1. ENTROPY, CARNOT ENGINE AND THERMOELECTRIC EFFECT Dr. Gopika Sood, Lecturer in Physics Govt. College For Girls, Sector -11, Chandigarh

2. BASIC DEFINITIONS 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*10 7 ergs/cal 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 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.

3. 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.

4. 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/V 2 ) (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

5. 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 C BA A(p i,V i, T i ) VV p p Isobaric Process: It is the process in which the pressure (p) remains constant and is represented by AB in the above Fig. Isochoric Process: It is the process in which the volume (V) remains constant and is represented by BC in the above Fig. Isothermal Process : It is the process in which temperature (T) of the system remains constant during the process 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 Cyclic Process : It is the process in which system taken from an initial state A(p i,V i, T i ) 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

6. 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. 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

7. SPECIFIC HEAT OF A GAS The two types of specific heats are, specific heat at constant volume (C v ) and specific heat at constant pressure (C p ). Since Volume is constant,  V=0, Dividing both sides of eq.1 by  T, we get C v = (  Q/  T) v =  U/  T Similarly, C p = (  Q/  T) p =  U/  T + p(  V/  T) p = C v + p. (  V/  T) pV = RT, (  V/  T) p = R/p Therefore, C p = C v + R For an adiabatic process in an Ideal gas,  Q=0, so from the first law dU + pdV = 0, Also, dU = C v dT + pdV = 0 ……. (2) pV = RT, dT = (1/R) (pdV + Vdp) Put this value in eq 2. and multiply by R/pV (C p /C v ) dV/V + dp/p = 0 Integrating,  ln V + ln p = constant  = C p /C v TV  -1 = Constant T  p 1-  = Constant

8. 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

9. CARNOT’S REVERSIBLE HEAT ENGINE CONTINUED

10. WORKING OF CARNOT’S ENGINE STEP :1 REVERSIBLE ISOTHERMAL EXPANSION: During this phase the gas expands quasi- statically 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, V B W 1 = Q  pdV = RT ln (V B /V A ) …… (3) V A 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 V C W 2 = Q  pdV = R/(  -1) (T – T’) …… (4) V B

11. WORKING OF CARNOT’S ENGINE CONTINUED STEP :3 REVERSIBLE ISOTHERMAL COMPRESSION: 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 p D. 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, V D W 3 = Q  pdV = - RT’ ln (V C /V D ) ….. (5) V C STEP :4 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 W 4 = (R/(  -1)) (T’ – T) ….. (6) W = W 1 + W 2 + W 3 + W 4 = W 1 + W 3 The net amount of work done by Carnot engine per cycle is

12. EFFICIENCY OF CARNOT’S ENGINE Considering the magnitudes of eq. 3 and 5 we get, Q’ T’ln (V C /V D ) ---- = ------------------- …….. (7) Q T ln (V B /V A ) For the isothermal processes p A V A = p B V B, p C V C = p D V D ….. (8) For the adiabatic processes p B V B  = p C V C , p D V D  = p A V A  …. (9) Multiply eq.’s 8 and 9 and cancelling the common factor, V B V C ---- = ----- …….. (10) V A V D Using eq. 10 in eq. 7 Q’ T’ --- = --- Q T or Q’ - Q T’ -------- = 1 - --- Q T Q’ - Q W  = -------- = --- Q Q Thus, the efficiency of CARNOT’S engine is independent of the working substance and depends only on the heat source and the heat sink

13. 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. A weak current of few milliamperes was observed to flow in the circuit when the copper-bismuth junctions were maintained at different temperatures. -------------- Milliammeter Cu Bi B A Cold Hot Heat being absorbed here Heat being evolved here

14. 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 A V p B 2 3 1 B A 1  ds + 2  ds =0...(11) A B 2  ds = - 2  ds …..(12) B A Since each path is reversible Considering eq. 11 and 12 B B 1  ds = 2  ds = (S B –S A ) A A The change in entropy of the body, in going from one equilibrium state to another is independent of the path chosen B B (S B -S A ) =  ds =  (  Q )/T A A

15. SUMMARY  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

16. Thank You