10. Principles of Thermodynamics - Electro-chemical systems#

In this section:

In next sections:

10.1. Chemical reactions#

\[a A + b B \rightleftharpoons c C + d D \ ,\]

or

(10.1)#\[\nu_{R_i} R_i \rightleftharpoons \nu_{P_j} P_j\]

10.2. First principle - energy balance#

Energy balance

Total energy balance

\[d E^{tot} = \delta L^{ext} + \delta Q^{ext} \ .\]

Macroscopic kinetic energy balance (theorem, from mechanics)

\[d K = \delta L^{ext} + \delta L^{int} \ .\]

Internal energy \(E := E^{tot} - K\)

\[d E = - \delta L^{int} + \delta Q^{ext} \ .\]
Internal energy, intensive and extensive variables
\[d E = - \delta L^{int,rev} + \underbrace{\delta^+ D + \delta Q^{ext}}_{=:T dS}\]

As \(dE\) is a state function, and \(\delta L^{int,rev}\) must be reversible by definition, the remaining terms must be reversible as well and completing the exact differential of the energy as a state function. The internal nature may have different natures, e.g. mechanical, elelctrical, magnetic, chemical

\[\delta L^{int,rev} = \delta L^{int,mech} + \delta L^{int,el} + \delta L^{int,magn} + \delta L^{int,el} + \dots\]

but all of these works can be written as the product of a intensive variable \(\mathbf{F}\) of the system and the variation of a extensive variable \(\mathbf{X}\), formally

\[\delta L^{int,rev} = \mathbf{F} \cdot d \mathbf{X} \ ,\]

  • mechanical work, depends on the nature of the system. As an example, for a gas \(\delta L^{int,rev} = - P d V\), or for an elastic solid \(\delta L^{int,rev} = - V \sigma : d \varepsilon\)

  • electrical work, being \(V\) the (uniform) electrical potential of the system, and \(Q\) its electrical charge (see Elecetrostatic energy: systems with uniform potential)

    \[- V d Q\]

  • chemical work, with \(\mu_i\) the molar chemical potential \(i\) of the substance (the amount of internal energy added to the system by a change in the number of moles of the \(i\) substance in the system)

    \[\mu_i d N_i\]

todo See and uniform the treatment in First principle, Second principle, Gibbs formalization, Thermodynamic potentials

Using \(\mathbf{X}, S\) as independent variables,

\[d E = \mathbf{F}(\mathbf{X}, S) \cdot d \mathbf{X} + T(\mathbf{X}, S) d S \ .\]

It the energy of the system is a order-1 homogeneous function of the extensive variables of the system, \(E(\lambda \mathbf{Y}) = \lambda E(\mathbf{Y})\)1, Euler theorem (Theorem 3.1) gives

\[E = T S + \mathbf{F} \cdot \mathbf{X} \ .\]

Explicitly writing the number of moles of the substances of the system out fror the extensive variables \(\mathbf{X}\), \(\mathbf{X} \rightarrow \left( \mathbf{X}, \mathbf{N} \right)\), and the chemical potentials out from the generalized intensive variables \(\mathbf{F} \rightarrow \left( \mathbf{F}, \boldsymbol\mu \right)\), the expression of the internal energy and its differential reads

\[\begin{split}\begin{aligned} E & = T S + \boldsymbol\mu \cdot \mathbf{N} + \mathbf{F} \cdot \mathbf{X} \\ dE & = T dS + \boldsymbol\mu \cdot d\mathbf{N} + \mathbf{F} \cdot d\mathbf{X} \ . \end{aligned}\end{split}\]

10.3. Law of Mass Action#

The law of mass action states that the forward velocity \(v_f\) (\(R \rightarrow P\)) and backward velocity \(v_b\) (\(R \leftarrow P\)) of reaction (10.1) reads

\[\begin{split}\begin{aligned} v_f & := -\frac{1}{\nu_{R_k}}\frac{d [ R_k ]_f}{dt} := \frac{1}{\nu_{P_k}}\frac{d [ P_k ]_f}{dt} = k_f \prod_i [ R_i ]^{\alpha_{R_i}} \\ v_r & := \frac{1}{\nu_{R_k}}\frac{d [ R_k ]_r}{dt} := -\frac{1}{\nu_{P_k}}\frac{d [ P_k ]_r}{dt} = k_r \prod_i [ P_i ]^{\alpha_{P_i}} \\ \end{aligned}\end{split}\]

Taking \(R \rightarrow P\) as the positive direction of the reaction, the velocity of the reaction is defined as \(v = v_r - v_f\), and the rate of change of the concentrations thus reads

\[\begin{split}\begin{aligned} \frac{d [ R_k ]}{dt} = \frac{d [ R_k ]_f}{dt} + \frac{d [ R_k ]_r}{dt} = \nu_{R_k} \left( v_r - v_f \right) = \nu_{R_k} \, v \\ \frac{d [ P_k ]}{dt} = \frac{d [ P_k ]_f}{dt} + \frac{d [ P_k ]_r}{dt} = \nu_{P_k} \left( v_f - v_r \right) = \nu_{P_k} \, v \ . \end{aligned}\end{split}\]

  • Equilibrium is reached when \(v_r = v_f\), and thus the rate of change of concentration of the substances is zero.

  • Extent of reaction \(\xi\), so that \(d \xi = v dt\).

Constant \(k_r\) and \(k_f\) usually can be written as

\[k = A \exp \left( - \frac{E}{kT} \right) \ ,\]

being \(E\) the activation energy, \(k\) Boltzmann constant, \(T\) the temperature, and \(A\) a constant summarizing all the other conditions for the reaction to occur.

References

1

Not all the systems are made by sub-systems that combine “extensively”: it’s likely to be impossible to write the energy of these systems as a homogeneous function of the extensive variables of the system.