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KALYANI STANDARD PHYSICR, , 170, , 7.8 ETROPY, the, , of, , topics heattransferene, Study of Carnot's reversible cycle has thrown much light, At the same time it has projected the necessity of a new parameter of thermodynamics ference., ics in, order, to meet the, on, , following requirements, , (0) It is important to study the thermal conditions of the working substance of the an, as the engine operates, since the engine delivers work only after its working substanene, , has, passed through these states., () Rank of an isothermal is determined by the temperature at which the system operates, It is due to the reason that the temperäture remains constant during isothermal change. H, , do we determine the rank of an adiabatic ? Naturally it is to be determined by something whieh, , hich, , remains constant during an adiabatic change., (ii) In case of an engine we can easily get the amount of energy transferred from, one, system to another. We feel the necessity of a parameter which should handle the direction, , bf, , transfer of energy., Motivated by these necessities, Classius, in 1854, introduced a new parameter called entrom, , ropy, , which provided satisfactory answers to the problems quoted above., , We have seen that in going from one adiabatic to another, a certain quantity of heat is, , either absorbed or rejected and the heat so absorbed or rejected depends upon the temperatur, at which it takes place. If 8Q is the amount of heat absorbed or rejected at a, temperature T., change in entropy 'SS' of the substance is given by, T, If the heat is, , absorbed, 8Q, positive. If the heat is rejected, 8Q and hence, &S is negative., While we can have an idea of change in entropy by, performing measurement of SQ, it is, not possible to realise the Physics of, of, We, can have a better, concept, entropy., insight about, entropy by considering its analogy with some other clearly understood concepts. From equation, (55) it may be noted that heat energy equals the product of, change in entropy and temperature., In mechanics we had seen that, potential energy also equals the, of weight and height, since both temperature and height are measured w.r.t. a certainproduct, zero level, temperature can, be considered to be a quantity, analogous to height. Thus entropy can be treated as analogous, to mass (being proportional to, weight). But mass is a quantity which measures inertia in linear, motion. So we may think of entropy as thermal inertia which, bears to heat motion a relaton, similar to that which mass and moment of inertia bear to linear and, rotational motion respectvely., In mathematics, entropy is a function defined by an equation while in statistics, it 1s, something which is proportional to the thermodynamic probability of the most probable state, and hence 8S is, , of the system. According to Boltzmann, the entropy of a substance, of disorder prevailing among its molecules just as the, temperature, hotness of the body., , is a measure of the degree, is the measure of degree o, , 7.9 CHARACTERISTICS OF ENTROPY, Following are some of the important characteristics linked with the concept of entropy, i) Change in entropy in a Carnot's eycle, , Case (a) Change in entropy for system as a whole, , Let the goure, working substance (gas) and the sink constitute one system. During first, , stroke heat, , temperature., , js lost by the source at temperature T1 while it is gained by the gas at same

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THERMODYNAMICS, , ., , 171, , Change in entropy of source =, , T, , Change in entropy of gas = +, , Net change in entropy of, , source, , and gas during, , first, , stroke, , Q, , =T, , T0, , Dring second stroke neither heat is absorbed nor rejected by any of the components of the system., Q=0, i.e.,, Change in entropy during second stroke 0, =, , Similarly it can be proved that net change in entropy during third and fourth stroke is aiso zero., Net change in entropy, , during complete cycle,, 8S = 0, , in entropy of working substance only, Case (6) Change, heat 1 at a temperature, The working substance gains, at, , a, , temperature T2 during, , third stroke while, , no, , T1, , in, , first stroke and, , heat is generated, , or, , lost, , rejects, , during, , heat, , second, , and fourth stroke., , first stroke, Change in entropy during, , Q1, , +, , =, , T, , second stroke, Change in entropy during, third stroke, Change in entropy during, , =, , fourth stroke, , 0, , =, , -, , T2, =, , 0, , Change in entropy during, cyele is, of the working substance during complete, 8S', in, entropy, Net change, , ST10Q1, , For Carnot's cycle,, , T, , +0=,, , T2, , Q2, Ta, , 8S = 0, , Carnot's, , substance of, as a whole of the working, the, system, Thus net,change in entropy of, by, reversible cycle is always zero., Consider a reversible cycle represented, reversible, cycle., be, of, to, any, can be supposed, iT Change in entropy shown, 7.7. This reversible cycle, in, Fig., as, P-V diagram, a closed, abed, efgh... For each, a number of Carnot's cycles, of, comprising, in entropy is zero. Therefore,, of these small cycles, the change, to the, of the cycle which is equal, net change in entropy for whole, must also be zero., individual, , Sum of the, , cycles, , entropy change of the, , 2, , A, , dS=0, dS =0, , broken, , or, , dQ-0, r, , Fig., into a, , 7.7 A, , cycle, , number, , curve, , ofreversible, , up, , cycles.

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STANDARD, RD pPHYSI, 172, When, , a, , change takes place, , to another, from state A, , state, , B, at constant, , tem, , in entropy, , J, , dS, , or, , Sn-S, , A, , erature, channange, , T, at temperature, , T., , Chan, , absorbed or rejected, nge in ent, ent, thein, is independent, ndent of, of the, conditions and hence, final, and, initial, the, on, depends only, of 1 gram of a neré, Let 's' be the entropy, (i) Entropy of a perfect gas., gas at, of gas 1s, temperature T. Entropy of 'm' gram, is the, , where, , amount, , of heat, , chosen, , ms =, , d o -d, , rding, , to, , u, , r, , the first law of thermodynamics,, ms, , dS=, , F, , dU+PdV., T, , ms -, , or, , For a perfect gas dU =m C,dT, where, , heat capacity at constant, equation of state for an ideal gas,, , C', , is the, , specific, , volume, per gram of the gas. According, ing to, , PV= RT, M, , PRT, P MV, , or, , Substituting for P and dU,, ms = mC, , or, , ms =m, , or, , ms, , C loge T+R log. V+ Constant, M, , =m C, loge T+, , R, , loge V+ Constant, , M is the molecular weight of the gas and R is the, gas constant, for one gram molecule of, gas., (iv), , Temperature-entropy diagram. A, between, entropy (along X-axis) and temperature (along graph, Y-axis), is very, convenient for estimation of the work done, an, by, plot the entropy of working substances against the engine. Let us, in, , temperature, Carnot engine (Fig., During first stroke entropy of, 7.8)., increases from OK to OL and the change is, line AB. During second stroke there is no represented by the, change in entropy, CAQ = 0), i.e., the entropy, remains constant, (= OL). The, change is represented by line BC. Similarly CD, and DA, represent the change in entropy during third and fourth stroke, se, , of, , a, , respectively., , SO, , Q1, , ad, 1S0, , c, , Q2, , Fig. 7.8 T-S Diagram.

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173, , THERMODYNAMICSS, , For first stroke, 6S-, , ST, , 1-Tas, , Forthird, , stroke,, , Ta, , 2-T2ös, 1-Q2= (T1 T2) 6s, -, , From graph,, , Ti -, , T2 = AD, , 8S, , and, , CD, , 1-2= AD x CD = Area ABCD, , 1-92, , is also the work done, during one, done, So, work, during one cycle of an engine is, , But, , T-S diagram for that eycle., (D), , Entropy, , the form of heat, , in a n irreversible, cycle. In, eddies etc. So work done in, , cycle of the engine., equal to the a r e a enclosed inside, , the closed, , of an irreversible cycle some energy is lost in, irreversible cycle is less than that done during, temperature limits. Let the system (reversible) gain heat, , or, , case, an, , a reversible cycie working between same, Q1 at T1 and reject heat e2 at temperature T2. Net gain in entropy 8S during the cycie is, , 8S 8S2, , Q2 Q1, -8S1=;T2, T1, 1-Q2Ti-T2, , Q1-2 Ti-T2, , Since, , Q1, , T1, , Ti, , only for a reversible cycle, , Q2 21-Ti, Te, 1 Q1, , or, , T2, Q7, Q, , Q, , or, , T2, , Thus in, , case, , of, , an, , Q1, , 2, ST2, irreversible, , 1>0, , (a, , positive quantity), , is, , a, , 1, , cycle there, , net increase in the, , entropy of the system.

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182, , KALYANI STANDARD PHYSIcS, From equation (35),, 0T, aV, , U, , aP, , Js avas asay, , 0T, , or, , PU, , as, , aP, , .(39), , av, , Similarly, proceeding from equations (36), (37), 0T, , aP, , and (38), , respectively, , we, , get, , as, , 40), , OP, , .(41), , 3T y, 3P, , ..42), , OT, , Equations (39), (40), (41) and (42), An aid to, memory, Following aid shall help in, , called, , are, , Maxwell's thermodynamic equations., , relations easily., reproducing, write down a Maxwell's relation, observe the following steps, i) Draw a circle and mark, one of its diameter, [Fig. 7.11}., (i) Starting from one end of, write T, V, S, P in that, diameter,, order. To remember this order, you may remember the sentence,, To, , Maxwell's, , T.V. Special Programme", (u) Take any one end of the diameter as, origin and let the letter, written there be numbered, as 1. Proceed in clockwise or, anti-clockwise manner and write down all the four letters in, the, , manner, , 2, 3, (iv) For clockwise rotation, the, is completed with a, while for anti-clockwise rotation, the equation, equation is completed with, T, , V, , P, , S, , This leads to the relation, 3T, , OV, S, , moving anti-clockwise words, , This leads, , T, V, , to, , the relation, , 0T, , Fi, , 1 An aid to, , -, , OP, as, , P, S, , memory, , for Maxwell's relations., , positive (+) sign, negative ( - sign, , (a) Starting from T, (i) moving clockwise we write, , ii), , +), , in, in, , between, between.

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THERMODYNAMICS, , 183, , (b) Starting from, ) moving clockwise we write, This leads to the relation, , (), , Moving anti-clockwise, , we, , get, , V, , This leads to the relation, , P, , 7, , 13 CLAUSIUS-CLAPEYRON'S EQUATION-"PHASE, Maxwell's relations, , One, , have found, , of them is in derivation of, , hange, , of, , phase of a substançe, , applications, , equation called, due to, , in, , solving, , a, , CHANGE", , number of, , a, , Whenever matter undergoes change from, , temperature., , state to another, it, The amount of heat is, called latent, , constant temperature., with latent heat, deals, equation, and, of heat at, , easily, , can, , dS, , From equation (77),, , problems, , Clausius-Clapeyron equation., change in its, one, , be obtained, , OP, , in, , absorbs certain, , heat., , Physics, , It deals with the, amount, , Clausius-Clapeyron's, , using Maxwell's, , relation., , 3T, , OP, OV, , :TdS-, , T, , Writing the equation for finite variation, AP, , TAV, , 4, Let, , and, , V, , AQ, , 1, , be the initial and final volume of a unit mass of, matter whieh, change of state. If L' is the specific latent heat for the change of state., , V2, , undergoes, , AQ=L and AV= V2- V, AP, , L, , .43), , TV2-V, , Equation (43) gives the rate of change of melting point or the boiling point of a substanee, with pressure and known, Case i) Variation of, , When, , as, , Clausius-, , Clapeyron's equation., , boiling point with pressure, , substance is converted from liquid state to vapour state, its, in vapour state is much larger than that in, liquid state (V1)., a, , V, , (V-V)>0, From equation (43),, , AT =TV2-V), L, , AP, , specific volume (V2), , 7V)

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KALYANI STANDARD PHYSICS, 184, of AT is, always positive, sign, due to a n i n c r e a s e, Boiling point of a liquid, same, , Since L, , and T, , are, , as, in, , that of AP., and, , pressiure, , decreases, , due, , to, , n, , increases, , .., , decrease in pressure., , Case, When, , (u, a, , of melting point with pressure, is positive, solid is converted into liquid (V- V), Variation, , some others., (a), , When, , (V2 - V), , solids like ice have, negative. Some, , is, , a, , in, , lesser, , sone, , specific, , cases, , and, , volume, , on, , negative, , in, , melting., , V2-V1<0, AT=, , Since, , TV2-11, AP, L, , If AP is positive, AT will be negative., , Melting point of, , a, , substance decreases due to, , an, , increase, , contracts on melting., (6) When, , (V2-Vi), , is, , positive., , Certain solids like, , wax, , expand, , in pressure, , on, , if, , the substanco, , melting., , V2-V1>0, If AP is positive, AT will be positive., , Melting point of, , a, , substance increases due to, , an, , increase, , in pressure, , conti acts on melting., , SOLVED EXAMPLE, , if the substance

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9.7, , PEREECT, , BLACK BODYY, , For all, , practical, , purposes, , a, , wavelengths of incident radiations, lamp black can be taken to, , surface coated with, , the, appears black because it absorbs all, , none., , A body, reflects, , and, be a, , heat radiations are absorbed by it. Ferry,, black body. Strictly speaking, about 98% of incident, enclosure, a suggestion that a hollow, in search of the conception of a'perfect black body, gave, black body., a, having a pin hole opening in it can be treated to be perfect, 'O. Inside the enclosure there, A hollow enclosure (of any shape) C has a pin hole opening, Incident radiation XO, entering, is a pin pointed conical projection N directly in front of 'O [Fig. 9.3)., it by to and fro reflections., strikes one of the sides of conical projection and gets lost inside, chance of a ray to come, Thus a radiation entering the enclosure is said to be absorbed. The only, is very small; hence the, if it is incident directly on the tip of N. Evidently this chance, body is said to be a perfect black body., , out 15

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KALYANI STANDARD PHYSICS-, , 214, , It is, , capable of absorbing all, , the, , wavelengths of radiations, , incident upon it. When heated, it emits radiations of al, , infinity. Radiations emitted, by the enclosure are called black body radiations. Since these, radiations are the strongest for a particular temperature, they, , wavelengths ranging, , from, , zero, , to, , are called full radiations or total radiations. Black body, radiations possess following properties:, ) The energy density is independent of nature or shape of, the walls of enclosure but depends upon the temperature., (i) The radiations are homogeneous in nature., Fig. 9.3 A perfect black body., , i ) The radiations are isotropic in nature., , iv) All bodies placed inside the enclosure also emit black, body radiations., , .8 EXPERIMENTAL STUDY OF BLACK BODY RADIATIONS, An experimental set up for the study of black body radiation is shown in Fig. 9.4a). 'S is an, electrically heated black body., A narrow slit collimates the radiations into a narrow beam which is incident on a stainless, , steel concave mirror M. If the distance between source and M1 is equal to the focal length of, M, reflected beam becomes parallel. This parallel beam is incident on one of the faces of a, prism ABC made up of rock salt. Each ray disperses into a number of rays. Rays of a particular, , wave-length are parallel to themselves in the emergent beam. The emergent beam falls on, another concave mirror M2 of stainless steel and different, wave-lengths are brought to focus, at different points. Detection of thermal radiations is done, by means of a thermopile P which, has galvanometer 'G connected in its circuit. A, graph between wave-length *' of the radiation, (along X-axis) and emissive power E (along y-axis) for a particular temperature of the source, is shown in [Fig. 9.4 (b)). A number of such, graphs are plotted for different temperatures of 'S., A, , M2, , E, , (a) Experimental set-up:, , 646kk, 1449, 1259, 904095k, , B, , Am, , (6) Results, , Fig., , 9.4, , Energy distribution, , Results. Following conclusions, , i), , Distribution, , wide range of, , were, , of energy of the, , wavelengths., , of, , perfect black body., drawn from the above, , spectrum, , a, , of black, , experimental study:, , body radiations, , is not, , uniform, , over, , a

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215, , RADIATION, , () For a partieular temperature, the emissive power E, , of perfect black body increases, , with an increase in wavelength, becomes maximum for a particular wavelength mand then, , decreasing, with the furthertoincrease in wavelength., starts, curves, of, different, (ii) m, , corresponding, , Dotted line A, , different temperatures., , joins the maximas, , the wavelength of radiation, for which the intensity is maximum, decreases with, , a rise in the temperature of body., , iv Area undereacheurve represents the radiant emittance of the body of that temperature., , It has been caleulated that area under each curve varies directly as the fourth power of the, absolute temperature of the body., , 9.9 LAWS FOR BLACK BODY RADIATIONS, VFollowing laws can be stated in respect of black body radiations., ) Stefan's law. It states that the total rate of emission of radiant energy from a black, surface is proportional to the fourth power of its absolute temperature., , According to stefan's law, total emissive power of a black surface, at a temperature T, is, , given by, E=oT, where 'o is Stefan's constant and its value is 5.67 x 10 Wm2K4, In case a black body at temperature T is surrounded by another black body, at, , temperature, , To. the totalemisive power F of the first one is given by a modified form of Stafan's law, called Stafan-Boltzmann law, according to which, , E =o (T*-T, ü) Wien's Displacement law. The emission from a black body consists of wave, lengths, Tanging from zéro to infinity. The emissive power of the body for different wave lengths is also, different. A relation connecting the intensity of radiation with the wave length is ealled law of, , distribution of intensity of black body radiation. One such attempt was made by Wien to atudy, the distribution of intensity of black bodyradiations at different temperatures. He observed, that, at a particular temperature, intensity due to a particular wave length à i s maximum., , A relation betweenm and the temperature 7 of the body is called Wien's displacement law, which can be stated as follows., , tstates that the product of wave length m for maximum intensity and the absolute, temperature of the body is constant., , mxT= constant = b, , L.e.,, , 6), where b' is called "Wien's constant' and its value is 0.2898 cm degree celcius. It is clear from, , equation (6) that wave-lethgth Am for maximum intensity gets displaced towards shorter, wave-length side due to an increase in temperature of the body. That is why the law has been, named as displacement law., , ijPlanck's law. Max Planck on the basis of his quantum theory of radiation gave a new, , distribution formula., According to this theory emission of radiation does not take place, continuously as postulated in classical physics. Instead it takes place, in a discrete fashion, in, terms of quantas of energy. Each quantum of energy called photon econtains energy hv where, , h' is called Planck's constant and V is the frequency of the resonator., , Based upon the concepts of quantum theory of radiation, Planck was able to show that, , fOT) should have the form, , fAT=-, , hc, , T-1, , .7)

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KALYANI STANDARD PHYSICS, 216, , Here, , 'c'is the, , velocity of light and h' is, , the energy, Boltzmann constant. He gave, , distribution, , formula as, , 8 mhc, , vd 5s, In terms, , T, , 1, , v, , of frequency using relation, d v 8h, , X, , .8), , hv, , ekT -, , Equations (7) and (8), , are, , known, , as, , d, , he, , 1, , of black, Planck's law of distribution, , radiations., , body, , SUMMARY, to, , heat travels from one point, transmission of heat in which, Radiation. It is the process of, medium., of light, without heating the intervening, another in straight lines, with velocity, , Thermal radiations. The energy emitted,, thermal radiation., temperature is called, , Emissive power, , Emissive power of, , (e)., , a, , in the form of radiations,, , body, , the energy radiated, in, , wave-length i is defined as, unit range of wave-length., , at, , a, , by, , a, , body due, , certain temperature T for, , vacuum,, , per second, per unit, , a, , area, , to its, , certain, , and per, , defined as, emittance of a body at a temperature T is, Radiant emittance (E). Radiant, unit area by the body., the total energy radiant, per unit time per, the ratio of emissive power 'en, of a, Kirchhoffs law. It states that at any temperature, , body, , to the, , absorbing, , to the emissive power, , 'a, for a particular wave-length is always, of a perfect black body for that wave-length., , constant and is, , power, , equal, , of radiations incident, Black body. It is a body which is capable of absorbing every type, black body., enclosure with a narrow opening in it serves as a perfect, upon it. A hollow, , Stefan's law. Total emissive power of a black body is given by, , E= oT, Wien's, , displacement law., , Mathematical form of Wien's, , mT=b., , Planck's law. According to Planck's law, , f(aT) =h, AeT, , 1, , e, , Energy distribution formuala is, Vdv 8 h, , hv, e, , kT, , 1, , displacement law, , is