Second Law of Thermodynamics, Gibbs’ Thermodynamics, and Relaxation Times of Thermodynamic Parameters

Abstract

The relationship between the second law of thermodynamics and Gibbs’ thermodynamics is discussed. The second law of thermodynamics is formulated more generally than Gibbs’ thermodynamics, which considers only strictly equilibrium values of thermodynamic functions. Gibbs’ approach generalizes the statistical mechanical theory of equilibrium for thermodynamic variables, except for the difference between the periods of relaxation of all thermodynamic parameters. For small systems, this approach consists of replacing the real physical nature of systems with the stratification of coexisting phases using a model with an interface of mobile phases in contact with a foreign (nonequilibrium) body. For solids, this results in confusion of concepts of the complete phase equilibrium of a system and the mechanical equilibrium of a deformed solid. These two problems are revealed using the molecular kinetic theory of condensed phases, ensuring a self-consistent description of three aggregate states and their interfaces. This theory allows the concepts of the times of the onset and completion of forming entropy in the considered system to be introduced. Allowing for experimental data on the ratios between the measured periods of relaxation for momentum, energy, and mass transfer processes in considering real processes not only ensures a solution to the two problems noted above; it also testifies to the redundancy of the Carathéodory mathematical theory to substantiate the introduction of entropy into multicomponent mixtures. A microscopic interpretation of the formation of entropy in closed systems is given that illustrates the essence of processes preceding the emergence of the reaction completeness parameter in de Donder and Prigogine approaches. Systems in which allowing for periods of relaxation alters existing theories are discussed.

Appendices

APPENDIX 1

Common knowledge of small systems in thermodynamics can be found in [1–9, 64, 65].

(1) For small phases, Laplace’s equation ${{p}_{{{\text{liq}}}}} = {{p}_{{{\text{vap}}}}} + 2\sigma {\text{/}}R$ associates pressure ${{p}_{{{\text{liq}}}}}$ inside a droplet with radius R. In vapor ${{p}_{{{\text{vap}}}}}$, $\sigma$ is ST.

(2) Kelvin’s equation relates pressure ${\text{ln}}\left( {P(r){\text{/}}{{P}_{0}}} \right) = - \frac{{\sigma {{V}_{L}}}}{{{{r}_{{{\text{pore}}}}}RT}}$ of saturated vapor Р(r) above the curved (with radius r_pore) and planar ${{P}_{0}}$ surfaces, where ${{V}_{L}}$ is the molar volume, R is the universal gas constant, and Т is temperature.

(3) The work of nucleation is expressed by Gibbs formula W = σA/3, where А is the surface area.

(4) In Gibbs’ phase rule n ≤ k + r (where k is the number of components, n is the number of phases, and r = 2 corresponds to Т and Р), the interface surface area serves as a third (r = 3) variable.

The analysis in [11] revealed a number of things according to the experimental data in (1).

(1) Laplace’s equation corresponds only to a mechanical subsystem in the absence of the chemical equilibrium; it implicitly means the presence of a foreign film between the coexisting phases and is incompatible with the chemical equilibrium condition of the entire system because contradicts experimental data (1).

(2) Kelvin’s equation is erroneous because it was obtained under the condition of the priority of mechanical over chemical equilibrium: ${{\tau }_{{{\text{imp}}}}} \gg {{\tau }_{{{\text{mas}}}}}$. This is forbidden in adsorption porometry and yields incorrect results when r_pore < 12–15 nm.

(3) The formula is erroneous. In deriving it, the initial concepts of phase and ST introduced by Gibbs himself for macrosystems were violated, and there must be no division by 3.

(4) The formula is erroneous because any phase boundary is a non-utonomous phase, due to the confusion of the mathematical formula for thermodynamic potentials and their physical meaning. The r value remains 2.

The general conclusion is that the analysis performed in [11] based on relationship (1) shows that thermodynamics cannot in principle be used for small systems and calculating the STs of interfaces.

APPENDIX 2

Analysis of principles of thermodynamics and the theory of elasticity of solids in [24, 25], performed using Gibbs’ representations of the phase equilibrium in heterogeneous systems, revealed their contradictions. (Below, we quote in italics the text from [23].)

(1) When there is deformation, the arrangement of molecules changes, and the body ceases to be in its original state of equilibrium. Forces therefore arise that tend to return the body to equilibrium. These internal forces that occur when a body is deformed are called internal stresses. If there is no deformation, there are no internal stresses.

The exclusion of internal stresses in a solid makes it impossible to describe internal deformations in thermodynamics. Initial postulate 1 of mechanics leads to the exclusion of the concept of nonequilibrium state of a solid in the absence of the external load. Gibbs used this postulate 1 due to the application of the concept of passive forces that replaces the concept of periods of relaxation of parameters in real systems.

(2) Deformed systems are assumed to be a generalization of thermodynamic equilibrium states corresponding to systems with no load.

Therefore, the all-round compression pressure Р is not the equilibrium pressure Р_e at the macroscopic boundary with the mobile phase at this temperature: Р = р + Р_e. The stress tensor р is introduced as an excess value to the equilibrium pressure!

(3) It is accepted that “the process of deformation occurs so slowly that thermodynamic equilibrium is established in the body at every moment, in accordance with external conditions. This assumption is almost always true in practice. The process will then be thermodynamically reversible” [23].

This formulation excludes the concept of chemical equilibrium in a system because a quasi-one-component homogeneous body is considered. It contradicts all real experimental values of periods of relaxation for solids τ_imp ⪡ τ_mass (1).

(4) “The expression for free energy F of a body as the function of the deformation tensor is obtained via series expansion in terms of the degree of smallness u_ik of deformations with no linear terms in u_ik power expansion of F. With an accuracy up to second order terms, the expression is obtained for the free energy of a deformed isotropic body: $F = {{F}_{0}} + \lambda u_{{ii}}^{2}{\text{/}}2 + \mu u_{{ik}}^{2}$, where $\lambda$ and $\mu$ are Lamé coefficients, and F₀ is the free energy of an undeformed body. Constant F₀, the free energy of an undeformed body, is usually omitted from consideration. F means only the free (elastic) energy of deformation: $F = \mu {{({{u}_{{ik}}} - {{\delta }_{{ik}}}{{u}_{{ll}}}{\text{/}}3)}^{2}} + Ku_{{ll}}^{2}{\text{/}}2$, where $K = \lambda + 2\mu {\text{/}}3$ is the bulk modulus of elasticity, and $\mu$ is the shear model.”

Constant F₀ is the free energy of an undeformed body and the main thermodynamic function depending on Т and density (without F₀, there is no equilibrium thermodynamics; it is completely excluded!). In practice, F₀ in mechanics can be any value, including nonequilibrium (which occurs in the overwhelming majority of cases). Its use means that the periods of relaxation of the mass transfer process are long, and such transfer does not occur during an experiment.

The essence of the mismatch is that any solid is considered to be equilibrium, though the free energy is excluded from consideration. We speak only about the elastic component of free energy, though condition τ ⪢ τ_mas is not satisfied.

APPENDIX 3

The Pauli equation is written for probability density matrix ρ(n, t) of finding weakly interacting particles in ensemble Δn of closely packed energy states (an analog of a phase cell in classical mechanics) in the form [38]

$$\frac{{\partial \rho (n,t)}}{{\partial t}} = \sum\limits_{n'} {{{\nu }_{{n'}}}{{P}_{{n'n}}}{\kern 1pt} \rho (n',t)} - \sum\limits_{n'} {{{\nu }_{n}}{{P}_{{nn'}}}\rho (n,t)} ,$$

(A3-1)

where ν_n is the number of states in Δn; P_nn', P_n'n are the probabilities of transitions in a unit of time from the state n to the state n', and vice versa. The first sum in the right side expresses the probability density increment due to transitions from n' cells to the n cell, and the second is a drop due to reverse transitions from the n cell to n' cells. The increment in time argument Δt corresponds to a time scale exceeding the period of one transition. This expression is written in the Markovian approximation, where the subsequent state of the system is completely determined by its current state at time t and does not depend on the occupancy of levels at times t ' < t. This structure of the equation describes the monotonic approach of the system to equilibrium.

If a monomolecular reaction in an inert gas thermostat with a constant concentration is considered an example, this equation is written as [38]

$$\frac{{\partial {{n}_{i}}}}{{\partial t}}\, = \,\sum\limits_j {\omega {{P}_{{ji}}}{{n}_{j}}(t)} \, - \,\sum\limits_j {\omega {{P}_{{ij}}}{{n}_{i}}(t)} \, - \,{{k}_{i}}{{n}_{i}}(t)\, + \,{{R}_{i}}(t),$$

(A3-2)

where n_i is the concentration of reacting molecules in the ith energy state at time t; P_ij is the probability of a transition when a reacting molecule collides with a thermostat molecule from the jth state to the ith energy state (related to one collision); P_ji is the probability of the reverse transition; k_i is the coefficient of the chemical reaction’s rate for the ith energy level; R_i(t) is the rate of external excitation of the ith level; and ω is the frequency of collisions.

Equations (A3-1) and (A3-2) are of a purely balanced nature and allow consideration from the common viewpoint of numerous transitions between levels, including chemical reactions. For chemical reactions, the behavior of vibrational degrees of freedom may be considered independent of the states of translational τ_tran, rotational τ_rot, and electronic degrees of freedom associated with dissociation τ_dis and ionization τ_ion(τ_tran ∼ τ_rot ⪡ τ_vib ⪡ τ_dis < τ_ion).

APPENDIX 4

Kinetic Equations in the LGM and Nonequilibrium Potentials

The physical reason for the nonequilibrium of states of solids is diffusion inhibition of the redistribution of components in their local volumes. The evolution of these processes is described by diffusion kinetic equations. The kinetic approach reflects the main property of the equilibrium state: its dynamic character. The LGM allows us to obtain kinetic equations on all time scales, and can be used for all aggregate states of compounds [45]. In the QCA, all probabilities of many-particle configurations describing the effect surrounding particles have on the rates of elementary processes are expressed through local concentrations $\theta _{f}^{i}$ and pair functions θ$_{{fg}}^{{ij}}$. The structure of a closed system of equations for unary and pair DFs corresponding to nearest neighbors is written as (α corresponds to the number of a stage in the multistage process)

$$\begin{gathered} \frac{d}{{dt}}\theta _{f}^{i} = \sum\limits_\alpha {P_{f}^{i}(\alpha )(U_{f}^{i}(\alpha ),U_{{fg}}^{{ij}}(\alpha ))} , \\ \frac{d}{{dt}}\theta _{{fg}}^{{ij}} = \sum\limits_\alpha {P_{{fg}}^{{ij}}(\alpha )(U_{f}^{i}(\alpha ),U_{{fg}}^{{ij}}(\alpha ))} . \\ \end{gathered} $$

(A4-1)

The right sides of the equations contain $P_{f}^{i}(\alpha )$ and $P_{{fg}}^{{ij}}(\alpha )$ contributions from the rates of elementary stages: $U_{f}^{i}(\alpha )$ = $U_{f}^{i}(\alpha \,|\,\theta _{f}^{i},\theta _{{fg}}^{{ij}})$ are the rates of elementary one-site processes i ↔ b at f-type sites; $U_{{fg}}^{{ij}}(\alpha )$ = $U_{{fg}}^{{ij}}(\alpha \,|\,\theta _{f}^{i},\theta _{{fg}}^{{in}})$ are the rates of elementary two-site processes i + j_α ↔ b + d_α on fg pairs of the neighboring sites. The structure of the equations was described in more detail in [45, 46, 84], and the right sides of (A4‑1) are explained in Appendix 5.

The presence of pair functions θ_ij allows us to consider the process’s prehistory determined not only by initial concentration distributions but the initial values of pair DFs as well. Pair functions play a key role in kinetic equations: they not only describe the effect of a process’s prehistory in dynamics but also ensure the self-consistency of the description of the equilibrium state and the rates of elementary stages in it [45, 46, 84].

Nonequilibrium Thermodynamic Potential

Knowing the solution to kinetic equations (A4-1) relative to $\theta _{f}^{i}$ and $\theta _{{fg}}^{{ij}}$, any thermodynamic functions depending on them as arguments can be calculated for each moment in time, including nonequilibrium thermodynamic potentials. Let us consider this problem using the example of Helmholtz energy F = E – TS, where Е is the energy of the system and S is its entropy. The energy and entropy of the system are expressed through the noted nonequilibrium functions (for simplicity, we consider a homogeneous system) as [11, 88]

$$E = N\left\{ {\sum\limits_{i = 1}^{s - 1} {\left[ {{{\theta }_{i}}{{\beta }^{{ - 1}}}\ln ({{a}_{i}}) + z\sum\limits_{j = 1}^{s - 1} {{{\varepsilon }_{{ij}}}{{\theta }_{{ij}}}} {\text{/}}2} \right]} } \right\},$$

$$S = N{{k}_{{\text{B}}}}\sum\limits_{i = A}^s {\left\{ {\mathop {{{\theta }_{i}}\ln ({{\theta }_{i}})}\limits_{_{{_{{_{{_{{}}}}}}}}}^{} } \right.} $$

(A4-2)

$$ + \left. {z{\text{/}}2\sum\limits_{j = A}^s {[{{\theta }_{{ij}}}(r)\ln {{\theta }_{{ij}}}(r) - {{\theta }_{i}}{{\theta }_{j}}\ln ({{\theta }_{i}}{{\theta }_{j}})]} } \right\}.$$

where ε is the parameter of nearest neighbor interaction; z is the number of neighboring sites; a_i is the constant of i-particle retention; 1 ≤ i ≤ s_c is the number of mixture components; and β = (k_BT)^–1.

Expressions (A4-2) thus describe the nonequilibrium Helmholtz energy at any time, including the equilibrium state of the system. When there are nonequilibrium states of solids, we can speak of nonequilibrium analogs of the equilibrium potentials of contacting solid and mobile phases, as was noted in [89]. The theoretical description at the microscopic level in the LGM shows that all thermodynamic potentials are expressed identically through unary and pair DFs, regardless of the system’s state. The distinction between equilibrium and dynamics is the way of describing unary and pair DFs themselves. In equilibrium, unary and pair DFs are related by equations with no time element. In dynamics, unary and pair DFs are explicitly related by kinetic equations through time argument (A4-1). Detailed equations of the diffusion type, which describe processes in solid matrices, were obtained in [11, 45, 85].

All equations for free energy and any other thermodynamic functions (including ST) that are expressed through unary and pair DFs under nonequilibrium conditions are thus expressions for calculating nonequilibrium analogs of equilibrium thermodynamic functions.

APPENDIX 5

Self-Consistency of Describing Equilibrium and the Dynamics of Elementary Stages

In kinetic theory, the problem of the self-consistency of expressions for the rates of elementary reactions (stages) and the equilibrium state of a reaction system is crucial for any phases. The essence of this statement is that by equating the expressions for reaction rates of any stage that proceeds in forward and backward directions, we should obtain equations describing the equilibrium distribution of molecules in this system.

This statement is well known from the example of the law of mass action for elementary stages of chemical reactions in the gas phase. The law of mass action is used for describing the rates of reactions, as was established empirically by Guldberg and Waage [90–92]. For reversible reactions we can write in general form $\sum\nolimits_i {{{\nu }_{i}}} \left[ {{{A}_{i}}} \right]\;\underset{{{{k}_{2}}}}{\overset{{{{k}_{1}}}}{\rightleftarrows}}\;\sum\nolimits_j {{{\nu }_{j}}} \left[ {{{A}_{j}}} \right]$, where the A_i and A_j in brackets denote different reacting particles, and ν_j and ν_i are the negative and positive values of the stoichiometric coefficient (the sign of which is determined by its place in the left or right sides of the equation); and k₁ and k₂ are the rate constants of forward and backward reactions. They are numerically equal to the rate of the reaction at unit concentrations of each of the reagents in the forward reaction.

The rate of the considered reaction within the law of mass action is written as $w = {{k}_{1}}\prod\nolimits_i {c_{i}^{{{{\nu }_{i}}}}} - {{k}_{2}}\prod\nolimits_j {c_{j}^{{{{\nu }_{j}}}}}$. In the equilibrium state the rate is zero (w = 0), from which it follows that the rate constants of forward and backward reactions are related with each other as ${{k}_{1}}{\text{/}}{{k}_{2}} = \prod\nolimits_j {c_{j}^{{{{\nu }_{j}}}}} {\text{/}}\prod\nolimits_i {c_{i}^{{{{\nu }_{i}}}}} = K$, where $K = {{k}_{1}}{\text{/}}{{k}_{2}}$ is the equilibrium constant of this stage. Thus, the equilibrium constant can be found differently: either by equilibrium or kinetic measurements. In this sense, we can assume that empirical rules for the description of reaction rates and equilibrium in the considered system provide the self-consistent description of this process at any time intervals, including the final deviations of the equilibrium state as well as the limiting equilibrium case itself.

However, the law of mass action assumes that there is the equilibrium distribution of molecules in the reaction system and that the chemical transformation stage is limiting, and there are no: (1) diffusion transport at the macroscopic level (uniform distribution over the macrovolume), (2) effect of external fields, (3) diffusion-controlled reaction at the molecular level, (4) effect of intermolecular interactions, and (5) a fraction of particles reacting in a unit time is so small that it does not distort the equilibrium distribution of molecules in the system.

For the majority of real processes it is needed to take into account the nonideality of the reaction system, and the kinetic theory in the LGM gave expressions for the rates of elementary stages in Eq. (A4-1), having expressed their summands $U_{f}^{i}(\alpha )$ (rates of elementary one-site processes i ↔ b (here h ∈ z_f)) and $U_{{fg}}^{{ij}}(\alpha )$ (rates of elementary two-site processes i + j_α ↔ b + d_α (h ∈ $z_{f}^{*}$)) through unary and pair DFs.

All rates of elementary stages $U_{f}^{i}(\alpha )$ and $U_{{fg}}^{{ij}}(\alpha )$ are calculated using the theory of absolute reaction rates for nonideal reaction systems, which are written with regard to the interparticle interaction in QCA [11, 45, 85]. Properties of the activated complex in the theory of absolute reaction rates for nonideal reaction systems depend on the interaction between particles in the transition and ground states. Apart from ${{\varepsilon }_{{ij}}}$ values in the ground state, it requires the knowledge of particle interactions in the transition state $\varepsilon _{{ij}}^{*}$. The formulas for rates $U_{f}^{i}(\alpha )$ and $U_{{fg}}^{{ij}}(\alpha )$ thus depend on both ${{\varepsilon }_{{ij}}}$ and energy $\varepsilon _{{ij}}^{*}$.

As an example, we give the expressions for rates $U_{{fg}}^{{iV}}(\alpha )$ of diffusion displacement when calculating the interactions of nearest neighbors:

$$U_{{fg}}^{{iV}}(\alpha ) = K_{{fg}}^{{ij}}(\alpha ){\kern 1pt} \theta _{{fh}}^{{iV}}\prod\limits_{\eta \in z_{f}^{*}} {S_{{f\eta }}^{i}} {\kern 1pt} \prod\limits_{\chi \in z_{g}^{*}} {S_{{g\chi }}^{V}} ,$$

(A5-1)

where $K_{{fh}}^{{ij}}(\alpha )$ is the rate constant of the elementary migration stage, and

$S_{{f\eta }}^{i} =$ $\sum\limits_{m = 1}^s {\theta _{{f\eta }}^{{im}}} \exp [\beta (\varepsilon _{{f\eta }}^{{im*}} - \varepsilon _{{f\eta }}^{{im}})]{\text{/}}\theta _{f}^{i}$,

$S_{{h\chi }}^{V} = \sum\limits_{m = 1}^s {\theta _{{h\chi }}^{{Vm}}} \exp (\beta \varepsilon _{{h\chi }}^{{im*}}){\text{/}}\theta _{f}^{V}$

are the cofactors of the function of nonideality of the jump rate. The product in (A5-1) is taken over the neighboring sites η (for central site f ) and χ (for central site g), excluding the bond fg, which is marked by the asterisk in $z_{f}^{*}$ and $z_{g}^{*}$. When the correlation effects are ignored, functions $S_{{f\eta }}^{i}$ are rewritten as $S_{{f\eta }}^{i} =$ $\sum\nolimits_{m = 1}^s {\theta _{\eta }^{m}} \exp [\beta (\varepsilon _{{f\eta }}^{{im*}} - \varepsilon _{{f\eta }}^{{im}})]$ in the random approximation and $S_{{f\eta }}^{i} = \sum\nolimits_{m = 1}^s {\exp [\beta (\varepsilon _{{f\eta }}^{{im*}} - \varepsilon _{{f\eta }}^{{im}})\theta _{\eta }^{m}]}$ in the mean field approximation.

A qualitative distinction of expressions (A5-1) for the rates is their dependence on pair DFs $\theta _{{f\eta }}^{{im}}$. They reflect the effects neighboring molecules have on the function of nonideality. Their presence also allows us to consider the prehistory of nonequilibrium states of the system. Within the limit of long times, the solution to kinetic equations (A4-1) in a closed system necessarily becomes equations for the equilibrium distribution of components.

In the LGM for any condensed phases, the requirements were found to satisfy the conditions for a self-consistent description of reaction rates and equilibrium of the system [11, 45, 46]. It was shown that the theory ensures a self-consistent description of dynamics and equilibrium on all spatial scales for any densities, temperatures, intensities of lateral interactions and external fields only when we allow for the effects of correlation, at least for short-range order.

Otherwise, the condition of self-consistency is violated. The molecular/mean field approximation, the random approximation, and the density functional approach therefore cannot be used to describe kinetic processes in dense phases. Equating their reaction rates in opposite directions, we do not obtain similar equations for isotherms of them in the same approximations.