The Thermodynamics of. The rst law ofthermodynamics relates the change in internal energy to the

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Willis Tobias Casey
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1 Lecture 7 The Thermodynamics of Electric, Magnetic and Other Systems The rst law ofthermodynamics relates the change in internal energy to the heat supplied and the work done. So far we have included only PV work and the N work done in adding particles to the system. More generally work can be done by any force acting on the system. Examples include: surface tension for a system with a well-dened surface linear tension for a system that can be stretched electric eects for a polarized system magnetic eects for a magnetized system gravitational eects and so on. Some examples are discussed in this lecture to indicate the wider applicability ofthe concepts of thermodynamics. 7.1 General formalism In the rst law ofthermodynamics, du = TdS + dw,wemaywrite the work done in a general form X dw = Y k dx k (7:1) k where the sum is over all the forces acting on the system, with Y k denoting the kth force and dx k the corresponding kth innitesimal displacement. In Lecture 1 the work includes only the term dw = ;PdV,andin Lecture 4 the additional term dn is included for an open system. Generalizing to a system with many dierent species P of particle, with i denoting an arbitrary species, gives dw = ;PdV + i idn i. Other explicit examples of forces that might 1

2 be included in (7.1) are dw = ;PdV + X i i dn i + da + KdL +dq + E dp + H dm + (7:2) where is the surface tension and A denotes the area of the system, K is the linear tension and L is the length of the system, is the electrostatic potential and q is the charge of the system, E is the electric eld strength and P is the polarization of the system, H is the magnetic eld strength and M is the magnetization of the system, and so on. 7.2 Surface tension An example of a system where surface temperature is important is one in which liquid droplets are embedded in some form of continuum, such a fog where there are water droplets in air, or a colloid such asmilkwhere there are droplets of one liquid in another liquid. The energy associated with the surface tension is a minimum where the droplet is a sphere, and the droplets may beassumed to be spherical. There must be a pressure dierence between the inside and the exterior of the droplets, because if there were not the tendency of the surface tension to make the droplet contract would be unopposed so that there would be no equilibrium. A purely mechanical calculation (that is more dicult that one might anticipate) shows that the pressure dierence needed to balance the surface tension is P = 2 R (7:3) for a sphere of radius R. Letusderive (7.3) using thermodynamic arguments. Let our system consist of the droplet (subscript d), the exterior medium (subscript e) and the (spherical) surface separating them. We cannot assume that the number of particles is constant because particles may evaporated from the droplet into the medium, or condense from the medium onto the droplet. In equilibrium, the chemical potential inside and outside are equal, and so we wish to choose a description of the medium in which is xed. The only state function we have introduced that involves as an independent variable is the thermodynamic potential (T V ). With V d =4R 3 =3, A d =4R 2 for a sphere, and some specic function of temperature, we have the correct variables to use (T V )asthestate function, with A regarded as a function of V. From the table in Lecture 4 we have =;PV for a system in a single volume. In the present case we have three contributions, =;P d V d ; P e V e + A d (7:4) due to the droplet, the exterior and the surface, respectively. In equilibrium, d!=dr must be equal to zero. We have dv d =dr =4R 2 = ;dv e =dr (the sum 2

3 V d + V e remains constant asthe radius of the droplet changes) and da d =dr = 8R. Hence,wehave d dr = ;(P d ; P e )4R 2 + 8R =0: (7:5) The result (7.3) follows, with P = P d ; P e. The internal energy associated with the surface tension is a little subtle. The internal energy may bewritten as U = TS ; PV + N + A, which follows from the table in Lecture 4 with the surface term, A, added, where now the subscript d is omitted. However, there is also a surface contribution to the entropy that needs to be made explicit. The contribution to the entropy from the surface term may beobtained from the thermodynamic potential S = ; : With = A this gives a nonzero contribution if is a function of temperature, which isalways the case. Hence, if we isolate the surface contributions to U and write them as U s,thenwe has U s = ; T d dt V A: (7:7) It also follows from this dependence of the entropy on the surface term that there is a quantity of heat absorbed when the surface area changes. This is dq = ;T d da (7:8) dt for a change in surface area by da. The sum ofthiscontribution to the heat and the contribution, da, tothe work is equal to the change in the internal energy implied by (7.7), as required by the rst law of thermodynamics. 7.3 Curie's law The application of thermodynamics to systems with electric and magnetic properties is usually discussed in the context of statistical mechanics, in connection with specic statistical models for such systems. In the context of classical thermodynamics we consider such systems only from a phenomenological viewpoint. Here only systems with magnetic properties are discussed. Systems with magnetic properties are classied as paramagnetic, diamagnetic, ferromagnetic, anti-ferromagnetic and superconducting. The rst two of these are the simplest systems in which the magnetization is proportional to the applies magnetic eld, with the constant of proportionality being positive for a paramagnetic substance and negative for a diamagnetic substance. The 3

4 magnetization, M, is dened formally as the induced magnetic dipole moment per unit volume of the material. Many paramagnetic substances are found to satisfy a phenomenological law, called Curie's law, of the form M = C H T (7:9) where H is the magnetic eld strength, T is the temperature and C is a constant that depends on the particular substance. Curie's law for paramagnetic substances plays a role in the thermodynamics of magnetism that is analogous to the role played by the equation of state in ordinary thermodynamics. (Statistical models provide an explanation for Curie's law, but we do not need to understand the microscopic explanation to discuss the thermodynamic implications of Curie's law.) 7.4 Description of the magnetic response In order to introduce magnetism into thermodynamics we need to include the relevant magnetic contribution to the internal energy. This is written as H dm in (7.2), and our rst task is to derive this. There are two aspects of the derivation: an identication of the work and a description of the magnetic response of the substance. The description of the response of the medium is the conceptually more dicult, so let us consider it rst. An important preliminary point isthat the fundamental electromagnetic elds are the electric eld strength, E, and the magnetic induction, B. The basic form of Maxwell's equations involves two elds,e, B, and the total charge and current densities,, J: re = ;@B=@t (7.10) rb = 0 J +(1=c 2 (7.11) re = =" 0 (7.12) rb =0 (7.13) It is conventional to introduce two other elds, the electric induction, D, and the magnetic eld strength, H. These are conventionally introduced by separating the charge and current densities into induced (ind) parts, which describe the response of the medium, and extraneous (ext) parts, which act as source terms. The induced parts are used to dene the polarization, P and magnetization, M, ind = ;r P J ind + rm: (7:14) Two additional elds are introduced by writing D = " 0 E + P H = B= 0 ; M: (7:15) 4

5 The middle two of Maxwell's equations then become rh = J ext + " (7.16) rd = ext (7.17) However, it needs to be emphasized that the separation into induced and extraneous part is model dependent, and so the form of Maxwell's equations involving D, H is also model dependent. A conventional description of the magnetic response is for the magnetization in terms of the magnetic eld strength, as in (7.9). However, the response is also sometimes written in terms of a relation between M and B, whichcanbe re-expressed as a relation between M and H using (7.15). 7.5 The work done by electric forces The derivation of the terms E dp and H dm in (7.2) requires some care. It is a little simpler to derive the result for the electric eects than for the magnetic eects, and it is appropriate to discuss this case rst. The electic eld, E, inwhichthe dielectric is placed must be generated by charges, and we mayassumethatthese charges are on some collection of conductors. Consider one such conductor. Let the potential at the conductor be and the charge on the conductor be q. Thework done to change q by q is W =q. Weidentify q as due to a change D in the electric induction. This D implies a surface charge and a net charge q = ; where the integral is over the surface of the conductor. theorem and E = ;r, we have q = ; Z I dv r(d) = da D (7:18) Z Then using Gauss' dv E D: (7:19) For a change D = P due to a change in the polarization, this gives the form indicated in (7.2). 7.6 Magnetic contribution to the energy Magnetic elds do no work, and it is not immediately obvious that there is any magnetic contribution to the energy. (Classical statistical mechanics cannot describe magnetic eects because of this, but quantum statistical mechanics overcomes the problem.) An indication of how the magnetic term in (7.2) arises in a specic model involves of a current owing in a wire wound around a rod. If there are N 5

6 windings and if the current is increased by di in a time dt,then the electromotive force is E = Nd=dt = NAdB=dt, where is the magnetic ux and A is the cross-sectional area of the rod. The rate work is done is EI, and so the work done in time dt is NAIdB. The magnetic eld strength due to a length L is H = NI=L,sothat the work done is VHdB,with V = AL. AtxedH, (7.15) implies db = dm, sothat the work done becomes VHdM. 7.7 Thermodynamics of magnetism There is an analogy between the thermodynamics of a paramagnetic material that obeys Curie's law and the thermodynamics of an ideal gas, with Curie's law playing a role analogous to that of the equation of state. As an example let us rst prove animportant property: the internal energy of a paramagnetic material that obeys Curie's law doesnotdepend on the magnetization. The counterpart for an ideal gas is: the internal energy of an ideal gas does not depend on its volume. (We knowthatthis is the case because U = C V T with C V = 3 Nk for a monatomicgas.) A formal proof of this follows 2 = which S = ;P V implies (=@V ) T =0as required. S = T V = P T (7:21) 6

7 Exercise Set 7 7.1) What is the extra pressure inside a soap bubble of radius (a) R =1cm, (b) R =1mm if the surface tension of the soap solution is =0:04Nm ;1? Note that a bubble has two surfaces. 7.2) A sample of a paramagnetic substance that obeys Curie's law has a total heat capacity C H =10 ;3 T 3 JK ;1, where T is in kelvin. The substance is cooled to 3 K in liquid helium, magnetized isothermally, and demagnetized adiabatically. If5 10 ;3 Jofheat is removed in the isothermal magnetization, what is the temperature at the end of the process? 7

Introduction. Statistical physics: microscopic foundation of thermodynamics degrees of freedom 2 3 state variables!

Introduction. Statistical physics: microscopic foundation of thermodynamics degrees of freedom 2 3 state variables! Introduction Thermodynamics: phenomenological description of equilibrium bulk properties of matter in terms of only a few state variables and thermodynamical laws. Statistical physics: microscopic foundation