Читать книгу Non-equilibrium Thermodynamics of Heterogeneous Systems - Signe Kjelstrup - Страница 36

The purpose of this appendix is to show how the description of de Groot and Mazur is compatible with the one used in this book. The analysis uses as starting point Chapter XIV in de Groot and Mazur [23]. We restrict ourselves to the case that the magnetic field and the magnetization are zero. The system is also electroneutral. As a final simplification, we shall assume the system to be in mechanical equilibrium. The last simplification leads to a form of the first law appropriate for the systems described in this book, Eq. (4.7). Vectors shall be indicated by bold letters in this appendix, meaning that they have an arbitrary direction, rather than being restricted to the x-direction as in most of the book.

For a system in which there are no magnetic field or magnetization, the Maxwell equations (in SI units) can be written as

(4.29)

where ε₀ is the dielectric constant of vacuum, E the electric field, j the electric current density and P the polarization density (in C/m²).

For the molar density of component j, we have the following conservation equation:

(4.30)

where vj and Jj are the velocity and the molar flux of component j. The potential energy density satisfies

(4.31)

where ψj is the potential energy per mole of component j, and Fj = − grad ψj is the external force acting on component j, due to this potential. For the energy of the electric field, we have

(4.32)

where we used the third Maxwell equation, Eq. (4.29c). Adding the last two equations results in

(4.33)

Integrating this relation over a time-independent volume V gives

(4.34)

where O is the surface of the volume, dr ≡ dxdydz is the volume element and subscript n indicates the component of a vector normal to the surface. This expression shows that the sum of the potential and electromagnetic energies changes due to two terms. The first gives the potential energy losses from mass flow through the surface. The second term describes the energy conversion inside the volume. The above expression for the time rate of change of the sum of the potential and electric field energy follows from conservation of mass and the Maxwell equations. No energy conservation has yet been used.

The total energy density per unit of volume e also contains the internal energy density per unit of volume u. We therefore have

(4.35)

In the description in this book, we use fluxes in the laboratory frame of reference. We do not use fluxes in the barycentric frame of reference. Therefore, we do not subtract kinetic energy from e in order to get u. As the total energy is conserved, one has

(4.36)

where Je is the total energy flux. In order to proceed, we need an expression for the total heat flux.

Consider the example of an electrolyte solution of NaCl. The salt is completely dissociated into ions and due to the reactions at the electrode surfaces the chloride ion carries the charge. The system is electroneutral and as a consequence

(4.37)

The time rate of change of these concentrations is given by

(4.38)

The electric current density is

(4.39)

where F is Faraday’s constant. Together with Eq. (4.38), it follows that

(4.40)

This also follows from the electroneutrality condition. Given that the sodium ions only move as part of the salt, it follows that the ion fluxes are given by

(4.41)

in terms of the salt flux and the electric current density.

The energy flux in the laboratory frame of reference consists of potential energy which flows along with the molar fluxes, energy of the charge carrier which is carried by the electric current density and of the total heat flux in the laboratory frame of reference

(4.42)

In order to see how the first law of thermodynamics reads for the internal energy, we subtract Eq. (4.33) from Eq. (4.36) and use Eqs. (4.35) and (4.42). This results in

(4.43)

The electric field is equal to minus the gradient of the Maxwell potential

(4.44)

The ϕ-potential we use in Eq. (4.7) is equal to

(4.45)

This is equal to minus the electrochemical potential of chloride ions divided by the charge of a mole of chloride ions. Electrochemical potentials were first defined by Guggenheim [66, 98]. Using this potential, Eq. (4.43) can be written as

(4.46)

Integrating this relation over a time-independent volume V gives

(4.47)

This is a more familiar form of the first law. The internal energy changes due to a total heat flux through the surface. The volume integral contains work added to the system. The first term is the electric work added, and the second is the work done by the displacement current. In the absence of external forces, and with one-dimensional flow conditions, Eq. (4.46) reduces to Eq. (4.7). Though we restricted the analysis to an example, the result is generally valid. We also refer to Sec. 10.6 in this context.

To further compare our analysis with that of de Groot and Mazur, we verify that the barycentric velocity in a description using charged components differs from that in a description using uncharged components, i.e. when the system is electroneutral. We do this for the example above, NaCl in water. The barycentric velocity in the description using charged components is

(4.48)

The electric current does not only carry charge, but also mass. This gives a difference between the barycentric velocities calculated using only neutral components and the one calculated using charged components. As we do not use fluxes in the barycentric frame of reference in this book, this does not lead to any complications in our analysis.

Also the measurable heat flux is different in the two descriptions. We have

(4.49)

The difference is the enthalpy of the charge carrier carried along by the electric current. We note that Jq and Js, both in the laboratory frame of reference, are the same whether one uses charged or uncharged components. In view of the fact that h_Cl− is not measurable, it is appropriate to call J_q^{′,uncharged} as measurable. De Groot and Mazur named J_q^′,charged the reduced heat flux.

Finally, we show, again for the same example, that both descriptions give the same entropy production. To simplify the system further we neglect the contribution due to the displacement current. This is the case considered by de Groot and Mazur [23] in Chapter XIII. On p. 344 in Eq. (40), they give the entropy production in a system with no magnetic field,

(4.50)

In the analysis of de Groot and Mazur, all fluxes are in the barycentric frame of reference. For a system in mechanical equilibrium, one may replace all of them by their value in the laboratory frame of reference. The difference is a term which is zero, according to Gibbs–Duhem’s equation. The fact that the expression is correct in an arbitrary frame of reference if the system is in mechanical equilibrium, is called Prigogine’s theorem [18]. Using Eq. (4.41) for the ion fluxes, we obtain

(4.51)

This is the expression we gave in Eq. (4.14) for the case that there is no polarization and no chemical reaction.