Transcript Entropy, Carnot Engine and Thermoelectric Effect

ENTROPY, CARNOT ENGINE AND
THERMOELECTRIC EFFECT
Dr. Gopika Sood,
Lecturer in Physics
Govt. College For Girls,
Sector -11, Chandigarh
BASIC DEFINITIONS
Entropy is a Statistical quantity as it measures the number of accessible
microstates. Entropy is also a Thermodynamical quantity.
Thus, ENTROPY IS AN IMPORTANT LINK BETWEEN THE STATISTICAL
PHYSICS AND THERMODYNAMICS
Heat : Heat is the energy of random motion, which flows between two systems due to
their temperature difference. It is denoted by the symbol Q.
Work : It is the transfer of mechanical energy to and from the system. It is denoted by W
Mechanical Equivalent of Heat : The amount of dissipated mechanical energy is directly
proportional to the amount of heat produced.
WQ
W = JQ
J= Mechanical Equivalent of heat = 4.18*107 ergs/cal
Thermodynamical Variables : Quantities which determine the state of system are known
as thermodynamical variables. These include, temperature (T), pressure (P), volume (V),
internal energy (U) and number of moles.
BASIC DEFINITIONS
CONTINUED
Thermally Isolated system : It is a system enclosed by perfectly insulating walls so that no heat flows
into or out of the system
Mechanically Isolated System : It is the system which is enclosed by perfectly rigid walls so that its
volume (V) remains unchanged.
Isolated System : It is the system that is both thermally and mechanically isolated from the surroundings.
Thermal Equilibrium : Two systems placed in contact with each other are said to be in thermal
equilibrium if no net transfer of heat takes place between them.
Mechanical Equilibrium : Two mechanically connected systems are said to be mechanical equilibrium if
they exert equal and opposite mechanical forces on each other.
Entropy : Like pressure, volume, temperature and internal energy, we have another thermodynamic
variable of a system, named Entropy.
Entropy is related to the disorder in the system. If all the molecules in a given sample of a gas are made
to move in the same direction with the same velocity, the entropy will be smaller than that in the actual
situation in which the molecules move randomly in all directions.
An interesting fact about entropy is that, it is not a conserved quantity.
 More interesting is the fact that entropy can be created but cannot be destroyed.
We define the change in the entropy of the system as S = Q/T.
EQUATION OF STATE
Internal Energy : The sum total of kinetic and potential energies of all particles in a system
is termed as the internal energy of the system.
Equation of state : The condition in which a particular material exists is described by
quantities such as pressure, volume, temperature and amount of substance. The volume V of
a substance is usually determined by its pressure P, temperature T, and amount of substance,
described by the mass m or number of moles n.
A relation between the values of any three thermodynamical variables for the system is called
its equation of state. For an ideal state,
pV= RT
For a gas obeying Vander Waal equation , the equation of state is
(P + a/V2) (V-b) = RT
Any of the thermodynamical variables (P, V, T, U) can be expressed in terms of other two.
Therefore, any two of these can be taken as independent variables and third being a
dependent variable
THERMODYNAMICAL PROCESS
Indicator Diagram : The plot between p and V (taking them across X and Y axis ) for a
gaseous system in equilibrium is called indicator diagram
Thermodynamical Process : It is the line joining a series of points in the indicator diagram
A
B
A(pi,Vi,Ti)
p
p
Adiabatic process: It is the process in
which no heat enters or leaves the
system, i.e., a process occuring in the
thermally isolated system
C
V
Isobaric Process: It is the process
in which the pressure (p) remains
constant and is represented by AB
in the above Fig.
Isothermal Process : It is the process in
which temperature (T) of the system
remains constant during the process
V
Cyclic Process : It is the process in which system taken from an
initial state A(pi,Vi,Ti) to a succession of states but always brought
back to the initial state. Such a state is represented by a closed path in
p-V diagram
Isochoric Process: It is the process in which the volume (V)
remains constant and is represented by BC in the above Fig.
LAWS OF THERMODYNAMICS
Zeroth Law of Thermodynamics: It states that there exists a useful quantity called
temperature which is a property of all thermodynamic systems such that if two systems
have the same temperature they must be in thermal equilibrium (i.e., when the two
systems are brought in thermal contact, no heat flows from one system to another).
First Law of Thermodynamics : It is simply the law of conservation of energy. It states
that the the change in the internal energy of the system is given by the amount of heat
energy received by it subtracted by the amount of the mechanical work done by it.
Mathematically,
U = Q - W = Q - pV
……..... (1)
Second Law of Thermodynamics: All natural processes tend to proceed in the direction of
increasing entropy, i.e., in the direction of increasing disorder.
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Some consequences of second law of thermodynamics :
Heat cannot, by itself, pass from colder to hotter body
No process is possible whose only result is the absorption of heat from a system at a single uniform
temperature and the conversions of this heat completely into mechanical work.
It is impossible to construct a heat engine which, operating in a complete cycle, will take heat from
a single body and convert the whole of it to mechanical work, without leaving changes in the
working system
SPECIFIC HEAT OF A GAS
The two types of specific heats are , specific heat at constant volume (Cv) and specific heat at constant
pressure (Cp) . Since Volume is constant, V=0, Dividing both sides of eq.1 by T, we get
Similarly,
Cv = (Q/T)v = U/T
Cp = (Q/T)p = U/T + p(V/T)p = Cv + p . (V/T)
pV = RT, (V/T)p = R/p Therefore,
Cp = Cv + R
For an adiabatic process in an Ideal gas, Q=0 , so from the first law
dU + pdV = 0, Also, dU = CvdT + pdV = 0 ……. (2)
pV = RT,
Integrating,
dT = (1/R) (pdV + Vdp)
Put this value in eq 2. and multiply by R/pV
(Cp/Cv) dV/V + dp/p = 0
 ln V + ln p = constant
TV-1 = Constant
 = Cp/Cv
T p1- = Constant
CARNOT’S REVERSIBLE
HEAT ENGINE
Sadi Carnot in 1824 introduced the concept of an
ideal heat engine.
 It is device which converts heat into mechanical
energy
It is an imaginary heat engine in which there is no
loss of energy due to friction etc.
All the processes taking place are assumed to be
completely reversible, that is why it is called
reversible heat engine.
Components and assumptions for Heat Engine
Heat Source is the object at high temperature
Heat Sink is the coldest available object
A working substance
The piston as well as the walls of the cylinder are
made of perfectly insulating material.
The bottom of the cylinder is made of perfect
conductor
The piston moves in the cylinder without any friction
CARNOT’S REVERSIBLE
HEAT ENGINE CONTINUED
WORKING OF CARNOT’S
ENGINE
STEP :1
REVERSIBLE ISOTHERMAL EXPANSION:
During this phase the gas expands quasistatically so that the piston moves from point A to B. Throughout the expansion the temperature (T)
remains constant. The volume of the gas inside the cylinder is proportional to the distance between the
piston and the bottom of the cylinder. The gas does some external work (W) at the expense of heat
extracted (Q) from the heat source. Since T is constant , Q = W and the work done is ,
VB
W1 = Q  pdV = RT ln (VB/VA)
VA
…… (3)
STEP :2
REVERSIBLE ADIABATIC EXPANSION: The piston continues to move towards on
account of its inertia. The gas continues to expand but now the expansion is adiabatic since the piston
and the walls of the are made of an insulator so no heat can enter and Q = 0, W = - U
VC
W2 = Q  pdV = R/( -1) (T – T’) …… (4)
VB
WORKING OF CARNOT’S
ENGINE CONTINUED
STEP :3
In this step, the gas, at temperature T’
is put in thermal contact with heat sink at the same temperature and is compressed isothermally until the
pressure becomes pD. In this process the work is done on the gas by external mechanical agent. Since T is
constant , U = 0 , Q = W and the work done is ,
REVERSIBLE ISOTHERMAL COMPRESSION:
STEP :4
VD
W3= Q  pdV = - RT’ ln (VC/VD) ….. (5)
VC
REVERSIBLE ADIABATIC COMPRESSION: Here the heat sink is replaced by a block of
perfect insulator so that the compression is carried out adiabatically. Work is done on the gas and its
temperature rises to T while the pressure and volume becomes p A and internal energy increases
W4 = (R/( -1)) (T’ – T)
The net amount of work done by Carnot engine per cycle is
W = W1 + W2 + W3 + W4 = W1 + W3
….. (6)
EFFICIENCY OF CARNOT’S
ENGINE
Considering the magnitudes of eq. 3 and 5 we get ,
For the isothermal processes
Q’
T’ln (VC/VD)
---- = ------------------Q
T ln (VB/VA)
pAVA = pBVB, pCVC = pDVD
….. (8)
For the adiabatic processes
pBVB = pCVC, pDVD = pAVA …. (9)
Multiply eq.’s 8 and 9 and cancelling the common factor,
Using eq. 10 in eq. 7
Q’
T’
--- = --Q
T
or
=
Q’ - Q
-------Q
Q’ - Q
-------Q
T’
= 1 - --T
W
= --Q
VB VC
---- = ----VA VD
…….. (7)
…….. (10)
Thus, the efficiency of
CARNOT’S engine is
independent of the working
substance and depends only
on the heat source and the
heat sink
THERMOELECTRIC EFFECT
Thermoelectric Effect : In Carnot cycle the heat is converted into electrical energy.
Also, the
passage of electric current through resistance causes heat. This is an irreversible phenomenon. If we
reverse the direction of flow of current, heat is still produced in the resistance but it cannot be used to
convert heat into electrical energy. However, there is an another process by which heat can be
converted into electrical energy and this phenomenon is called THERMOELECTRIC EFFECT.
Milliammeter
Cu
Cu
Bi
Heat being
evolved here
----------------------------------------------------Hot
A
----------------------------------------------------- Cold
B
Heat being
absorbed here
A weak current of few milliamperes was observed to flow in the circuit when the
copper-bismuth junctions were maintained at different temperatures.
ENTROPY CHANGE IN A
REVERSIBLE PATH
It is possible to move from the one equilibrium state (A) to another state (B) by infinite number
of ways, i.e., it is possible to go from A to B through a reversible process (by path 1 and 2). The
net change in entropy is
B
A
1  ds + 2  ds =0 ...(11)
A
B
Since each path is reversible
A
B
2  ds = - 2  ds
B
A
A
p
1
3
= (SB –SA)
2
B
…..(12)
Considering eq. 11 and 12
B
B
1  ds = 2  ds
A
A
B
B
(SB-SA) =  ds =  (Q )/T
A
A
V
The change in entropy of
the body, in going from one
equilibrium
state
to
another is independent of
the path chosen
SUMMARY
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Entropy links the two branches of “Science”, Statistics and Thermodynamics
First Law of Thermodynamics relates the change in internal energy to the
amount of heat received to the mechanical work done in a system
Second Law of Thermodynamics says that all natural processes move in the
direction of increasing entropy
Work done in a cycle of a cyclic process is numerically equal to the area
enclosed by the closed curve representing the p-V diagram of the process
The efficiency of “The Carnot Engine” is  = (Q’ –Q)/Q = W/Q
Thank You