CHAPTER 1 THEORETICAL ASPECTS OF HEAT CAPACITY OF SOLIDS AND LIQUIDS

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1 CHAPTER 1 THEORETICAL ASPECTS OF HEAT CAPACITY OF SOLIDS AND LIQUIDS

2 1.1 CERTAIN THEORETICAL ASPECTS OF HEAT CAPACITY OF SOLIDS - BRIEF INTRODUCTION Heat capacity is considered as one of the most important thermal properties of solids and liquids. In the case of solids, the theory involved in the lattice dynamics can be understood from the knowledge of heat capacity of solids. The heat capacity of a body is simply defined as the ratio of the constant power P applied to the thermally isolated sample to the resultant rate of increase of temperature T. An appropriate mathematical expression is given below. C 1.1 AT represents the change in temperature rise of the body due to the supply of an amount of heat equal to A.Q. The value of C depends on how the thermodynamical variables of the body change during the addition of heat. As far as solids are concerned, the important property is the heat capacity at constant volume Cv defined as C v LT AT * the heat capacity at constant pressure C which is deterp mined experimentally, is defined as 1

3 2 c - Lt up (42) at - 0 vat'p (1.3) If *U * denotes the internal energy of the then it becomes (AQ)V = (AU)v system, (1.4) V thus, eqn. (1.2) takes the form r = Lt /A_U \ _ /3 U\ v AT 0 VAT v '3 T; v... (1.5) From fundamentals of thermodynamics, we can write (AQ) - A(U + pv)p = (AH)... (1.6) where H denotes the enthalpy of the system. Eqn. 1.3 can be written now as C P Lt /AH\ AT -* 0 kat'p (M) v3t'p (1.7) The entropy changes are related to the heat capacity of the system by the following relationship ds)v - C I-Y dt 00 <AS>P c f -E. dt J T... (1.9) The above relations are valid provided there is no phase change involving latent heat in the temperature interval over which the integration is carriedout.

4 3 The heat capacity per gram is normally considered as the specific heat of the substance and is expressed as C (cals/gm) = C/m (1.10) m is the mass of the sample. Similarly, the molar heat capacity of the substance (C^) is obtained by multiplying the specific heat with the molecular weight of the sample. For elements, the correspondings atomic heat capacity CA is obtained by multiplying the specific heat with its atomic weight. Practically it is not possible to keep the solid at constant volume during temperature change and hence Cp is the one which is experimentally measured. But Cv is determined theoretically and is made use of in the lattice dynamics study. The molar heat capacities at constant pressure (CMp) and constant volume (C^v) are thermodynamically related as C Mp CM Mv B (1.11) In the above expression Vm, «, 6 and T represent the molar volume, thermal expansion coefficient, compressibility and temperature in K at which the physical parameters involved are determined. The exact evaluation of the difference in the heat capacities is

5 4 not possible over a wide range of temperatures due to the fact that most of the determinations of 6 have been made near room temperatures. Therefore, one has to make approximations which depend on theoretical or emperical equations of state of solids. As these approximation methods are not exact, values of Cv derived in this way are less accurate than the experimentally determined C data. P Generally used approximation [1] is based on the Hie Gruneisen equation of state f or solids from which Gruneisen parameter y is given by 1.12 The value of y is taken as independent of temperature and ranges from 1 to 3. With this assumption, y can be calculated from values of av and 6 at room temperature and the eqn.(1.3)can be expressed as C * = 1 + ya 1 v T 1.13 The heat capacity of solids at constant volume Cv according to Dulong and Petit is given by Cv = = 3R = 6 Cals/gm atom/ C This shows that the atomic heat C of an element in the v

6 5 solid state is constant and is equal to 6 calories per gin atom per C. But experimental studies show that the specific heat of a solid decreases as temperature is lowered and becomes vanishingly small near the absolute zero. Classical theories cannot give any explanation for the experimentally observed heat capacity behaviour of solids below the laboratory temperatures. Quantum mechanical treatment of the heat capacity of solids has been made by Einstein [2] by assuming the crystal as an aggregate of atomic oscillators, all of which vibrate with the same frequency vq. Einstein's theory could not explain temperatures. the heat capacity of solids at very low Assuming that the atomic oscillators of the crystal are strongly coupled and that these exist in two transverse and one longitudinal modes of vibration with the same maximum cut off frequency. Debye [3] developed the theory of specific heats which could explain satisfactorily the heat capacity of solids near absolute zero. The theory of heat capacities has been further modified by Born-Vankaraman [4] by considering that the maximum cut off wavelength for the two modes of vibrations is the same. As mentioned earlier the heat capacity at room temperature is close to 31. Above room temperature Cp

7 6 and Cv increase only slightly. Below room temperature, the heat capacity falls rapidly and approaches zero 3 value as a T variation in insulators and T in metals. In magnetic solids, there is a large contribution to the heat capacity near the temperature at which the magnetic moments become ordered. There are two contributions to the heat capacity of metals namely lattice heat capacity and electronic heat capacity. At high temperatures, the lattice contribution to the heat capacity of a metal dominates over the contribution arising due to the conduction electrons. At low temperatures the contribution to the heat capacity due to the conduction electrons dominates over the lattice contribution. At high temperatures, the heat capacity of a metal can be expressed as C = 3Nk + ( ) Tp and T represent the Fermi temperature and the temperature of the specimen in K at which Cy is determined. N and k represent the Avagadro number and Boltzman constant respectively. At temperatures below Debye temperature and very much below, Cv can be written as the sum of electronic and lattice contributions.

8 7 Thus Cv can be expressed as below, where A and B are constants. Cv - AT + BT The heat capacity of a material gives important information on the thermodynamic stability of various phases of the material existing under different conditions. It also detects second order transitions. The heat content, free energy and entropy of the material can be evaluated from the knowledge of heat capacity.

9 8 1.2 CERTAIN THEORETICAL ASPECTS OF HEAT CAPACITY OF LIQUIDS - BRIEF INTRODUCTION The heat capacity at constant pressure of a liquid (c ) is an useful parameter in the study of solutions. For example, it is important in correlating thermodynamic data e.g., in the prediction of the temperature dependence of the enthalpies of different processes in solution (dilution, reaction etc.) and in relating adiabatic to isothermal compressibilities. But a main use of heat capacities of liquids is to derive from them apparent or partial molar heat capacities (0 * r t ^ or C D). p, D These functions elucidates the heat capacity of the solute itself and all changes in the heat capacity of the medium caused by the introduction of the solute (solute-solvent and solute-solute interactions). With aqueous solutions, the study becomes still interesting, since ^ being related to the second derivative of the chemical potential with respect to temperature will be very sensitive to all structural changes in the aqueous medium. The parameter ^ is an important one, but, relatively little is known about it in comparison with other thermodynamic quantities. It is defined by the following expression [5]

10 9 Cp, 0 " M Co - <Cp " Co)/m i*16 where M, C. C, m denote the molar mass of the solute, the SDecific heat capacity of solution, the specific heat capacity of the pure solvent, and the molality of the solution resdectively. The heat capacity measurements on aqueous solutions of non-electrolytes yield useful information concerning the solute-solvent interactions. Heat capacities of organic compounds in solution are also quite interesting. The enthalphy and entropy (and even heat capacity) changes for important equilibria and kinetic processes are sensitive to temperature. The heat capacity behaviour of organic solutes as a function of structure, concentration, and medium are having great practical importance of structure - reactivity correlations and the widespread tendency to intemret even small reactivity differences in terms of potential energies. Cp of a liquid may generally be determined by either direct methods [6-111 or by adiabatic decomposition or piezoelectric method [12]. In the direct method, a small quantity of heat is elecically supplied to the calorimeter vessel containing the experimental liquid and the resulting rise in temperature is measured. From a knowledge of the thermal capacity

11 10 of the calorimeter, which may be obtained either from the C of the material used in its construction or by a P second experiment with a liquid of known heat capacity, the heat capacity at constant pressure of the experimental liquid can be determined. Radiation losses can be avoided by maintaining the shield around the calorimeter at the same temperature as that of the experimental liquid. However, if the shield is maintained at a constant temperature midway between the initial and final temperatures of an experimental run, and if the heating rate is steady, the radiation losses can be effectively eliminated since the losses in the first half of the experimental run would get cancelled with those in th second half. In the adiabatic decompression method [12], a small quantity of liquid is subjected to a pressure of 10 to 20 atmospheres and after the equilibrium is attained, the pressure is reduced and the resulting change in temperature gives the adiabatic thermal pressure coefficient (9T/9p). b Using the standard thermodynamic relation C P (ix> /(i!> ^tv^p s 1.17 the specific heat at constant pressure can be evaluated. In the above equation (9v/9T)p can be obtained from a

12 11 knowledge of the variation of molar volume with temperature. Measurement of heat capacity of liquid or solid using direct method calls for a constant current source to supply power to the heater. Generally, in calorimetry it is desirable to be able to operate with relatively high heater voltages at low currents in order to minimise Joule heating in the leads which connect with the sample. In the present work, a new constant current and voltage sources are developed with programmability. The advantage of IC LM 317 (adjustable positive voltage regulator) in various modes is explored in the present study. A brief review of the instrumental aspects of heat capacity measurements with the present technique are discussed in the following chapters.

13 12 REFERENCES 1. F. Seitz The Modern Theory of Solids, McGraw-Hill, New York, 1940, p A. Einstein Ann. Phys. (LPZ), 22 (1907) 180, P. Debye Ann. Phys. (LPZ), 39 (1912) M. Born and Vankarman 5. P. Picker, P.A. Leduc, P.R. Philip, and J.E. Desnoyers 6. F. Kawaizumi, T. Otake, H. Nomura and Y. Miyahara 7. F.T. Gucker, Jr., and F.C. Ayres 8. M. Leblane and E. Mobius 9. V.C. Williams and M. Sivetz Phys. Z., 13 (1912) 297. J. Chem. Thermodynamics, 3 (1971) 631. Nippon Kagaku Kaishi, 10 (1970) J. Am. Chem. Soc., 59 (1937) 447. Math-Phys-Klasse., 85 (1933) 75. Rev. Sci. Instrum., 17 (1946) J. Sturtevant J. Am. Chem. Soc., 59 (1937) N.S. Osborne, H.F. Stimson and D.C. Ginnings J. Res. Natl. Bur. Standards 18 (1937) J.S. Burlew J. Am. Chem. Soc., 62 (1940) 681, 690, 696.

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