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Maxwell's Contributions to Thermal Physics

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Maxwell's Contributions to Thermal Physics GENERAL I ARTICLE Maxwell's Contributions to Thermal Physics Sisir Bhanja Introduction James Clerk Maxwell (1831-1879) was one of the great­ est theoretIcal physicists of the nineteenth century. He is remembered for his pioneering contributions to di­ verse fields of mathematical physics. His work in ther­ Sisir Bhanja did his Ph D mal physics bears two distinct approaches: from the Indian Associa­ tion for the Cultivation of 1. Thermodynamics or the macroscopic description, Science, Kolkata. Apart from his doctoral work on and the physics of molecular collisions, he did some 2. Kinetic theory or the molecular and statistical ap­ work on Mossbauer proach. effect. He retired some time back from the We discuss both these approaches in this article. Department of Physics, St Xavier's College, Kolkata, Maxwell's Work in Thermodynamics where he taught for a number of years. Maxwell's main work in the field of thermodynamics ~urrently, he is pursuing consists of the group of thermodynamic relations (in­ studies in gravitation and volving the entropy'S', the volume 'V' and the temper­ cosmology. ature 'T') which are named after him. He derived these relations by combining the two laws of thermodynamics, namely, a) The First Law: dQ = dU + PdV and b) the Second Law: dQ = TdS, where dU is the change in the internal energy U. Com­ bining these two laws, we have dU = TdS - PdV. Keywords Maxwell's thermodynamic re- . Maxwell noted that since dU is a perfect differential, lations, velocity distribution law, coefficient of viscosity. -RE-S-O-N-A-N-C-E--I-M-a-y-2-0-03--------------~-----------------------------~- GENERAL I ARTICLE for U = U(x, y), where x, yare two independent vari­ ables, from which we obtain J (T, S) = J (P, V) , x,y x,y where the Jacobian J(T,S) = 8(T,S) = (8T/8X 8T/8Y) x, y 8(x, y) 8S/8x 8S/8y . The four relations then appear as follows: Relation 1 : Putting x = T and y = V, (8S/8V)T = (8P/aT)v· Relation 2 : Putting x = T and y = P, (8S/8P)T = -(8V/8T)p. Relation 3 : Putting x = Sand y = V, (aT/8V)s = -(8P/8S)v. Relation 4 : Putting x = Sand y == P, (aT/8P)s = (8V/8S)p. These relations due to Maxwell are enormously useful in providing relationship "between measurable quanti­ ties and those which either cannot be measured or are difficult to measure", as is shown with two illustrations given below. From Relation 2 we have, on multiplying through by T and noting that T dS = dQ, (8Q/8P)T = -T(8V/aT)p = - V-I (8V/aT)p VT, = -0 VT, I where 0 = V- (8V/8T)p is the isobaric coefficient of volume expansion. -58------------------------~------------RE-S-O-NA-N-C-E-I-M-a-Y-2-00-3 GENERAL I ARTICLE Thus, for substances which expand on heating, a > 0, so that (8Q/8P)T < 0 which implies that an amount of heat must be taken away from the substance when the pressure is increased in order that the process re­ mains isothermal. In other words, when a substance with a positive expansion coefficient is compressed, heat is generated. For substances which contract on heat­ ing, cooling should take place. Nature is replete with such examples and Maxwell's relations provide the due explanations. Our second illustration follows from Relation 1 multi­ plying through by T which leads to (8Q/8V)T = T(8P/aT)v. The left-hand side describes a process in which supply of heat does not change the temperature but alters the volume isothermally, as is seen in a change of state. The above relation thus tells us where 112 and VI are the respective volumes and L, the latent heat of fusion. The quantities on the left-hand side are easily observable experimentally and allow us to calculate (8P/aT)v. As an example, we calculate the change in the freezing point of water due to change in the pressure. We note the following: freezing point of water under normal conditions is 273.16K, L, the latent heat of fusion of ice is SOcal/gm = SO x 4.1S6 x 107erg/gm, the specific volumes V2 = 1.0001cm3 /gm for water and Vi = 1.090Scm3/gm for ice. For a change of pressure 8P = 1atmosph~re = 1.013 x 106dynescm -2, the consequent change of the melting -RE-S-O-N-A-NC-E--I-M-a-Y-2-00-3------------~~-------------------------5-9 GENERAL I ARTICLE point 8T = -O.0075°C, i.e., the melting point is lowered for substances such as water which contract on melting. In the reverse case, i.e., for substances which contract on solidification, the melting point rises consequent to an increase of pressure. These relations are of immense use not only to physics, also in engineering practices. Kinetic Theory The thermodynamic approach does not allow a calcula­ tion of the quantities involved, based on the first princi­ ples e.g., to calculate U, we must know the mechanism that contributes to U. The microscopic description of molecular motion, as given in the pioneering work of Maxwell, really solves these problems. Maxwell's work in this area provided a breakthrough by demonstrat­ ing the real importance of molecular motion (which still had its sceptics) in understanding thermal phenomena and the importance of statistics. Maxwell deduced the law of distribution of velocities from considerations of probabilities "far in advance of anything previously at­ tempted on the subject" H~s ideas, though simplified, did hit upon some of the essentials, as are elaborated below. (a) The gas molecules are point masses. The system is considered to be in equilibrium, external disturbances being totally absent. (In fact, the equilibrium follows from the elastic collisions between molecules which are in continual motion, as was pointed out by L Boltz­ mann, another great player in this field who was almost contemporary to Maxwell). (b) To meet the requirements of statistical considera­ tions, the system must be assumed to be made up of a very large number of molecules; there should be an equal number of molecules in any given volume 8V within the system. In other words, the system should be homoge- --------~-------- 60 RESONANCE I May 2003 GENERAL I ARTICLE ( c) The motion of the molecules is totally random such that the molecules have the same velocities in all direc­ tions. As a consequence of this isotropy the components of velocities along any system of coordinates are equal and it is immaterial which coordinate system the results are expressed in. (d) The velocity components (u, v, w) along any three coordinates (x, y, z) are uncorrelated to each other. The quantities (u, v, w) being uncorrelated, their probabili­ ties are also independent of each other. (e) The probability function describing the molecules with a particular velocity component v in the interval v to v + dv is dependent only on v and the element dv. All these assumptions were obviously consistent with the observed facts about the randomness of molecular motion leading to uniform bulk properties like density, temperature, etc. throughout the entire volume of the gaseous system in equilibrium. Now, with 0 as the origin of a rectangular Cartesian system of coordiantes (x, y, z) and (u, v, w) as the corresponding triad of ve­ locity components, the resultant velocity c is related to (u, v, w) by (1) To calculate the probability of a molecule to have a ve­ locity between c and c + dc, the same can be calculated component-wise. If dnu be the number of molecules per unit volume which have the x-component of velocity be­ tween u and u + du, then the corresponding probability function P(u, du) = dnu/n = f{u) du, (2) where n is the total number of molecules per unit vol­ ume. On similar considerations, the corresponding pro b­ abilities for a molecule to have the y-component of veloc­ ity between v and v+dv and the z-component of velocity -RE-S-O-N-A-N-C-E-I-M-a-Y-2-0-0-3------------~-----------------------------61 GENERAL I ARTICLE between wand w + dw may be written respectively as P(v, dv) = dnv/n = f(v) dv and P(w, dw) = dnw/n = f(w) dw. (2a) The functional form has been taken the same for all three velocity components by virtue of assumption (c). So, the effective probability of a molecule having its ve­ locity components between u and u + du, v and v + dv and wand w + dw simultaneously, is given by P(u, v, w, du, dv, dw) = dnu,v,w/n = f(u) f(v) f(w) du dv dw (3) so that the number of molecules within the specified velocity ranges will be dnu,v,w = n f(u) f(v) f(w) du dv dw. (4) The same set of velocity components make a single re­ sultant velocity vector c. All the velocity vectors start from a common origin O. The lengths of these vectors give the magnitudes and their directions are the direc­ tions of motion of the molecules at any given time. If spheres of radii e and e + de are drawn centred at 0, the origin, the vectors ending between these two con­ centric spheres would represent all the velocities present that have values between e and e + de. The chance that this single velocity vector c ends in the volume element du dv dw is, by assumption (e), ¢(e) du dv dw, which may be alternatively written as F(e2)du dv dw = f(u) f(v) f(w) du dv dw (5) valid for any arbitrary range of values of du, dv, dw, i.e.
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