Abstract
In this chapter, we will discuss the equations describing the equilibrium composition of a system based on classical thermodynamics, limiting our presentation to the essential ingredients of the theory. The axiomatic approach (Callen 1985) will be used, leaving an alternative derivation in Chap. 3 in the framework of statistical thermodynamics.
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Notes
- 1.
See (Capitelli and Giordano 2002; Capitelli et al. 2005b) where the equivalence of the use of different potentials for describing chemical equilibrium is discussed.
- 2.
This property is a direct consequence of the mass conservation law.
- 3.
The following results will be closely examined in Chap. 3.
- 4.
It must be noted that in this example the advancement degree defined in (1.27) is related to the dissociation degree by the simple relation α d = {ξ} _ {n 0}.
- 5.
This assumption is valid only for ideal gases.
- 6.
This assumption, which will be eliminated in the next chapters, disregards the internal structure of atoms and molecules as well as differences in the translational energies of products and reactants.
- 7.
- 8.
As for the case of dissociation, the assumption \(\Delta {\bar{H}}^{0} \approx I\) disregards the internal structure of atoms as well as differences in the translational energies of products and reactants. Debye–Hückel corrections, which introduces the lowering of ionization potential are also disregarded in (1.57).
- 9.
In general, the system of equilibrium equations is very complex and require computer programs to be solved.
- 10.
Many of the quantities defined in this section will be recalculated in the Chap. 3 in the framework of statistical thermodynamics.
- 11.
Following the Boltzmann statistical distribution.
- 12.
This assumption is true only for an ideal gas, failing however when real effects are important.
- 13.
It must be noted that the total mass of a reacting mixture is constant. This property can be demonstrated in general considering that the atomic constituents of the mixtures are rearranged during the chemical processes, remaining however in the system. A demonstration for the dissociation process is given in note 19.
- 14.
See note 12.
- 15.
The value of the formation enthalpy depends on which species are considered as reference for energy.
- 16.
The energy equipartition considers two conditions: each degree of freedom is independent from the others, the temperature is sufficiently high that the quantum behavior of internal levels is negligible. The use of the energy equipartition theorem, while correctly describing translational and rotational degrees of freedom, strongly overestimates the contribution of vibrational energy at low temperature, as discussed in Chap. 5.
- 17.
It should be noted that the vibrational degree of freedom gives to enthalpy a term RT due to kinetic and potential energy contribution.
- 18.
In general, the number of vibrational mode is given by 3N atoms − r − 3 being r the number of rotational axis.
- 19.
The total mass as a function of the dissociation degree is given by\(M({\alpha }_{d}) = {n}_{0}(1 - {\alpha }_{d}){M}_{{A}_{2}} + 2{\alpha }_{d}{n}_{0}{M}_{A}\). Considering that M A = M A 2 ∕ 2we have \(M({\alpha }_{d}) = {n}_{0}(1 - {\alpha }_{d} + 2{\alpha }_{d}/2){M}_{{A}_{2}} = {n}_{0}{M}_{{A}_{2}}\).
- 20.
This choice is done only on the basis of simplicity of the equations. Under global equilibrium, all the potentials are equivalent.
- 21.
For an ideal, non-reacting gas, it is \({z}_{\gamma } = \frac{\rho } {P}{\left (\!\frac{\partial P} {\partial \rho } \!\right )}_{T} = 1\). In any case, it is ≾ 1, therefore often the isentropic coefficient is written as \(\gamma = \frac{{c}_{p}} {{c}_{v}}\) (Anderson 2000).
- 22.
The frozen isentropic coefficient is used to calculate the frozen speed of sound (Anderson 2000), which is used for non-equilibrium flows, where the chemical composition is calculated solving a master equation (Colonna and Capitelli 2001a,b).
- 23.
From the theorem on derivatives \({\left (\!\frac{\partial P} {\partial \rho } \!\right )}_{T} ={ \left (\! \frac{\partial \rho } {\partial P}\!\right )}_{T}^{-1}\).
- 24.
The coefficients can be related to interaction potential between the particle in the mixture as will be discussed in the Chap. 7.
- 25.
It should be noted that the ∑ n = 0 ∞ b n = (1 − b) − 1.
- 26.
The parameter b is called co-volume and should be the minimum volume that can be occupied by one mole of molecules. Such coefficient is independent of the temperature. On the other hand, a is related to the attractive force and it depends on the temperature, with the asymptotic relation lim T → ∞ a = 0. From (1.147), we can say that the Van der Waals equation neglect the contribution of the attractive forces to higher order virial coefficients.
- 27.
In this section, we consider \(\dot{X} ={ \left (\frac{\partial X} {\partial T} \right )}_{{n}_{i}}\).
- 28.
For a general function F[x, y(x)], the derivative {dF} {dx} = {∂F} {∂x} y + {∂F} {∂y} x {dy} {dx}.
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Capitelli, M., Colonna, G., D’Angola, A. (2012). Classical Thermodynamics. In: Fundamental Aspects of Plasma Chemical Physics. Springer Series on Atomic, Optical, and Plasma Physics, vol 66. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8182-0_1
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