### B.Sc. Sem II Thermodynamics-II

• 1. 1 | U I I , S e m e s t e r - I I ( P A N a g p u r e ) Thermodynamics - I Thermodynamic system: A definite quantity of matter (solid, liquid or gases) bounded by some closed surface is called thermodynamic system. Example: Gas contained in a cylinder with movable piston. Thermodynamic Variables: The variables which determine the thermodynamic behavior of a system are called thermodynamic variables. The quantities like pressure (P), volume (V) and temperature (T) are thermodynamic variables. There are some other thermodynamic variables, such as internal energy (U), entropy (S), etc. all other thermodynamic variables can be expressed in terms of P, V and T. Thermodynamic process: A thermodynamic process is said to be taking place, if the thermodynamic variables of the system change with time. In practice following types of thermodynamic processes can take place Isothermal process: A thermodynamic process that takes place at constant temperature is called isothermal process. Isobaric process: A thermodynamic process that takes place at constant pressure is called isobaric process. Isochoric process: A thermodynamic process that takes place at constant volume is called isochoric process. Adiabatic process: A thermodynamic process in which no heat enters or leaves the system is called adiabatic process. Indicator diagram: A graphical representation of the state of a system with the help of two thermodynamic variables (PV or TV or PT) is called indicator diagram of the system. If thermodynamic variables P and V are used for graphical representation of the state of a system as shown in figure, then it is called PV-indicator diagram of the system.
• 2. 2 | U I I , S e m e s t e r - I I ( P A N a g p u r e ) Importance of PV indicator diagram: A system can be taken from state A to state B by many ways, as shown in above figure. Let the coordinates of state A and B are (P1,V1) and (P2,V2) respectively, where P1 and V1represents pressure and volume of state A and P2 and V2 represents pressure and volume of state B. 1. Starting from point A, pressure continuously decreases from P1 to P2 along the curve AB so that volume increases from V1 to V2. The work done by the system is given by 2 1 1 area(ABFEA) V V W dW PdV    2. Starting from point A the volume V1 is kept constant in going from A to D , the pressure decreases from P1 to P2 and then P2 is kept constant from D to B. The work done in this process i.e. along ADB path is given by 2 2 2 1W ( ) area(DBFED)P V V   3. Again starting from point A, the pressure P1 is kept constant in going from A to C and then volume V2 is kept constant from C to B. The work done in this process i.e. along path ACB is given by 3 1 2 1W ( ) area(ACFEA)P V V   Thus area under the PV curve in the PV-indicator diagram gives the work done by the system in the given process, going from initial state to final state. Also 1 2 3W W W  . Hence work done by the system depends upon the path and independent of initial and final states of the system. Heat flow: Whenever two systems at different temperatures are brought into thermal contact, heat must flow from the system at higher temperature to the system at lower temperature till the whole system attains common temperature which lies between temperatures of the two systems. Thermal equilibrium: When all the parts of the thermodynamic system are at the same temperature, (which is the same as that of the surroundings) the system is said to be in thermal equilibrium. Zeroth law of thermodynamics: The Zeroth law of thermodynamics deals with the thermal equilibrium of more than two systems. When a body A is brought in thermal contact with a cold body B, heat flow from A to B and after some time the flow stops. Then the bodies are then said to be in thermal equilibrium with each other.
• 3. 3 | U I I , S e m e s t e r - I I ( P A N a g p u r e ) The Zeroth law of thermodynamics states that: “If two systems (A & B) are separately in thermal equilibrium with a third system (C), then they are also in thermal equilibrium with one another”. If [A]  [C] & [B]  [C], then [A]  [B] Internal energy: The energy content of the system is called its internal energy. It is the sum of the following forms of energies of the system: 1. Kinetic energy due to translational, rotational and vibrational motion of the molecules, all of which depends on temperature. 2. Potential energy due to intermolecular forces, which depends on the separation between the molecules. 3. The energy of the 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. The internal energy is a state function i.e. it is depends upon the state of the system. It is independent on the path by which it proceeds from initial state to final state. If the state of the system is changed from initial state 1 to final state 2. Then change in internal energy is dU =U2 – U1 , where dU is perfect differential. First law of thermodynamics: When certain amount of heat Q is supplied to a system which does external work W in passing from state 1 to state 2, the amount of heat is equal to sum of external work done W by the system and increase in the internal energy (U2 – U1) of the system. Q = W + (U2 – U1) For very small change in the state of the system, Q W dU   where and are not perfect differential butQ W dU  is a perfect differential because U is a function of the state of the system. The amount of heat Q or Q is taken positive if heat is supplied to the system, and negative if heat is removed from it. Similarly the work W or W is positive when the external work is done by the system in expansion and negative if the work is done on it in compression. Significance of first law of thermodynamics: 1. The law establishes the relation between heat and work. 2. It is applicable to any process by which a system undergoes a physical or a chemical change. 3. It introduces the concept of the internal energy and provides the method of determination of internal energy.
• 4. 4 | U I I , S e m e s t e r - I I ( P A N a g p u r e ) Applications of first law of thermodynamics: Isothermal process: A thermodynamic process that takes place at constant temperature is called isothermal process. According to first law of thermodynamics Q W dU   As in isothermal process temperature remains constant, there is no change in internal energy i.e. dU = 0. Q W   Thus in an isothermal process the heat supplied to an ideal gas is equal to the work done by the gas. Isobaric process: A thermodynamic process that takes place at constant pressure is called isobaric process. For isobaric process work done is given by 2 1 2 1( ) V V W PdV P V V   According to first law of thermodynamics Q = W + (U2 – U1)  2 2 1 1 2 1 = + ( )Q U PV U PV Q H H       where H = U+PV is called enthalpy. Thus heat absorbed at constant pressure is equal to increase in enthalpy of the system. Isochoric process: A thermodynamic process that takes place at constant volume is called isochoric process. Thus for isobaric process dV = 0 0dW P dV   Therefore according to first law of thermodynamics 2 1( )Q dU U U    Thus heat absorbed at constant volume is equal to increase in internal energy of the system.
• 5. 5 | U I I , S e m e s t e r - I I ( P A N a g p u r e ) Adiabatic process: A thermodynamic process in which no heat enters or leaves the system is called adiabatic process. Thus for isobaric process 0Q  Therefore according to first law of thermodynamics 2 1( )W U U   Thus in adiabatic process increase (or decrease) in internal energy is equal to external work done on (or by) the gas. Cyclic process: If a system undergoes series of changes and finally returns to an exact original state, the process is called cyclic. In this process 2 1( ) 0U U  Q W  Thus in cyclic process external work done by the system is equal to amount of heat absorbed by the system Reversible and Irreversible processes: Reversible process: If a thermodynamic system passes from initial equilibrium state to final equilibrium state through large number of changes and when brought back to its initial state, it retraces the original path in opposite direction then it is called reversible process. Condition for reversible Process: 1. There should not be dissipative effects such as friction, viscosity and electric resistance in the process. 2. All the processes taking place in the cycle must be infinitesimally slow. 3. There should not be any loss of energy due to conduction, convection or radiation. Examples: 1. All isothermal and adiabatic changes are reversible when performed slowly. 2. Ice melts when certain amount of heat is absorbed by it. The water so formed can be converted in to ice if the same quantity of heat will be given out. 3. If the resistance of the thermocouple is neglected, in such case Peltier heating or cooling is reversible. Irreversible process: If a thermodynamic system passes from initial equilibrium state to final equilibrium state through large number of changes and when brought back to its initial state, it does not retraces the original path in opposite direction, then it is called irreversible process. Examples: 1. All the natural spontaneous processes are irreversible. 2. Joule Thomson effect. 3. Transfer of electricity through resistance. 4. Spontaneous expansion of gas in vacuum.
• 6. 6 | U I I , S e m e s t e r - I I ( P A N a g p u r e ) Carnot’s ideal heat engine: In 1824, the French engineer Carnot hypothized heat engine which is free from all practical imperfections. Such an engine cannot be realized in practice. It has maximum efficiency and it is an ideal heat engine. It consists of following 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. It can enclose one mole of an ideal gas as the working substance. 2. Source: A reservoir maintained at constant temperature T1 K having perfectly non- conducting walls, a perfectly conducting top. It has infinite heat capacity so that any amount heat can be extracted from it, keeping its temperature constant at T1 K. 3. Sink: A reservoir maintained at constant temperature T2 K having perfectly non- conducting walls, a perfectly conducting top. It has infinite heat capacity so that any amount heat can be given to it, keeping its temperature constant at T2 K. (where T1 > T2) 4. Stand: A perfectly non-conducting platform acts as a stand for adiabatic processes. Carnot cycle: Carnot’s heat engine works in reversible cycle called as Carnot cycle. The Carnot cycle is represented in the PV diagram in the figure below. It consists of four steps: isothermal expansion, adiabatic expansion, isothermal compression and adiabatic compression.
• 7. 7 | U I I , S e m e s t e r - I I ( P A N a g p u r e ) Work done in Carnot cycle: Let us consider one mole of ideal gas enclosed in the cylinder. Let initial equilibrium state of the gas is represented by the point A (P1V1T1). 1) Isothermal expansion: The cylinder is placed on source at temperature T1 K and gas is allowed to expand isothermally to state represented by the point B (P2V2T1). Let Q1 be the heat absorbed by the gas from source. The external work done by the gas from A to B is given by W1 = Area (ABGEA) = Q1 2) Adiabatic expansion: The cylinder is now placed on non-conducting stand and gas is allowed to expand adiabatically to state represented by the point C (P3V3T2). The external work done by the gas from B to C is given by W2 = Area (BCHGB) 3) Isothermal compression: The cylinder is now placed on sink at temperature T2 K and gas is compressed isothermally to state represented by the point D (P4V4T2). Let Q2 be the heat rejected by the gas to sink. The external work done on the gas from C to D is given by W3 = Area (DCHFD) = Q2 4) Adiabatic compression: The cylinder is now placed on non-conducting stand and gas is compressed adiabatically to its original state represented by the point A (P1V1T1). The external work done on the gas from D to A is given by W4 = Area (DFEAD) Hence the total work done by the gas in one complete cycle is W = Area (ABCDA) But net amount heat absorbed by the gas is (Q1 – Q2). Therefore, W = (Q1 – Q2) Efficiency (η): The efficiency of heat engine is defined as the ratio of net work done (W) by the heat engine in one cycle to the heat (Q1) absorbed by the engine from source. 1 2 1 1 2 1 1 Q QW Q Q Q Q         Application of Carnot Cycle for Ideal (Perfect) Gas: Consider one mole of ideal gas as a working substance in Carnot engine. Let T1 and T2 be the temperatures of the source and sink respectively (T1>T2). 1. Isothermal expansion: In isothermal expansion at T1 K from A to B, there is no change in internal energy. Hence according to first law of thermodynamics, heat absorbed by the gas is equal to external work done by the gas 2 1 1 1 V V Q W dW  
• 8. 8 | U I I , S e m e s t e r - I I ( P A N a g p u r e ) 2 1 1 V V Q PdV  For one mole of ideal gas, PV= RT 2 2 1 1 1 1 1 V V V V RT dV Q dV RT V V     2 1 1 2 1 1 1 ln( ) ln V Q RT V V RT V           ……………. (1) 2. Adiabatic expansion: In adiabatic expansion (dQ =0) from B to C, the work done by the gas is given by 2 2 1( )VW C T T   2 1 2( )VW C T T   …………………. (2) where: CV is molar specific heat at constant volume. 3. Isothermal compression: In isothermal compression at T2 K, there is no change internal energy. Hence according to first law of thermodynamics, heat rejected by the gas is equal to external work done on the gas 4 3 2 3 V V Q W dW   4 3 2 V V Q PdV  For one mole of ideal gas, PV= RT 4 3 2 2 V V RT Q dV V    4 2 2 4 3 2 3 ln( ) ln V Q RT V V RT V           3 2 2 4 ln V Q RT V          (-ve sign indicates that work is done on the gas) For adiabatic process, we have 1 constantTV   1 1 1 31 1 2 2 3 2 2 1 1 1 1 4 2 4 1 1 2 1 VT TV T V T V T V T V TV T V                               3 34 2 2 1 1 4 V VV V V V V V     2 2 2 1 ln V Q RT V          …………………. (3)
• 9. 9 | U I I , S e m e s t e r - I I ( P A N a g p u r e ) 4. Adiabatic compression: In adiabatic compression (dQ =0) from D to A, the work done on the gas is given by 4 1 2( )VW C T T   …………………. (4) where: CV is molar specific heat at constant volume. Hence, the net amount of heat absorbed by the gas per cycle = Q1 – Q2 The net work done by the gas is given by 1 2 3 4W W W W W    1 3W W W  = Q1 – Q2 2 2 1 2 1 2 1 1 – ln ln V V Q Q RT RT V V               2 1 2 1 2 1 – ( )ln V Q Q R T T V          …………………. (5) From equation (1) and (5) 2 1 2 11 2 1 2 1 1 1 2 1 2 1 1 ( )ln – ln – V R T T VQ Q Q V RT V Q Q T T Q T                      2 2 1 1 1 1 Q T Q T      Characteristics of Carnot’s reversible heat engine: 1. The efficiency depends upon the temperature of the heat source (T1) and the heat sink (T2). 2. The efficiency is independent on the nature of working substance. 3. The reversible engines working between same to temperature have the same efficiency. 4. As T1> T1 - T2 , the efficiency of the Carnot’s engine is always less than one. Limitation of first law of thermodynamics: The first law of thermodynamics states the equivalence of heat and work. It simply tells whatever work is obtained; an equivalent amount of heat is used up, or vice versa. The law does not state anything about the extent of heat that can be converted into work and conditions under which condition this reverse process can take place.
• 10. 10 | U I I , S e m e s t e r - I I ( P A N a g p u r e ) Second law of thermodynamics: Limitations of first law of thermodynamics lead to the formulation of second law of thermodynamics. It can be stated in a number of ways, which means the same thing. Clausius’s statement: It is impossible for any self acting machine to transfer heat from a body at lower temperature to a body at higher temperature without using external agency. In other words “heat cannot flow itself from a colder body to a hotter body”. Kelvin’s statement: It is impossible to get continuous supply of work from a body, by cooling it to a temperature below its surrounding. Planck’s statement: It is impossible to construct an engine which, operating in complete cycle will produce no effect other than raising the weight and cooling of a heat reservoir. Kelvin-Planck’s statement: It is impossible to construct an engine operating in complete cycle will produce no effect other than absorbing heat from a source and conversion into an equivalent amount of work without rejecting some heat to the sink. Cornot’s Theorem: From the second law of thermodynamics two important conclusions are derived; these conclusions together constitute Cornot’s theorem which may be stated in the following forms. 1. No heat engine can be more efficient than perfectly reversible engine working between same two temperatures. 2. All the reversible engines working between same two temperatures have same efficiency, whatever the working substance. Proof of First part To prove the first part of this theorem let us consider two engines I and R working between temperatures T1 and T2, where T1>T2. Here I is irreversible and R is reversible engine. Let in each cycle engine R absorbs Q1 heat from source at temperature T1 and rejects heat Q2 to sink at temperature T2. Let in each cycle engine I absorbs Q1’ heat from source at temperature T1 and rejects heat Q2’ to sink at temperature T2. Suppose I is more efficient than R (i.e. ηI > ηR). 1 2 1 2 1 1 ' ' ' Q Q Q Q Q Q    Let both engines does same work (W) in one cycle. 1 2 1 2' 'Q Q Q Q    1 1 1 1 'Q Q   1 1 'Q Q  and 2 2 'Q Q Now consider that two engines are coupled together so that work done by engine I drives engine R backwards.
• 11. 11 | U I I , S e m e s t e r - I I ( P A N a g p u r e ) Thus, engine I absorbs Q1’ heat from source and rejects heat Q2’ to sink; engine R in its reverse cycle absorbs Q2 heat from sink and rejects heat Q1 to source. Therefore net gain of heat by source at temperature T1 = 1 1 'Q Q and net loss of heat by sink at temperature T2 = 2 2 'Q Q Thus self acting machine formed by two coupled engines transfer heat from a body (sink) at lower temperature to a body (source) at higher temperature. This contradicts with the second law of thermodynamics. Hence our assumption: engine I is more efficient than engine R is wrong and we conclude that no heat engine can be more efficient than perfectly reversible engine working between same two temperatures. Proof of second part Here we consider two reversible engines R1 and R2, and assume that R2 is more efficient than R1. Then by proceeding in the same way as above, we can show that R2 cannot be more efficient than R1. Therefore all reversible engines working between same two temperatures have same efficiency. Thus efficiency of a perfectly reversible engine depends only on the the temperatures between which engine work, and it is independent of the nature of working substance. Entropy: Consider reversible Carnot cycles bounded by two adiabatic M, N and isothermals at temperature T1, T2 and T3 as shown in figure. In the reversible Carnot cycle ABCDA, during the isothermal process at T1 along AB heat absorbed is Q1 and during the isothermal process at T2 along CD heat rejected is Q2. 2 2 1 1 1 1 Q T Q T      2 2 1 2 1 1 1 2 or Q T Q Q Q T T T    In the reversible Carnot cycle DCEFD, during the isothermal process at T2 along DC heat absorbed is Q2 and during the isothermal process at T3 along EF heat rejected is Q3.
• 12. 12 | U I I , S e m e s t e r - I I ( P A N a g p u r e ) 3 3 2 2 1 1 Q T Q T      3 3 32 2 2 2 3 or Q T QQ Q T T T    Thus in general we have 31 2 1 2 3 Constant QQ Q T T T     i.e. when we go from one adiabatic to another by a reversible isothermal process, the ratio Q T is constant. If two adiabatic are very close to each other, then heat rejected or absorbed is small ( Q ) then Constant Q T   This constant is called change in entropy dS. Q dS T    For finite reversible change in state of the substance from A to B, the change in the entropy of the substance is B A S B A B S A Q dS S S T      The measure of thermal energy per unit temperature of the system which is not available for doing useful work is called entropy of the system. It is also defined as a measure degree of randomness or disorder in the system. It is extensive property depends upon the mass of the system. Its SI unit is joule/kelvin.