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THERMODYNAMICS
System : A definite quantity of the matter bounded by some closed surface is known as
system.
Eg: A gas contained in a cylinder with a movable piston constitutes the system.
A system which can exchange both energy and matter with its surroundings is called
as an open system. Most of the engineering devices are open system.
A system which can exchange energy only and not matter with the surroundings is
called as closed system. We would always consider a system of constant mass.
When the system is not influenced by any way with its surroundings i.e. neither heat
nor mechanical work exchanged it is said to be isolated system.
Surroundings : All those things which are outside the system and influence its behavior are
known as surroundings of the system.
Boundary: The surface which separates the system from the surroundings is called the
boundary.
Heat: Heat is a form of energy which is transferred from one body to other body due to
temperature difference between them. Consider the case of two bodies (say A and B, A being
at high temperature) are placed in contact with each other. It is observed that after some time
the bodies acquire the same temperature. This temperature is somewhere between the two
initial temperatures. This means that something has been transferred from A to B. this
something is called heat.
Work : When a force applied on a system such that it produces a displacement in the system,
the work is said to be done. (W=F.S)
The work done by the system should be taken positive (+)
The work done on the system should be taken negative (-)
Heat absorbed by the system should be taken positive (+)
Heat rejected by the system should be taken negative (-)
When the work done by one part of the system on another part of the same system is
called the internal work.
In an actual gas, there exists intermolecular attraction between the molecules. When
the gas expands, the work is done against these mutual attractions between the molecules.
Hence, the work done by the gas during expansion against molecular forces of attraction is
internal work. External work is a macroscopic concept. This involves an attraction between a
system and surroundings. In thermodynamics, external work is very important while internal
work has no place.
Dr. P. Venkata Ramana, AUCE (A)
Equilibrium:
When the property of a system is defined, it is understood that the system is in
equilibrium. If a system is in thermal equilibrium, the temperature will be same throughout
the system. If a system is in mechanical equilibrium, there is no tendency for the pressure to
change. In a single phase system, if the concentration is uniform and there is no tendency for
mass transfer or diffusion, the system is said to be in chemical equilibrium. A system which
is simultaneously in thermal, mechanical, and chemical equilibrium is said to be in thermal
equilibrium.
A system is said to be in an equilibrium state if its properties will not change without
some perceivable effect in the surroundings. The equilibrium generally requires all properties
to be uniform throughout the system.
Zeroth law of thermodynamics :
Temperature is a property of a system which determines the degree of hotness.
Obviously, it is a relative term. Eg: A hot cup of coffee is at a higher temperature than a
block of ice. On the other hand, ice is hotter than liquid hydrogen.
If two systems (say A and B) are in thermal equilibrium with a third system (say C)
separately (that is A and C are in thermal equilibrium; B and C are in thermal equilibrium)
then they are in thermal equilibrium themselves (that is A and B will be in thermal
equilibrium.
Let us say TA, TB and TC are the temperatures of A, B and C respectively.
A and C are in thermal equilibrium. TA=TC
B and C are in thermal equilibrium. TB=TC
Consequence of zeroth law of thermodynamics
A and B will also be in thermal equilibrium TA=TB
All temperature measurements are based on this law.
Dr. P. Venkata Ramana, AUCE (A)
Adiabatic change:
When a thermodynamic system undergoes a change in such a way that no exchange of
heat takes place between it and the surroundings, the change is known as adiabatic change.
Under adiabatic change, there is a change of temperature. In this change, the heat developed
or lost cannot be given out or taken up from the surroundings.
Eg: when a gas is compressed suddenly or allowed to expand very rapidly, there is no time
available for the exchange of heat with the surroundings. Boyle’s law does not hold good for
this change.
Under this change, the pressure P and volume V of a gas are related by
PV
γ = constant, where γ is the ratio of two specific heats of a gas, ie. γ=Cp/CV
Work done in an Isothermal process:
When an ideal gas expands isothermally, work is done by the gas against the external
surroundings and an equivalent amount of heat flows from outside into the gas (temperature
remains constant during isothermal process). Similarly, when the gas is compressed
isothermally, work is done on the gas, in this case an equal amount of heat flows out from the
gas.
Let one gram mole of an ideal gas expand isothermally. Let P 1 and V 1 be the pressure
and volume respectively at initial state A and P 2 and V 2 be the pressure and volume
respectively at final state B. During isothermal expansion, both pressure and volume change.
Let P and V be the pressure and volume at any intermediate state C. Here we assume that
pressure P remains constant for a small increase in volume V. the work done by the gas
during this small expansion is PdV.
The net work done by the gas when it expands from V 1 to volume V 2 is given by
2
1
V
V
W PdV
For isothermal process, PV = RT or V
RT
P =
2
1
V
V
V
dV W RT
1
2 log V
V
W RT e
1
2
- 3026 log 10 V
V
W RT
Work done in an adiabatic process:
Let one gram mole of an ideal gas expand adiabatically from initial state A (P 1 , V 1 ) to
final state B (P 2 , V 2 ). During adiabatic expansion both pressure and volume change. Let P
and V be the pressure and volume at some intermediate state C. for a small change in volume
dV from C to D, let pressure remain constant. The work done by the gas during the small
expansion is PdV.
The net work done by the gas during expansion from volume V 1 to volume V 2 is
given by
2
1
V
V
W PdV
The second statement of this law is a particular form of the general law of
conservation of energy. Let us suppose a quantity dQ of heat is supplied to a system. This is
used in three ways
- A part is used in raising the temperature of the system which is equivalent of
increasing its internal kinetic energy of system ( dUk ).
- A part is used in doing internal work against molecular attraction which is equivalent
of increasing the potential energy of the system ( dUp ).
- The remaining part is used in doing external work dW.
dQ = dUk + dUp + dW
dQ = dU + dW where dU = dUk + dUp = increase in total internal energy of the system
This is the mathematical form of first law of thermodynamics. This can be stated that
in all transformations, the heat energy supplied must be balanced by external work done plus
the increase in internal energy.
Significance of the first law:
There are three basic ideas related to the mathematical form of the first law:
- Heat is a form of energy in transit.
- Energy is conserved in thermodynamic system
- Every thermodynamic system in equilibrium state possesses internal energy which is
a function of the state of the system.
Important points in applying first law of thermodynamics :
The following points should be kept in mind
- All the quantities must be expressed in the same unit
- The increase in internal energy should be taken as positive while the decrease in
internal energy should be taken as negative.
- The work done by the system should be taken positive while the work done on the
system should be taken negative.
- Heat absorbed by the system should be taken positive while heat rejected by the
system should be taken negative.
Limitations of first law of thermodynamics:
- No restriction of the flow of heat: The first law establishes definite relationship
between the heat absorbed and the work performed by a system. The first law does
not indicate whether heat can flow from a cold end to a hot end or not.
Eg: We cannot extract heat from the ice by cooling it to a low temperature. Some
external work has to be done.
- Does not specify the feasibility of the reaction: First law does not specify that process
is feasible or not. Eg: When a rod is heated at one end then equilibrium has to be
obtained which is possible only by some expenditure of energy.
- Practically it is not possible to convert heat energy into an equivalent amount of work.
Applications of first law of thermodynamics:
- Isothermal process : An isothermal process is one which the temperature of the
system does not change. i.e. dT=0. Hence, there is no change in the internal energy of
the system ( dU=0 ). So, by first law of thermodynamics, dQ = dW (for isothermal
process).
- Adiabatic process : an adiabatic process is on which takes place in such a manner that
no heat enters or leaves the system, i.e dQ=.
Hence, from first law of thermodynamics, 0 = dU + dW
dU =− dW
- Isochoric process : a process taking place at constant volume is known as isochoric
process. In such a process, the work done is zero. Hence, from first law of
thermodynamics, dQ = dU
- Isobaric process : a process taking place at constant pressure is known as isobaric
process. Eg : the boiling of water to steam or the freezing of water to ice. In isobaric
process, a gas either expands or contracts to maintain a constant pressure and hence a
net amount of work is done by the system or on the system. The amount of heat dQ is
partly used in increasing the temperature dT and partly used in doing external work.
dQ = CP dT + PdV
Reversible Process:
A reversible process is one which can be reversed in such a way that all changes
occurring in the direct process are exactly repeated in the opposite order and inverse sense
and no changes are left in any of the bodies taking part in the process or in the surroundings.
If heat is observed in the direct process, the same amount of heat should be given out in the
reverse process.
If work is done on the working substance in the direct process, then the same amount
of work should be done by the working substance in the reverse process.
The conditions for reversibility are
It consist the following parts:
- Working substance : the working substance is an ideal gas enclosed in a cylinder
piston arrangement (cylinder with perfectly non conducting walls and perfectly
conducting base fitted with frictionless non conducting piston).
- The source of heat : A hot body of high thermal capacity maintained at a high
temperature T 1 K serves as source. The high thermal capacity of the source is
necessary so that on taking heat from it no change of temperature occurs.
- The sink : A cold body maintained at a lower temperature T 2 K serves as a sink.
Usually this is surrounding atmosphere.
- Insulating stand : To make the whole system perfectly, a non conducting stands is put
on the base of the cylinder.
Carnot’s cycle:
The working substance is made to undergo a cycle operation made up of four parts:
(i) Isothermal expansion
(ii) Adiabatic expansion
(iii) Isothermal compression
(iv) Adiabatic compression
All these four operations are shown in figure.
1. Isothermal expansion AB: consider that one gram mole of a gas is contained in the
cylinder. Let initial state of the gas is denoted by A on indicator diagram. Where it has
temperature T 1 , pressure P 1 and volume V 1. The cylinder is placed on the source and the
piston is moved slowly upward so that the gas expands isothermally (temperature of working
substance remaining the same). In this operation, the pressure of the gas falls and the volume
increases. Here it should be remembered that fall of temperature is compensated by supply of
heat from the source. The final state of the gas is represented by point B. Thus, the isothermal
expansion of the gas is represented by the curve AB on indicator diagram.
Let the quantity of heat absorbed from the source in this process be Q 1. This is equal
to the amount of work done W 1 by the gas in the expansion from state (P 1 , V 1 ) to final state
(P 2 , V 2 ).
2
1
1 1
V
V
Q W PdV
2
1
1 1 1
V
V V
dV Q W RT (Since PV=RT)
1
2 1 1 1 log^ V
V
Q W RT e =Area ABGEA …… (1)
4
3
2 3
V
V
Q W PdV
4
3
2 3 2
V
V
V
dV Q W RT
3
4 2 3 2 log^ V
V
Q W RT e
4
3 2 3 2 log^ V
V
Q W RT e = Area CDFHC ….. (3)
4. Adiabatic compression DA: this is the final stage of the process where the cylinder is
detached from the sink and placed on the insulating stand. The gas is compressed such that it
undergoes a slow adiabatic compression till the state A is again reached. Now the
temperature rises from T 2 K to T 1 K (the temperature of the source). This change is shown by
the curve DA on the indicator diagram. The work done W 4 on the gas during this process will
be
1
4
4
V
V
W PdV
( )
1 2 4 −
RT T
W = area ADFEA ….. (4)
From eqn (2) and (4) W 2 =W 4 Hence net work = W 1 - W 3
Quantity of heat absorbed by the gas Q 1 - Q 2 =W 1 - W 3
Efficiency of the Engine:
In order to calculate the efficiency of heat engine, we should calculate the work done
and the amount of heat taken. Let Q 1 be the heat taken during isothermal expansion at
temperature T 1 K and Q 2 be the quantity of heat rejected at temperature T 2 K to the sink, then
the efficiency is defined as
heattakenfromsource
workdonebytheengine =
heattaken
heatconvertedintowork
Q 1
W
1
1 2
Q
Q − Q
1
2 1 Q
Q
The net work done by the gas
W = W 1 + W 2 + W 3 + W 4
( ) ( )
log 1
log
1 2
4
3 2
1 2
1
2 1 −
RT T
V
V
RT
RT T
V
V
W RT e e
4
3 2 1
2 1 log^ log V
V
RT
V
V
W RT e e ……. (6)
Points B and C lie on same adiabatic, hence
1 2 3
1 1 2
− −
TV T V
1
2
3
2
1
−
V
V
T
T
Again, the points A and D lie on the same adiabatic, hence
1 2 4
1 11
− −
TV T V
1
1
4
2
1
−
V
V
T
T
From eqns (7) and (8), we get
1
1
4
1
2
3
− −
V
V
V
V
1
4
2
3
V
V
V
V
1
2
4
3
V
V
V
V
Substituting this value in eqn (6)
1
2 2 1
2 1 log^ log V
V
RT
V
V
W RT e e
( )
1
2 1 2 log^ V
V
W RT T e
Q 1
W
Clausius statement of second law :
The action of refrigerator is just reverse of heat engine. The Clausius statement of
second law is based on the working of refrigerator. In refrigerator, the transfer of heat takes
place from a cold body to a hot body with the aid of an external agency. No refrigerator so far
constructed which can transfer heat from a cold body to a hot body without the aid of external
agency. This consideration led Clausius to state the second law as:
“ It is impossible for a self acting machine unaided by any external agency to
convert heat from a cold body to a hot body. ”
Equivalence between two laws:
Consider a refrigerator which transfer heat Qʹ from a cold body to a hot body without
the aid of any external agency. This is the violation of Clausius statement.
Consider an engine working between same source and sink. The engine extracts heat
Q from the hot body and rejects Qʹ to the cold body. Now suppose that heat engine and
refrigerator are coupled together to form a self acting machine. The net amount of heat taken
from hot body is Q-Qʹ. This is converted into work and no part is delivered to the sink. This
is violation of Kelvin’s statement.
The violation of Clausius statement is also the violation of Kelvin’s statement and
vice-versa; this shows that both the statements are equivalent.
Carnot’s Theorem:
Statement : According to this theorem no engine can be more efficient than reversible engine
working between the same two temperatures. This may also be stated as all reversible engines
working between the same two temperatures have the same efficiency, whatever may be the
working substance.
Proof : Let us consider a reversible engine A and irreversible engine B working with the same
source and sink. Here A works in the forward direction while B works in the backward
direction.
Let engine A absorb an amount Q from the source, convert a part of it W into work
and transfer the balance Q-W to the sink.
Similarly, let engine B absorb an amount of heat Qʹ from the sink, a certain amount of
work Wʹ is done on that working substance and transfers the net amount Qʹ+Wʹ to the source.
Here we assume that the source is left with no change, i.e., the amount of heat which
A receives from the source is returned by B to the source.
Qʹ+Wʹ=Q or Wʹ=Q-Qʹ
If possible, let the reversible engine A be more efficient than reversible engine B,
Efficiency of A > Efficiency of B
Q
W
Q
W
or W W ….. (1)
Now imagine that the two engines are coupled by a belt in such a way that engine A
works directly and it derives the engine B reversibly. The compound engine extracts an
amount of heat (Q-Wʹ) from the sink and gives an amount of heat Q to the source. Therefore,
the net amount of heat from the sink
=( Q − W ) −( Q − W )
=( W − W ) ….. (2)
This is positive value. The above fact shows that by coupling engine, the heat
observed from the sink (Q-Wʹ) is greater than heat given back to the sink (Q-W) while the
source is unaffected. In this way any amount of heat may be taken from the sink, work done
and no change may be left with source. This is impossible according to second law of
thermodynamics which state that it is impossible for any device to extract heat from a simple
reservoir and converting the whole into work. Hence our assumption is wrong. No engine
working between a source and sink can be more efficient than a reversible engine.
Coefficient of performance:
The coefficient of performance (K) is defined as the ratio of the heat taken in from the
cold body to the work needed to run the refrigerator.
W
Q
K
2
Dr. P. Venkata Ramana, AUCE (A)
Clausius showed that there is something which remains constant during adiabatic
process just as temperature remains constant in isothermal process. This constant property is
termed as entropy. Thus, the thermal property of a body which remains constant during an
adiabatic process is called as entropy. It is denoted by the symbol ‘S’.
Physical significance of Entropy:
The entropy of a substance is a real physical quantity and is a definite function of the
state of the body like pressure, temperature, internal energy etc.
temperature
heatenergy Change ofentropy =
Change of entropy temperature = heatenergy ….. (1)
We know that gravitational potential energy = mgh
Gravitational potential energy α mass x height ….. (2)
From equation (1) and (2) temperature is equivalent to height and entropy corresponds
to mass or inertia. Hence, entropy may be defined as a quantity which bears to heat motion a
similar relation as mass bears to linear motion and moment of inertia in rotational motion.
Entropy is also termed as thermal inertia since more entropy results in less amount of heat
energy being converted into work.
Entropy is also a measure of randomness or disorder of molecules of the system. The
increase in entropy implies a transition from a more ordered arrangement of molecules to less
ordered arrangement i.e., from order to disorder.
Entropy measurement:
For a Carnot heat engine working at T 1 and T 2 , it has been observed that the heat
absorbed (Q 2 ) and heat rejected (Q 1 ) are related as
2
2 1
2
2 1
T
T T
Q
Q Q −
2
1
2
1 1 1 T
T
Q
Q
2
1
2
1
T
T
Q
Q
2
2
1
1
T
Q
T
Q
T
Q
=constant
Therefore, we can say that for any particular system, the ratio of heat absorbed or lost
isothermally and reversibly to the absolute temperature at which this takes place a constant
parameter. If we consider the heat absorbed at T 1 , then the above equation
1
1
2
2
T
Q
T
Q
1
1
2
2
Q
T
Q
=^0
T
Q
Consider the Carnot cycle comprising two processes, an irreversible process 1 - 2 and a
reversible process 2 - 1. If the a reversible cycle process, consists of many Carnot cycles, for
each cycle, we have
=^0
T
Q
Where δQ is the extremely small amount of heat absorbed at temperature T during the
course of an isothermal and reversible process. The total entropy change of the cyclic process
1 → 2 →1 can be fragmented into two components
1 2 2 1
=^ +^ =
→ → T
Q
T
Q
T
Q
→ →
1 2 2 1 T
Q
T
Q
1 → 2 2 → 1
T
Q
T
Q
From the above equation the quantity Q/T is a state function. This quantity is called as
entropy. The entropy is a extensive property measured in Joule per Kelvin per mol (JKmol
If S 1 and S 2 be the entropies at points 1 and 2 respectively, then
