zero.
Thus, for two systems A and B in thermal contact with each other, and the composite system surrounded by
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Dept. of Physics, GIT
adiabatic walls, the heat gained by one system is equal to the heat lost by the other system.
Internal Energy (U)
The energy content of a system is called its internal energy. It is the sum of following forms of energy of the
system:
i.
Kinetic energy due to translational, rotational and vibrational motion of the molecules, all of which depend
only on the temperature,
ii.
potential energy due to intermolecular forces, which depends on the separation between the molecules, and
iii.
the energy of electrons and nuclei.
In practice, it is not possible to measure the total internal energy of a system in any given state. Only
change in its value can be measured.
If the state of the system is changed from an initial state 1 to a final state 2, by supplying heat Q to the
system and if W is the work done by the system during the change, then increase in the internal energy of the system
is given by
(U2-U,) = Q-W
Where U1 is the internal energy in state 1, and U2 the internal energy in state 2.
Significance of the First Law:
The first law of thermodynamics is important because
i.
It is applicable to any process by which a system undergoes a physical or chemical change.
ii.
It introduces the concept of the internal energy, and
iii. It provides method for determining the change in the internal energy.
Limitations of the First Law:
The first law of thermodynamics is based on the principle of conservation of energy of a system. Though it
is applicable to every process in nature between the equilibrium states, it does not specify the condition under which
a system can use its heat energy to produce a supply of mechanical work. It also does not say how much of the heat
energy can be converted into work.
Isochoric process (i.e. volume constant)
If a system undergoes a change in which the volume remains constant, the process is called isochoric. At
constant volume, no external work is done i.e. W = 0
 Heat absorbed is given by  Q = ΔU
This expression may be used to define the internal energy of a system.
Thus, the increase in the internal energy of a system is equal to the heat absorbed by the system, at constant
volume. The work done in isochoric process is zero.
Dr. M Chaitanya Varma
Dept. of Physics, GIT
Isobaric process (i.e. pressure constant)
If a system undergoes a change in which the pressure is kept constant, the process is called isobaric. Suppose Q
is the heat absorbed by a system at constant pressure P and suppose its volume increases from Vl to V2. Then from
the first law, we have
Q = (U2-U,) + W
or
Q = (U2-U1) + P(V2-V1)
or
Q = (U2 + PV2) - (U1-PV1)
= H2-H1
H = U + PV.
From equation (4.6), we conclude that the heat absorbed at constant pressure is equal to increase in quantity
H, called as enthalpy. Like the internal energy U, the quantity H is a function of thermodynamic variables. The
quantity H is also called the heat function at constant pressure.
Adiabatic process
When a system undergoes from an initial state to a final state in such a way that no heat leaves or enters the
system, the process is called adiabatic. In this process,
Q0

U 2  U1  W
For an ideal gas
U 2  U1  Cv d r

W  Cv d r
Here W is the work done by the system. Thus, when a system expands adiabatically, its internal energy decreases. If the system is compressed adiabatically then W is negative. Thus, in case of compression
U 2  U1  (W )  W
Therefore, in an adiabatic compression of a system its internal energy increases. The processes that take
place suddenly or quickly are adiabatic processes.
Cyclic process
For the cyclic process, the law can be stated as
dQ  dU  dW
In a cyclic process, the system is restored to the initial state at the end of
the cycle.
Since, the internal energy is a state function,

 dU
0
Dr. M Chaitanya Varma
Dept. of Physics, GIT
 dQ
or

 dW
 Total work obtained = Net Heat supplied. Thus no work is obtained if no heat is supplied or work can be obtained
only at the cost of energy. From indicator diagram, we write
Q  Area ACBDA
Isothermal process
If a system is perfectly conducting to the surroundings and the temperature remains constant j throughout
the process, it is called an isothermal process. Consider a working substance at a certain % pressure and temperature
and having volume represented by the point A (Fig. 4.6).
Pressure decreased and work is done by the working substance at the cost of its internal energy and suffers a fall
in temperature. But the system is perfectly conducting to the surroundings. It absorbs heat from the surroundings
and maintains a constant temperature.
Thus, from A to B, the temperature remains constant. The curve AB is called the isothermal curve or
isothermal. In going from B to A back, the system gives out extra heat to the surroundings and maintains the
temperature constant.
Thus, during isothermal process, the temperature of the working substance remains constant. It can absorb heat
or
give
heat
to
the
surroundings.
The
equation
for
isothermal
process
is
PV = RT = constant (for one gram molecule of a gas)
For an gram molecules of a gas,
PV = nRT
 For an ideal gas undergoing isothermal process,
(U2-Ux) - 0
 From the first law of thermodynamics, we get
Q =W
i.e. in an isothermal process the heat supplied to an ideal gas is equal to the work done by the gas.
The Indicator Diagram
It is very convenient to represent the behaviour of an engine by an indicator diagram. This helps to understand
the performance of heat engines. Suppose that a system is taken from an initial equilibrium state A to the final
equilibrium state B along path AB and the process is quasistatic (Fig 4.7).
Each point on the path traced represents the state of p the system in terms of coordinates (P, V). Such a trace
obtained in an actual engine is called indicator diagram or P - V diagram.
The work done by the gas in expanding against pressure P is PdV, where dV is the infinitesimal change in
volume. For finite process AB, the work done
=  PdV = area ABba
Dr. M Chaitanya Varma
Dept. of Physics, GIT
Thus, the indicator diagram directly indicates the work done by the engine during each cycle of operation,
the work being equal to the area enclosed by the indicator diagram. The work done is +ve if the indicator
diagram is traced in clock-wise direction and -ve if the direction is anticlock wise.
Work done During an Isothermal Process
When a gas is allowed to expand isofhermally, work is done by it. Let the initial and final volumes be V1 and V2
respectively. Fig 4.8 represents the indicator diagram. The area of the shaded strip represents the work done for a
small change in volume dV when the volume changes from V1 to V2.
Work done during an Adiabatic Process
During an adiabatic process, the system is thermally insulated from the surroundings. The gas expands from
volume V1 to V2. as shown by indicator diagram [Fig. 4.9]. The work done by the gas for an increase in volume dV
is P dV.
Work done when the gas expands from Vl to V2 is given by
V2
W   PdV
V1
= Area ABba
During an adiabatic process
PV   constant = K
P
Or
K
V
V2
dV
V
V1
W K

K
1 
 1
1 
  1   1 
V1 
V2
Since A and B lie on the same adiabatic,


P1V1  P2V2  K
W

1
1 
1
1 
w
 K
K 
  1   1 
V1 
V2
 P2V2 P1V1 
  1   1 
V1 
 V2
1
P2V2  P1V1 
1 
Taking T1 and T2 as the temperatures at points A and B respectively and considering one gram molecule of the gas
P1V1  RT1
Dr. M Chaitanya Varma
Dept. of Physics, GIT
P2V2  RT2
Substituting these values in equation (4,10)
W
1
RT2  RT1 
1 
W
R
T2  T1 
1 

R
T  T 
 1 1 2
Hence the work done in adiabatic process depends only upon the initial and final temperatures T 1 and T2.
Thus the work done along any adiabatic between two isothermals is independent of the particular adiabatic.
Slopes of Adiabatics and Isothermals
In an isothermal process
PV = constant
Differentiating
PdV VdP  0
dP P
 ...
dV V
Or
In an adiabatic process
PV   cons tan t
Differentiating,
PV  1dV  V  dP  0
dP P
 ...
dV V
 dP 


Therefore, the slope  dV  of an adiabatic is  times the slope of the isothermal.
Hence, the adiabatic curve is steeper than the isothermal curve at a point A where the two curves intersect each
other.
Reversible and Irreversible Process
Reversible Process
A reversible process is one which can be retracted in opposite direction by changing the external conditions
infinitesimally.
The process will not be reversible if there is any loss of heat due to friction, radiation or conduction. If the changes
take place rapidly, the process will not be reversibly. The energy used in over coming friction can not be retraced.
Dr. M Chaitanya Varma
Dept. of Physics, GIT
The conditions of reversibility for any heat engine or process can be stated as follows:
1.
The pressure and temperature of the working substance must not differ appreciably from those of the
surrounding at any stage of the cycle of operation.
2.
All the processes taking place in the cycle of operation must be infinitely slow.
3.
The working parts of the engine must be completely free from friction.
4.
There should not be any loss of energy due to conduction, convection or radiation during the cycle of
operation.
It should be remembered that the complete reversible process or cycle of operation is only an ideal case. In
an actual process, there is always loss of heat due to friction, conduction, convection or radiation.
Irreversible Process
In nature all changes are irreversible because of the following reasons: (1) The conditions for
thermodynamic equilibrium i.e. mechanical, thermal or chemical equilibrium are not satisfied because a natural
process does not take place quasi-statically. (2) Dissipative effects, such as friction, viscosity, inelasticity, electric
resistance, eddy-formation etc. are always present.
Examples of some natural irreversible processes are: spontaneous expansion of a gas into an evacuated
space, spontaneous conduction of heat along a metal bar which is hot at one end and cold at the other, transfer of
heat by radiation from a hotter to a colder body, transfer of electricity through a resistor. Nonetheless reversible
processes are most important in thermodynamics because they can be handled analytically.
Definition of Efficiency
The efficiency, , of a heat engine is defined as the ratio of the mechanical work done by the engine in one
cycle to the heat absorbed from the high temperature source, thus

Q1  Q2
Q1
where Q1 is the heat absorbed from the source at high temperature, Q2, is the heat rejected to a sink at low
temperature and (Q1 – Q2) is the mechanical work done by the engine in one cycle. Since (Q1 - Q2) < Q1, the
efficiency can never be 100%.
Carnot's Ideal Heat Engine
Sadi Carnot conceived a theoretical engine which is free from all practical imperfections. Such an engine
cannot be realised in practice. It has maximum efficiency and it is an ideal heat engine. Sadi Carnot's heat
engine requires the following important parts:
1.
A cylinder having perfectly non-conducting walls, a perfectly conducting base and is provided with a
perfectly non-conducting piston which moves without friction in the cylinder. The cylinder contains
one mole of perfect gas as the working substance.
Dr. M Chaitanya Varma
Dept. of Physics, GIT
2.
Source. A reservoir maintained at a constant temperature T1 from which the engine can draw heat by
perfect conduction. It has infinite thermal capacity and any amount of heat can be drawn from it at constant
temperature T1.
3.
Heat insulating stand. A perfectly non-conducting platform acts as a stand for adiabatic processes.
4.
Sink. A reservoir maintained at a constant lower temperature T2 (T2 < Tx) to which the heat engine can reject
any amount of heat. The thermal capacity of sink is infinite so that its temperature remains constant at
T2, no matter how much heat is given to it.
CARNOTS CYCLE
In order to obtain a continuous supply of work, the working substance is subjected to the following cycle of
quasi-static operations known an Carnot's cycle (Fig. 4.16).
1. Isothermal expansion. The cylinder is first placed on the source, so that the gas acquires the temperature T1 of
the source. It is then allowed to undergo quasi-static expansion. As the gas expands, its temperature tends to fall.
Heat passes into the cylinder through the perfectly conducting base which is in contact with the source. The gas
therefore, undergoes slow isothermal expansion at the constant temperature T1.
Let the working substance during isothermal expansion goes from its initial state
A( P1,V1,T1 ) to the state
B( P2 ,V2 , T1 ) at constant temperature T1 along AB. In this process, the substance absorbs heat Q1 from the source at T1
and does work W1 given by
v2
Q1  W1   PdV  RT1 log e
v1
2.
V2
 area ABGEA
V1
Adiabatic expansion. The cylinder is now removed from the source and is placed on the insulating stand.
The gas is allowed to undergo slow adiabatic expansion, performing external work at the expense of its internal
energy, until its temperature falls to T2, the same as that of the sink.
This operation is represented by the adiabatic BC, starting from the state B (P2, V2, T1) to the state C (P3, V3, T2). In
this process, there is no transfer of heat, the temperature of the substance falls to T2 and it does some external work W2
given by
V3
V3
V2
V2
W2   P.dV  K 
dV
V
( During adiabatic process,

PV  P V
KV31  KV21
 3 3 2 2
1 
1 
PV 
= constant = K)
( P2V2  P3V3  K )
Dr. M Chaitanya Varma
Dept. of Physics, GIT

RT2  RT1
R(T1  T2 )


1 
 1
area BCHGB
3. Isothermal Compression. The cylinder is now removed from the insulating stand and is placed on the sink
which is at a temperature T2. The piston is now very slowly moved inwards so that the work is done on the gas. The
temperature tends to increase due to heat produced by compression since the conducting base of the cylinder is in
contact with the sink, the heat developed passes to the sink and the temperature of the gas remains constant at T2. Thus
the gas undergoes isothermal compression at a constant temperature T2 and gives up some heat to the sink.
This operation is represented by the isothermal CD, starting from the state" C (P3, V3, T2) to the state D (P4, V4, T2).
In this process, the substance rejects heat Q2 to the sink at T2 and work W3 is done on the substance given by
V4
Q2  W3   PdV  RT2 log e
V3
  RT2 log e
V3

V4
V4
V3
area CHFDC
(- ve sign indicates that work is done on the working substance)
4. Adiabatic Compression. The cylinder is now removed from sink and again placed on the insulating stand.
The piston is slowly moved inwards so that the gas is adiabatically compressed and the temperature rises. The
adiabatic compression is continued till the gas comes back to its original condition i.e. state A (P1, V1, T1), thus
completing one full cycle.
This operation is represented by adiabatic DA, starting from D (P4, V4, T2) to the final state A (P1, V1, T1). In this
process, work W4 is done on the substance and is given by
V1
W4   P.dV
V4

R(T1  T2 )
 1
= area DFEAD
(- ve sign indicates that work is done on the working substance. Since W2 and W4 are equal and opposite, they
cancel each other.)
Work done by the engine per cycle
During the above cycle, the working substance absorbs an amount of heat Q 1 from the source and rejects Q2
to the sink.
Hence, the net amount of heat absorbed by the gas per cycle
 Q1  Q2
The net work done by the engine per cycle
 W1  W2  W3  W4
 W1  W3
From the graph, the net work done per cycle
Dr. M Chaitanya Varma
Dept. of Physics, GIT
= area ABGEA + area BCHGB – are CHFDC – area DFEAD
= area ABCDA
Thus, the area enclosed by the Carnot’s cycle consisting of two isothermals and two adiabatics gives the net
amount of work done per cycle.
In the cyclic process.
Net heat absorbed
= Net work done per cycle.
Q1  Q2  W1  W3
 RT1 log e
V
V2
 RT2 log e 3
V1
V4
Since the points A and D lie on the same adiabatic DA
T1V1 1  T2V4 1
T2  V1 
 
T1  V4 
 1
Similarly, points B and C lie on the same adiabatic BC
T1V2 1  T2V3 1
T2  V2 
 
T1  V3 
 1
From equations (5.20) and (5.21),
 V1 
 
 V4 
 1
Or
V1 V2

V4 V3
Or
V2 V3

V1 V4
V 
  2 
 V3 
 1
Substituting in equation (5.19), we get
Net work done
 Q1  Q2  RT1 log e
V2
V
 RT2 log e 2
V1
V1
W  (Q1  Q2 )  R(T1  T2 ) log e
V2
V1
Efficiency
The efficiency of the heat engine is the rate of quantity of heat converted into work (Useful output) per cycle to
Dr. M Chaitanya Varma
Dept. of Physics, GIT
the total amount of heat absorbed per cycle.
Efficiency,


useful output W

input
Q1
(Q1  Q2 )
Q1


RT1 log e
V2
V1
V2
V1
T1  T2
T1
 1
or
R(T1  T2 ) log e
T2
T1
From equation, we conclude that the efficiency depends only upon the temperature of the source and sink and is
always less than unity. The efficiency is independent of the nature of working substance. From equation,
we get
 1
T2
,
T1
  1, if T2  oK i.e. the temperature of the sink is at absolute zero degrees. In practice, it is never possible
to reach absolute zero and hence 100% conversion of heat energy into mechanical work is not possible.
Again, the efficiency is minimum or zero when
temperature of sink, then
T1  T2 i.e. the temperature of the source is equal to the
  0 i.e. the engine does not work.
The Carnot's heat engine is perfectly reversible. It can be operated in the reverse direction also. Then it works as a
refrigerator. The heat Q2 is taken from the sink and external work is done on the working substance and heat Q2 is rejected to
the source at a higher temperature (principle of a refrigerator).
Moreover in the Carnot's heat engine, the process of isothermal and adiabatic expansions and compressions are
carried out very-very slowly i.e. quasi-static. This is an ideal case. Any practical engine can not satisfy these
conditions. Therefore, all practical engines have an efficiency less than the Carnot's engine.
Effective Way to Increase Efficiency
The expression for the efficiency of a Carnot's engine is
 1
T2
T1
The efficiency  can be increased by one of the following ways:
(i) Using a heat-source at constant temperature T1 and heat-sink at temperature as low as possible, or
(ii) Using a heat-sink at constant temperature T2 and a heat-source at temperature as high as possible.
In practice, it is not convenient to use heat sink at a temperature below that of the atmosphere. Therefore, the
Dr. M Chaitanya Varma
Dept. of Physics, GIT
more effective way to increase the efficiency is to use a heat-source at temperature as high as possible.
Carnot's Engine and Refrigerator
Carnot's cycle is perfectly reversible. It can work as a heat engine and also as a refrigerator.
When it is operated as a heat engine, it absorbs heat Q1 from the source at a temperature T1 does an amount of work
W and rejects heat Q2 to the sink at temperature T2, (T2 < T1).
When it is operated as a refrigerator, it absorbs heat Q2 from the sink at temperature T2. W amount of work is done
on it by some external means and rejects heat Q1 to the source at temperature T1 (T1 > T2).
In the second case of heat flows from a body at a lower temperature T2 to a body at a higher temperature T1 with
the help of external work done on the working substance. This action is that of a refrigerator. In every cycle, heat Q2 is
extracted from the cold body. This will not be possible if the cycle is not completely reversible.
The amount of heat absorbed at the lower temperature is Q2. The amount of work done on the working substance
by the external agency (input energy) = Wand the amount of heat rejected = Q1. Here Q2 is the desired refrigerating
effect in each cycle.
..". Coefficient of performance, P 
Q2
Q2

W Q1  Q2
Suppose 200 joules of energy is absorbed at the lower temperature and 100 joules of work is done on it by external
help. Then, 200 + 100 = 300 joules are rejected at the higher temperature. The coefficient of performance.
P
Q2
W

Q2
Q1  Q2

200
2
300  200
Therefore, the coefficient of performance of a refrigerator is 200%.
In the case of heat engine, the efficiency cannot be more than 100% but in case of a refrigerator, the coefficient of
performance can be much higher than 100%. If the working substance is an ideal gas, then
Q1 Q2 Q1  Q2


T1 T2
T1  T2
Or
Q2
T2

Q1  Q2 T1  T2
Dr. M Chaitanya Varma
Dept. of Physics, GIT
Thus, equation can also be expressed as
Coefficient of performance, P 
T2
T1  T2
Second Law of Thermodynamics
The first law of thermodynamics states the equivalence of heat and energy. It simply tells that whatever work is
obtained, ah equivalent amount of heat is used up, or vice versa. It does not say anything about the direction in
which the change might occur or about the range or limit to which it can be possible. The first law shows that
perpetual motion of the first kind is impossible i.e. energy can not be created out of nothing or production of energy
without disappearance of an equivalent energy of another form is not possible.
The first law has no answer why heat always flows from a body at high temperature to a body at lower temperature
and does not flow in the reverse direction. It can not explain why vast amount of available heat cannot be converted
into mechanical work. It we could control and make use of the limitless store of heat viz-solar energy, we could have
an inexhaustible supply of useful energy. Thus, we could have a perpetual motion machine not forbidden by the first
law. This is called perpetual motion of the second kind. In practice, however there is no engine which can convert
the heat from the single source in to useful work without rejecting some heat to a heat sink at a lower temperature i.e.
perpetual motion of the second kind i.e. production of useful energy from the internal energy of one body is
impossible.
It was the quest of several such questions which led to the formulation of second law of thermodynamics. The
second law is a generalization of certain experiences and observations and is concerned with the direction in which
energy transfer takes place. The law has been stated in a number of ways, which means the same thing.
Lord Kelvin's Statement
In a heat engine the working substance extracts heat from the source, converts a part of it into work and rejects
the rest to a sink at a lower temperature. The temperature of the source must be higher than the surroundings and
engine will not work when the temperatures of source and sink are the same. No engine has ever been constructed
which converts all the heat absorbed from the source into work without rejecting a part of it to the cold body. As the
engine absorbs more and more heat from the hot body, the latter suffers a continuous fall in temperature and if a
continuous supply of work is desired, the hot body will in the long run become as cold as its surroundings. Then no
heat flow will be possible, the engine will stop working and hence no mechanical work will be obtained. It means that
we can not obtain a continuous supply of work from a single supply of it i.e. the presence of colder body is a must for
the continuous conversion of heat into work. Such considerations led-. Lord Kelvin to state that
"It is impossible to get a continuous supply of work from a body by cooling it to a temperature lower than that
of its surroundings."
Planck's Statement
"It is impossible to construct an engine which, working in a complete cycle, will produce no effect other than
Dr. M Chaitanya Varma
Dept. of Physics, GIT
the raising of a weight and the cooling of a heat reservoir."
Thus, it is impossible to construct an engine which working in a complete cycle, will produce no effect other than the
absorption of heat from a reservoir and its conversion into an equivalent amount heat absorbed to a sink at lower
temperature.
These two statements can be combined into one equivalent statement, known as the Kelivn-Planck's
statement of the second law of thermodynamics.
Kelvin-Planck Statement
"It is impossible to construct an engine which, operating in a cycle, has the sole effect of extracting heat from a
reservoir and performing an equivalent amount of work."
Clausius's Statement
According to Clausius "It is impossible for a self-acting machine working in a cyclic process, unaided by
external agency, to transfer heat from a body at a lower temperature to a body at a higher temperature."
In other words it may be stated as "Heat cannot flow of itself from a colder body to a hotter body."
This statement is based upon the performance of refrigerator - a heat engine working in the backward direction.
This statement means that natural flow of heat is always from a hot body to a cold body. If heat is to be transferred from
cold body to hot body, work will have to be done by external agency. A refrigerator is a device which transfers heat
from a colder body to hotter body by doing external work on the working substance. The compression is brought about
by an external agency i.e. 'electricity' by performing work on the working substance.
Carnot's Theorem
Statement
From the second law of thermodynamics two important results are derived; these conclusions are taken together
to constitute Carnot's theorem which may be stated in the following forms.
(a) 'No engine can be more efficient than a perfectly reversible engine working between the same two temperatures.'
(b) 'The efficiency of all reversible engines, working between the same two temperatures is the same,
whatever the working substance.'
Proof:
First Part: To prove the first part of the theorem, we consider two engines R and / working between the
temperatures Tl and T2 where Tx > T2 [Fig. 4.18]. Of these two engines R is reversible and / is irreversible.
Suppose / is more efficient than R. Suppose in each cycle, R absorbs the quantity of heat Qx from the source at
Dr. M Chaitanya Varma
Dept. of Physics, GIT
Tl and rejects the quantity of heat Q2to the sink at T2. Suppose in each cycle / absorbs the quantity of heat
Q1' from
the source at 7", and gives up the quantity of heat Q2' to the sink at T2. Let the two engines do the same amount of
work W in each cycle. According to the assumption / is more efficient than R.

Q1'  Q2' Q1  Q2

Q1'
Q1
Or
W W

Q1' Q1
Or
Q1  Q1'
And
Q1'  Q2'  Q1  Q2
Or
Q2  Q1'  Q1  Q1'
Now because
Q1  Q1'
Q2  Q2'
Now suppose the two engines are coupled together so that / drives R backwards and suppose they use the same
source and sink. The combination forms a self-acting machine in which / supplies external work W and R absorbs this
amount of work in its reverse cycle. I in its cycle absorbs heat
Q1' from the source and gives up heat Q2' to the sink. R
in its reverse cycle, absorbs heat Q2 from the sink and gives up heat
Q1 to the source.
The net result of the complete cycle of the coupled engines is given by
Gain of heat by the source at
T1  Q1  Q1'
Loss of heat by the sink at
T1  Q2  Q2'
External work done on the system = 0
Thus, the coupled engines forming a self-acting machine unaided by any external agency transfer heat
continuously from a body at low temperature to a body at a higher temperature.
This conclusion is contrary to the second law of thermodynamics, according to which in a cyclic process heat
cannot be transferred from one body to another at a higher temperature by a self-acting machine. Hence our
assumption is incorrect and we conclude that no engine can be more efficient than a perfectly reversible engine
working between the same temperatures.
Second Part: The second part of the theorem may be proved by the same arguments as before. For this purpose,
Dr. M Chaitanya Varma
Dept. of Physics, GIT
we consider two reversible engines R1 and R2 and assume that R2 is more efficient than R1 Proceeding in the same way
we can show that R2 cannot be more efficient than R1 Therefore, all reversible engines working between the same two
temperatures have the same efficiency.
Thus, the efficiency of a perfectly reversible engine depends only on the temperatures between which the
engine work, and is independent of the nature of the working substance
Concept or Entropy
In order to describe the condition of a working substance completely, Clausius felt the needs of another variable of
state, in addition to volume, pressure, temperature, internal energy etc. This
Change in Entropy
Let us consider reversible carnot’s cycles bounded by same two adiabatic L and M and isothermals T 1 , T2 and T3 as
shown in an indictor diagram (Fig 5.1) for an ideal gas. Then all along the adiabatic L and M, there is change in
volume and temperature with change in pressure. Let ABCD and DCEF represent the Carnot’s reversible cycles.
During Crnot’s cycle ABCD, an amount of heat Q1 is absorbed in going from A to B at constant temperature T 1 and
amount of heat Q2 is rejected at constant temperature T 2. Then efficiency of Carnot’s engine is given by

1
Q1  Q2 T1  T2

Q1
T1
Q2
T
 1 2
Q1
T1
Q2 T2

Q1 T1
Q1 Q2

T1 T2
Similarly considering the Carnot’s cycle DCEF in which an amount of heat Q 2 is absorbed at constant temperature
T2 and heat Q3 is rejected at constant temperature T 3.
Q2 Q3

T2 T3
From equations (5.1) and (5.2) we have
Q2 Q2 Q3


=…… constant
T1 T2 Q3
In general, if Q is the amount of heat absorbed or rejected at a temperature T in going from one adiabatic to the
other, then
Dr. M Chaitanya Varma
Dept. of Physics, GIT
Q
=constant
T
If the two adiabatic are very close to each other and if
Q
is the small quantity of heat absorbed at constant
temperature T in going from one adiabatic to another, then
Q
=constant
T
This constant ratio is called the ‘change in entropy’ between the states represented by the two adiabatic. It is denoted
by

Q
S 
T
The change in entropy for a finite reversible change in the state of working substance from A to B is given by
SB
B
dQ
T
A
 ds  
SA
B

dQ
T
A
SB  S A  
B
The expression
S
dQ B
A T  S ds is a function of the thermodynamic coordinates of a system. This function is
A
represented by symbol S and is called entropy. Hence ‘entropy of a system is a function of the thermo dynamical
coordinates defining the state of the system viz., the pressure, volume, temperature or internal energy and its change
between two states is equal to

dQ
Between the states along any reversible path joining them.” dS is an exact
T
differential. Entropy is an extensive property since it depends on the mass of the working substance.
Physical Concept of Entropy
In fact, it is very difficult to form a physical concept of entropy as there is nothing physical to represent it.
Moreover it cannot be felt like temperature, pressure etc. But according to definition
Change of entropy o= Heat added r subtracted 
dQ
T
Absolute temperature
One can say that heat has the same dimensions as the product of entropy and absolute temperature. Since
the gravitational potential energy of a body is proportional to the product of its mass and height above some Zero
level, likewise we may take temperature analogous to height and entropy as analogous to mass or inertia. Thus, we
may take entropy as thermal inertia which bears to heat motion , a relation similar to that which mass bears to
linear motion or moment of inertia bears to rotational motion.
Dr. M Chaitanya Varma
Dept. of Physics, GIT
Change in Entropy in Adiabatic process
In an adiabatic process, no heat is allowed to enter or leave the system. Hence Q  0
Thus, there is no change of entropy during an adiabatic process, or the entropy remains constant during an adiabatic
reversible reversible process. Hence the adiabatic curves on the p-V diagram are called as isoentropics or curves of
constant entropy.
Change of Entropy in Reversible Cycle
Consider a complete reversible Carnot’s cycle ABCD as shown in Fig. 5.2 for an ideal gas formed by two
isothermals i.e. AB at a temperature T 1 and CD at temperature T2 and the two adiabatics BC and DA.
(i) Isothermal expansion AB; Let Q1 be the quantity of heat absorbed by the working substance in going
from state A to state B during isothermal expansion AB at a constant temperature T 1. The increase in
entropy of the working substance is given by
B
Q1
A
1
 dS   T
(ii) Adiabatic expansion BC; in going from state B to state C along the adiabatic BC, there is no change in
entropy of the working substance, but the temperature falls from T 1 to T2 due to expansion.
C
 dS  0
B
(iii) Isothermal compression CD; IN going from state C to state D along the isothermal CD, the working
substance rejects heat Q2 to the sink at temperature T2. The entropy of the working substance decreases and
the change in entropy is given by
D
Q2
C
2
 dS   T
(iv) Adiabatic compression DA; in going from D to A along the adiabatic DA, there is no change in entropy
but temperature rises from T2 to T1.
A
 dS  0
D
Thus the net gain in entropy of the working substance in the whole cycle ABCDA
B
C
D
A
A
B
C
D
 dS   dS   dS   dS
dS 
Q1 Q2

T1 T2
But for a reversible Carnot’s cycle]
Dr. M Chaitanya Varma
Dept. of Physics, GIT
Q1 Q2
Q Q

or 1  2  0
T1 T2
T1 T2
Substituting we get
dS 
Q1 Q2

0
T1 T2
Where the integral sign with a circle refers to a complete cycle.
Thus in a cycle of reversible process, the entropy of the system remains unchanged or remains constant. In
other words, the total change in entropy is always zero.
Principle of Increase of Entropy
Let us consider an engine performing irreversible cycle of changes in which the working substance absorbs
heat Q1 at temperature T1 from the source and rejects heat Q2 to the sink at temperature T2.
' 
Q1  Q2
Q
1 2
Q1
Q1
But according to Carnot’s theorem, this efficiency is less than that of a reversible engine working between the same
two temperatures T2 for which efficiency is given by
  1
T2
T1
 ' 
Thus,
1
Q2
T
1 2
Q1
T1
Q2 T2
Q2 Q1


or
Q1 T1
T2 T1
Q2 Q1

0
T2 T1
Or
Considering the whole system, the source loses (-ve sign ) the entropy by an amount
sign) entropy
Q1
and the sink gains (+ ve
T1
Q Q 
Q2
. Therefore, the net change in entropy for the whole system is  2  1 , which is clearly
T2
 T2 T1 
greater than zero I.e. positive ( Equation 5.6). Thus, there is an increase in entropy of the system
during an
irreversible process.
To make it clearer, let us consider the natural process of conduction of heat from a body A at temperature
T1 to another body B at a lower temperature T 2. This process is irreversible. Since heat always flows from higher to
a lower temperature the quantity of heat thus transmitted be Q, then
Dr. M Chaitanya Varma
Dept. of Physics, GIT
Decrease in entropy of hotter body=
Q
T1
Increase in entropy of colder body =
Q
T2
 Net increase in entropy of the system
dS 
Q Q

T2 T1
= +ve quantity as
T1  T2 .
Thus, we conclude that the entropy of a system increase in all irreversible processes. This is known as the
law or principle of increase of entropy. All natural processes taking place in the universe are irreversible. It means
the entropy of the universe increases.
Change of entropy in Irreversible Process
The thermo dynamical state of a system can be defined with the help of the thermo dynamical coordinates
of the system. The state of a system can be changed by altering the thermo dynamical coordinates. Changing from
one state to the other by changing the thermo dynamical coordinate is called a process.
Consider two states of a system i.e, state A and state B. change of state from A to B or vice versa is a process
and the direction of the process will depend upon a new thermo dynamical coordinate called entropy. All processes
are not possible in the universe.
Consider the following processes;
1.
Let two blocks A and B at different temperatures T1 and T2 (T1>T2) be kept in contact but the system as a whole is
insulated from the surroundings. Conduction of heat taken place between the blocks, the temperature of A falls and
the temperature of B rises and thermo dynamical equilibrium will be reached.
2.
Consider a flywheel rotating with an angular velocity w. Its initial kinetic energy is ½ I .
2
After some time the wheel comes to rest and kinetic energy is utilised in overcoming friction at the bearings. The
temperature of the wheel and the bearing rises and the increase in their internal energy is equal to the original kinetic
energy of the fly wheel.
3.
Consider two flasks And B connected by a glass tube provided with a stop cock. Let A contain air at high pressure
and B is evacuated. The system is isolated from the surroundings. If the stop cock is opened, air rushes from A to B,
the pressure in A decreases and the volume of air increases.
All the above three examples though different, are thermo dynamical processes involving change in
thermidynamical coordinates. Also, in accordance with the first law of thermodynacs, the principle of conservation
of energy is not violated because the total energy of the system is conserved. It is also clear that, with the initial
Dr. M Chaitanya Varma
Dept. of Physics, GIT
conditions described above, the three processed will take place.
Let us consider the possibility of the above three processes taking place in the reverse direction. In the first case, if
the reverse process is possible, the block B should transfer heat to A and initial conditions should be restored. In the
second case, if the reverse process is possible, the heat energy must again change to kinetic energy and the fly wheel
should start rotating with the initial angular velocity w. In the third case, if the reverse process is possible the air in
B must flow back to A the initial condition should be obtained.
But, it is a matter of common experience, that none of the above conditions for the reverse processes are reached. It
means that the direction of the process cannot be determined by knowing the thermo dynamical coordinates in the
two end states. To determine the direction of the process a new thermo dynamical coordinates has been devised by
Claudius and this is called the entropy of the system. Similar to internal energy, entropy is also a function of the
state of a system. For any possible process, the entropy of an isolated system should increase or remain constant.
The process in which there is a possibility of decrease in entropy cannot take place.
To conclude processed in which the entropy of an isolated system decreases do not take place or for all processes
taking place in an isolated system the entropy of the system should increase if remain constant. It means a process
is irreversible if the entropy decreased when the direction of the process is reversed. A process is said to be
irreversible if it cannot be retraced back exactly in the opposite direction. During an irreversible process, heat energy
is always used to overcome friction. Energy is also dissipated in the form of conduction and radiation. This loss of
energy always takes place whether the engine works in one direction or the reversed direction. Such energy cannot
be regained. In actual practice all the engine are irreversible. If electric current is passed through a wire, heat is
produced. If the direction of the current is reversed, heat is again produced. This is also an example of an irreversible
process. All chemical reactions are irreversible. In general, all natural processed are irreversible.
Physical Significance of Entropy
Although it is very difficult to conceive the idea of entropy as there is no physical method to demonstrate it i.e., it
cannot be felt like temperature, pressure, volume and does not produce any effect which can be demonstrated; it has
a great significance in thermodynamics. Entropy is a real physical quantity defined by the equation
dS 
Q
T
B
or S B  S A 
dQ
T
A

And possesses the following important properties;
1. Just like temperature remains constant in isothermal process, the entropy remains constant I n adiabatic process.
2. It is a definite single valued function of the thermodynamic variables describing the state of a working substance.
3. In every natural process (I.e. irreversible change) there is always an increase in entropy.
4. The second law of thermodynamic can be stated in terms of entropy of a system.
5. Due to increase in entropy, unavailable energy increases.
6. According to Freeman Dyson, the entropy is a measure of disorderness. This disorderness can be evaluated by
using the relation.
S  k log W
Dr. M Chaitanya Varma
Dept. of Physics, GIT
Kelvin’s Thermodynamic Scale of Temperature
The efficiency of reversible Carnot’s engine depends only upon the two temperatures between which it works and is
independent of the properties (nature) of working substance. Using this property of Carnot’s reversible engine which
solely of absolutely depends on temperature and nothing else, Lord Kelvin in 1848 suggested a new scale of
temperature, known as ‘Absolute scale, called the Kelvin’s work or thermodynamic scale and showed that it
agrees with the ideal gas scale.
Theory;
Suppose a reversible engine absorbs heat Q1 at temperature
 2 and rejects heat Q2 at temperature  2
(Measured on
any arbitrary scale), then since the efficiency of the engine is a function of these two temperatures

Q1  Q2
 f (1 , 2 ) or
Q1
1
Q2
 f (1 ,  2 )
Q1
Q1
1

 F (1 ,  2 )
Q2 1  f (1 ,  2 )

Where F is some other function of
1
and
2 .
Similarly, if the reversible engine works between a pair of temperature
heat
2
and
 3 ( 2   3 )
absorbing a
Q2 and rejecting Q3 , we can write
Also if it works between
Q1
 F (1   3 )
Q3
1 and  3 (1   3 ), then
Multiplying equation (i) and (ii), we have
Q1 Q2 Q1


 F (1 , 2 )  F (1 , 2 )
Q2 Q3 Q3
Now comparing with equation (iii), we have
F (1 ,  2 )  F (1 ,  2 )  F ( 2 ,  3 )
This is called the functional equation. It does not contain
chosen that
2
2
on the left hand side, therefore, function F should be so
disappears from the right hand side also. This is possible if
F (1 , 2 ) 
 (1 )
 ( 2 )
and F ( 2 ,  3 ) 
 ( 2 )
 ( 3 )
 is another unknown function of temperature.
 (1 )  ( 2 )  (1 )
F (1 , 3 ) 


From equation (iv),
 ( 2 )  ( 3 )  ( 3 )
Where
Equation (i) can now be written as
Q1
 (1 )
 F (1 , 2 ) 
Q2
 ( 2 )
Dr. M Chaitanya Varma
Dept. of Physics, GIT
Since
1   2
and
Q1  Q2 , the function  (1 )   ( 2 ). Thus, function  ( ) is a linear function of  and can
be used to measure temperature.
Thus, Lord Kelvin suggested
 ( )
Q1 1

Q2  2
should be taken proportional to  i.e.
Or
 (1 )1 and  ( 2 ) 2 , we have
1 Q1

 2 Q2
This equation shows that the ratio of the two temperatures on this scale is equal to the ratio of the heat absorbed to
the heat rejected. This temperatures scale is called Kelvin’s thermodynamic (or absolute or work) scale of
temperature.
Third Law of thermodynamics; Nernst’s Heat Theorem
Statement;
According to Nernst, The heat capacities of all solids tend to zero as the absolute zero of temperature is approached
and that the internal energies and entropies of all substances become equal the, approaching their common value
asymptotically tending to zero.
In terms of entropy, the theorem may also be stated as;
It states that ‘’at absolute zero temperature, the entropy tends to zero and the molecules of a substance of a system
are in perfect order (well arranged)’’.
In all heat engines, there is always loss of heat in the form of conduction, radiation and friction.
Therefore, in actual heat engines
Q1
Q
is not equal to 2 .
T1
T2
Q Q 
 1  2  is not zero but it is a positive quantity. When cycle is repeated, the entropy of the system increases
 T1 T2 
and attains a maximum value, a stage of stagnancy is reached and no work can be done by the engine at this stage. In
the universe; the entropy is increasing and ultimately the universe will also reach a maximum value of entropy when
no work will be possible. With the increase in entropy, the disorder of the molecules of a substance increases. The
entropy is also a measure of the disorder of the system. With decrease in entropy, the disorder deceases. At absolute
zero temperature, the entropy tend entropy tend to zero and the molecules of a substance or a system are in perfect
order (well arranged). This is known as the third law of thermodynamics.
Dr. M Chaitanya Varma
Dept. of Physics, GIT