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.
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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.
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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
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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-
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( 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
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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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Thus we get a remarkable result that the product of a set of functions of some variables is equal to a function of the sum of the squares of the variables, which at once indicates a logarithmic relationship between the vari ables. Maxwell immediately took this relationship as the form of the probability function or the distribution law. To deduce the distribution law we need to identify the functional forms for F(c2 )andf(u). This is achieved by choosing a particular value of c keeping c constant 2 and hence F(c ) = a constant. We must have
2 d[F(c )] = 0 = d[f(u) f(v) f(w)]. (7)
Since c2 = u2 + v2 + w2 = a constant, we must also have
dc2 = 0 = udu + vdv + wdw. (8)
Using the method of undetermined multiplier in the form of a constant A, (7) and (8) yield
[f'(u)/ f(u) + Au]du + [f'(v)/ f(v) +
Av]dv + [f'(w)/ f(w) + Aw]dw = O. (9) It is to be borne in mind that the velocity components are independent of each other and that du, dv, dw are totally arbitrary. The coefficients of du, dv, dw in (9) must be zero separately and we have
f'(u)/ f(u) = -AU
f'(v)/ f(v) = -AV f'{w)/ f(w) = -AW (10) which, on integration, yield
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Equation (11) has two unknown constants A and a. The constant A can be determined from the normalisa tion condition that the probability of finding a molecule within the velocity range -00 < u( or v, w) < +00 must be equal to one. This gives +00 1-00 f(u)du = Aay'1r = 1, or, 1 A= aVi. (12)
From the above we see
2 2 feu) (l/ay'7r) exp( _u /( ) = (l/ay'7r)fg(u), 2 2 f(v) (l/ay'7r) exp(-v /a ) == (l/ay1r)fg(v), 2 2 few) (l/ay'7r) exp( _w /( ) = (l/ay'1r)fg(w), (13)
2 where fg(x) = exp(-x2/a ), the suffix 9 expresses the Gaussian nature. From the definition of probability it is clear that the probability of finding molecules in the range of velocities u, u + du, v,v + dv and w, W + dw is feu) f(v) few) du dv dw (l/ay'7r)3 fg(U) fg(v) fg(w) du dv dw - (l/ay1r)3 fg( Ju2 + v 2 + w2 )du dv dw - (1/ay'1f)3 fg(e)du dv dw (14)
2 in conformity with (6), so that F(e ) of (6) is simply Figure 1. (1/aft)3 fg(e). FUrther defining an element of the type shown in Figure 2 1, we find that the volume element du dv dw = e de dnc , where dOc is the solid angle subtended at the origin in the velocity space by the surfaGe element dSc . Suppose we want to know the probability of finding a molecule in the velocity range e and e + de. This can be calculated by integrating over dnc .
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Since f dOc = 4 7r, we find this probability to be p(c)dc = (l/aJ7r)3 fg(c)47r c2 dc. (15)
Significance of ri In the Gaussian distributions f(u), f( v), f( w) given above, the quantity a cannot be found independently. These functions help us calculate the average of any X(u, v, w) as (x) = ! X(u, v, w) f(u) f(v) f(w)du dv dw. We thus find the following averages:
2n 1 2n (lu + 1) = i: lu +1lf(u} du =
2 10"0 u2n+1 feu) du = n!o.2n+l / .j7[(u2n+1} = 0;
2n 2n 1 (u ) = a r(n + 2)/yr:ff. (16) These results show that a describes the moments of u and hence to know f (u), one of the moments of the distribution must be known. By putting n = 1, we 2 2 find (u ) = a / 2 so that the average kinetic energy per molecule must be
(17) where k is the Boltzmann constant and T the absolute temperature. For this identification of the average ki netic energy with temperature, the reader may look up the Feynman Lectures, Volume 1, pp.491-495. Thus (18) It is thus seen (Figure 2) that as T increases, the width of the curve broadens while the height shrinks. This implies
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a x 10"" 3 ..... - ..... ~-..... ---.. -,....----~--.--... ---~-~-.--:"'--...,.----, I .-. 3.5 2.5 ...
'300 kelvin 2· 2.S· .300 kelvin ~1.5 ,. , \ .. ~ I.S \ l' ,10001c.e1vin '.. ',1000 , kelvin . , 1· '. , I O.S· --.'- --, , ...... "~ ... __ 6~kcM" 0.5'· " '. •.. !>OOOketvin .>..... 0:'/-··-.--·/- '-" ...., 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 o 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 u in anise.: x 10' c inanfsa:; ). 10fl
Figure 2. implies that an increase in the temperature randomizes the system. Its role vis-a-vis the entropy of a system can be pondered over and a student may also try as an exercise as to what happens when T = 0 (suppose we reach it). While (16) and (17) give us the averages, we may ask - what is the most probable velocity? Note any study of the velocity distribution actually involves constructing a histogram np( c )dc. The most probable velocity. occurs at c = c* where p(c) has a maximum. From (16) we find that c* = n. Thus for a Maxwellian distribution we must have
These have been checked experimentally by various me ans, e.g., rotating sector velocity discriminations or sha pes of spectral lines and show excellent agreement with Maxwell's distribution. Some Important Results of Maxwell's Law ofVe locity Distribution 1. It can be used to derive an expression for the pressure P of a gas from first principles and hence explains the perfect gas equation of state PV = N kB T, which was known empirically for a long time. Maxwell's Law really established the microscopic basis of this equation.
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2. It could be used to calculate the number of mole cules crossing a unit area (from one side) per unit time. This is given by n(c2)~ / [(67f)
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tween different parts of the gas. The problem is that if molecules are considered to be point masses T goes to infinity and the gas can never equilibrate! The way out is to assume that the gas molecules have a finite size, say they are spheres of diameter (J". This way, one of Maxwell's assumptions is relaxed and we still assume that this will have negligible effect on the velocity distribution. The basis for the calculation is that any molecule may collide with another provided their centres are separated by a distance (J", i.e. there can be v = n(1r(J"2)(lcl - C21) = l/rcoll collisions per unit time. By considering all the molecules to have the same velocity i.e., ICII = 1C21 = c, Clausius had found L = 3/(41rno-2), which Maxwell showed to be in error. The use of his own distribution law of velocities for both Cl and C2 yielded the correct result L = 1/(V2n1r(J"2). In what follows, we describe how viscosity can be calcu lated from these ideas. Maxwell was the first (1860) to turn his attention to an analysis of the viscosity of gases from the kinetic theory point of view, by calculating the transfer of momentum on collision. Unlike in the previous cases which were examples of gases in equilibrium having no mass-flow motion in a specified direction, the idea of viscosity was seen to arise out of varying speeds of different layers of gas and the consequent transfer of momentum from one layer to another. This type of motion involves the dissipation of internal energy and hence is necessarily irreversible and leads to greater molecular chaos and in crease of the entropy of the system till equilibrium is reached. Let us consider a gaseous system in the space between two parallel plates of which one is at rest, while the other is moving parallel to the first one in the di rection of Y The gas layers will be at rest relative to each boundary, at the respective boundary. The speed of mass-flow motion obviously will fall off linearly from the moving plate to the fixed one, thus generating a
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~f I
I I L ~O'o) ..,.. gradient dv / dz in the Z-direction. Consequently, when the steady state of motion is reached the upper plate loses momentum to the gas and thus experiences a vis cous drag which is the rate of loss of momentum per second. If A = area of the plate, F = viscous force on the moving plate then the viscosity-coefficient rJ of the gas is defined by F / A = f - rJdv / dz, the negative sign indicating that the viscous force tends to oppose the motion. In the adjoining Figure 3, V(z) = Va + (dv/dz)Z. Let us consider a small volume element dV having ndV molecules in it and the mean collision frequency may be considered as z = c/ L, where c = mean speed for a gas having Maxwellian velocity distribution. Because of the mass-flow motion parallel to the Y axis, this is not an equilibrium situation and hence not a fit case for using Maxwell distribution law. However, the mass-flow ve locity (rv a few cm per second) being much smaller than the rms c (rv40000 cm/sec), we may assume it to have little disturbance on the Maxwell velocity distribution. In time dt, ndV molecules in dV experience nzdV dt col lisions i.e. this number of molecules start new paths in time dt, leaving dV All directions being equally proba ble (in ideal Maxwellian situation), the number of mole cules, which leave dV and reach the small area dx dy in time dt = number of paths starting in dV in time dt x
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Figure 4.
OJ \J) >~v .t /f':.r r-'---~I ·~-f}-;··~;ZL··_---~··7\ ~/ ,! ,..) // / I
~L:-~/~/.'/~.//_/_---r-~_--.: ...~-==r-·~
the elementary solid angle (dx dy cos B/r2) subtended at dV by dx dy x the fraction of molecules exp( -r / L ) moving towards dx dy and travelling a distance r (or more) without suffering a collision and being diverted =nz dx dy/(47rr2) exp (-r/L)dV dt (Figure 4). Assuming that, on an average, a molecule possesses the y-component of velocity appropriate to the layer of its last collision, each of the above molecules carries an amount of momentum m(vo + r cos B dv/dz) from dV through the area element dx dy. Hence the net momen tum-transfer going through dx dy downward from the space above the xy plane is given by
2 P dx dy.j.= -m1: 10: h: (Va + r COS 0 dv/dz)
[nz dx dy COS B/(47rr2)]exp(-r/L)dV dt
00 = -nmz dx dy dt / ( 47r )[ Va 10 exp( -r / L )dr
{~/2 {2~ Jo sin B cos e de Jo d¢
00 + dv / dz 1 r exp ( -r/ L )dr 10"/2 sin 0 cos20dO l" d¢].
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In a similar manner, the momentum transfer from below the x - y plane upward from a similarly placed volume element is given by P dx dy t= nmz dx dy dt/(47r)
[va fo'''' exp (-r / L )dr
2 fo" /2 sin e cos e de 10 " dr/> - dv / dz fo'''' r exp (-r / L )dr
10,,/2 sin e cos2 e dJ) l" dr/>J. The negative sign with the second term within the bracket being due to the fact that below the xy plane, V(z) = Va - r cos () (dv/dz). Adding the two terms above, the net momentum trans fer per unit time per unit area on integration i.e. the net viscous drag/unit area
P/dx dy dt = -[47r/3(47r)] nmzL2 dv/dz -1/3nm(c/ L)£2dv/dz = -1/3nmc L dv/dz = -rJ dv/dz. so that rJ = 1/3nmcL. The treatment given above by Maxwell's distribution law is flawed in the sense that z = c/ L was taken con stant during integration, in keeping with Maxwell's dis tribution law. This was corrected by Tait and he ob tained a value 5.1 % higher than that deduced by Maxwell.
Maxwell's expression for rJ may be put in the form rJ = 1/3 nmcx 1/( V27r a2 n = mc/(3V27ra2) so that rJ should vary as Vm, since c varies as J1/m. For gases this holds quite accurately because the variation of a for different gases is comparatively small.
The expression rJ = mc/(3J27ra2 ) is independent of pressure. Investigations showed this to be true for pres sures from a few mm of Hg up to several atmospheric
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pressures, a result which was striking as it was quite difficult to comprehend, before experiments were per formed, that the viscosity of a dense gas should be the same as that of a rarefied gas. There are, however, risks of its failure (i) at low pressure when the mean fee path L becomes comparable to the size of the container (ii) very high pressures due to strong intermolecular forces.
Again since c = (8kBT/,rrm) 1/2, 'r] should vary as v'T Experiments showed that 'r] increases more rapidly than this with temperature. This might be due to the erro
neous assumption that (J" is independent of the temper ature. Almost an identical treatment, incorporating energy tran-: sfer allows us to calculate the thermal conductivity K and mass transfer gives us the diffusion coefficient D. It is found that K = 'r] Cv and D = 'r] / p. These re sults, though qualitative in nature, correctly point to the role of collisions in determining the kinetic coeffi cients of gases. To conclude, we note that Maxwell's thermodynamic relations greatly enriched the field. The work was not
Address for Correspondence pathbreaking though. Formulation of the kinetic theory Sisir Bhanja establishes Maxwell as a visionary, who gave a message DA Salt Lake to the future: the statistical approach may not give all Sector 1 the details, but one should see if it gave the necessary Kolkata 700 064 results. Many of Maxwell's models could be improved Email: sisbhan [email protected] upon but it was possible to build these new structures because Maxwell had -given us the right foundation.
When we try to pick out anything by itself, we find it is tied to everything else in the [?{l. I I' universe.
John Muir (1938-1914) US Naturalist, Explorer
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