Non-equilibrium Thermodynamics of Heterogeneous Systems. Signe KjelstrupЧитать онлайн книгу.
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
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(4.32) |
where we used the third Maxwell equation, Eq. (4.29c). Adding the last two equations results in
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(4.33) |
Integrating this relation over a time-independent volume V gives
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(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
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(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
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(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
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(4.37) |
The time rate of change of these concentrations is given by
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(4.38) |
The electric current density is
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(4.39) |
where F is Faraday’s constant. Together with Eq. (4.38), it follows that
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(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
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(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
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(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
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(4.43) |
The electric field is equal to minus the gradient of the Maxwell potential
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(4.44) |
The ϕ-potential we use in Eq. (4.7) is equal to
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(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
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(4.46) |
Integrating this relation over a time-independent volume V gives
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(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
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