Examples of coupled irreversible processes like the thermoelectric phenomena, the transference phenomena in electrolytes and heat conduction in an anisotropic medium are considered. For certain cases of such interaction reciprocal relations have been deduced by earlier writers, e,g. , Thomson's theory of thermoelectric phenomena and Helmholtz' theory for the e.m. f. of electrolytic cells with liquid junction. These earlier derivations may be classed as quasi-thermodynamic; in fact, Thomson himself pointed out that his argument was incomplete, and that: his relation ought to be established on an experimental basis. A general class of such relations will be derived by a new theoretical treatment from the principle of microscopic reversibility. ()$1-2.) The analogy with a chemical monomolecular triangle reaction is discussed; in this case a a simple kinetic consideration assuming microscopic reversibility yields a reciprocal relation that is not necessary for fulfilling the requirements of thermodynamics ($3). Reciprocal relations for heat conduction in an anisotropic medium are derived from the assumption of microscopic reversibility, applied to fluctuations. ($4.) The reciprocal relations can be expressed in terms of a potential, the dissipation-function. Lord Rayleigh's "principle of the least dissipation of energy" is generalized to include the case of anisotropic heat conduction. A further generalization is announced. ($5.} The conditions for stationary flow are formulated; the connection with earlier quasithermodynamic theories is discussed. ($6.) The principle of dynamical reversibility does not apply when (external) magnetic fields or Coriolis forces are present, and the reciprocal relations break down.
A general reciprocal relation, applicable to transport processes such as the conduction of heat and electricity, and diffusion, is derived from the assumption of microscopic reversibility. In the derivation, certain average products of fluctuations are considered. As a consequence of the general relation S=k log W between entropy and probability, different (coupled) irreversible processes must be compared in terms of entropy changes. If the displacement from thermodynamic equilibrium is described by a set of variables a&, , a, and the relations between the rates u&, , a and the "forces" BS/dn1, , 8S/du are linear, there exists a quadratic dissipationfunction,
Introdzution. The shapes of colloidal particles are often reasonably compact, so that no diameter greatly exceeds the cube root of the volume of the particle. On the other hand, we know many coiloids whose particles are greatly extended into sheets (bentonite), rods (tobacco virus), or flexible chains (myosin, various Iinear polymers).In some instances, a t least, solutions of such highly anisometric particles are known to exhibit remarkably great deviations from Raoult's law, even to the extent that an anisotropic phase may separate from a solution in which the particles themselves occupy but one or two per cent of the total volume (tobacco virus, bentonite). We shall show in what follows how such results may arise from electrostatic repulsion between highly anisometric particles.Most colloids in aqueous solution owe their stability more or less to electric charges, so that each particle will repel others before they come into actual contact, and effectively claim for itself a greater volume than what it actuaily occupies. Thus, we can understand that colloids in general are apt to exhibit considerable deviations from Raoult's law and that crystalline phases retaining a fair proportion of solvent may separate from concentrated solutions. However, if we tentatively increase the known size of the particles by the known range of the electric forces and multiply the resulting volume by four in order to compute the effective van der Waal's co-volume, we have not nearly enough to explain why a solution of 2 per cent tobacco virus in 0.005 normal NaCZ forms two phases. Some care is needed when we apply the general principles of statistical thermodynamics to solutions of colloidal particles. On one hand, any force acting on a particle of whatever size is important as soon as the work of the force is comparable to kT. On the other hand, the presence of one colloidal particle will usually affect the free energy of dilution of the electrolyte present by a large multiple of kT. This difficulty must be circumvented by a11 theories and experiments pertaining to the distribution of colloidal particles. One suitable piece of experimental apparatus is an osmometer whose membrane is impermeable to the colloidal particles, but permeable to a11 small molecules and ions of the electrolytic çolvent. The osmotic pressure measured across çuch a membrane will be exactly proportional to the number of particles if the solution behaves like an ideal gas. The analogy can be extended to real gases and real soiutions, whereby the gas pressure still corresponds to osmotic pressure.
7General Kinetic Theory and Conventions.
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ANNALS NEW YORK ACADEMY OF SCIENCESThe imperfection of an ideal gas can be computed when we know the forces between the molecules for every coníiguration. For that purpose, we have to evaluate the integral
B(T) = /ebuikT d r / N !where u stands for the potential of the forces and d.r denotes a volume element in configuration-space. The free energy of the gas in terms of this integral iswhere the additional function po(T) ...
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