Some Basic Concepts and Definitions
In a macroscopic system characterized by a specific temperature, pressure, and Gibbs free energy, spontaneous reactions occur as the system moves towards equilibrium Under constant temperature (T) and external pressure (p), the change in Gibbs free energy serves as a crucial criterion for these reactions, ensuring that the process aligns with the principles of thermodynamics.
For a reaction at equilibrium, a reversible process, the necessary and sufficient condition is
Another important property of ∆Gis that it is a Lyapunov function in that it obeys (1.1) and (1.3) d∆G dt ≥0 (1.3) wheret is time, until equilibrium is reached Then (1.2) and (1.4) hold d∆G dt = 0 (1.4)
A Lyapunov function indicates the direction of motion of the system in time (there will be more on Lyapunov functions later).
Thermodynamics plays a crucial role in predicting the maximum work achievable by a system, such as during a chemical reaction For systems maintained at constant temperature and pressure, the change in Gibbs free energy indicates the maximum work available, excluding any pressure-volume work.
Systems that are not in equilibrium can exist in various transient states, either moving towards equilibrium or settling into a non-equilibrium stationary state Additionally, they may exhibit more complex dynamic behaviors, such as periodic oscillations of chemical species, known as limit cycles, or chaotic behavior An illustrative example can clarify these two initial conditions.
In a chemical reaction sequence represented as A ⇔ X ⇔ B, the forward and backward rate coefficients for the first reaction (A ⇔ X) are denoted as k₁ and k₂, while k₃ and k₄ represent the corresponding rates for the second reaction In this context, A serves as the reactant, X functions as the intermediate, and B is the final product For simplicity, it is assumed that the chemical species behave as ideal gases, with the reactions taking place in a schematic apparatus at a constant temperature.
We could equally well choose concentrations of chemical species in ideal solutions, and shall do so later Now we treat several cases:
1 The pressuresp A andp B are set at values such that their ratio equals the equilibrium constantK p B p A =K (1.6)
1.1 Some Basic Concepts and Definitions 5
Fig 1.1 Schematic diagram of two-piston model The reaction compartment (II) is separated from a reservoir of species A (I) by a membrane permeable only to
A reservoir of species B is separated by a membrane that only allows B to pass through, while the pressures of both A and B are maintained by constant external forces on the pistons To facilitate the reactions at significant rates, catalysts C and C are exclusively located in region II.
If the whole system is at equilibrium then the concentration ofX is
X eq = k 1 k 2 A=k 4 k 3 B, (1.7) andKcan be expressed in terms of the ratio of rate coefficients
At equilibrium ∆G = 0, or in terms of the chemical potentials à A à B =à X
2 The pressures ofAandBare set as in case 1 If the initial concentration of
Xis larger thanX eq then a transient decrease ofX occurs untilX =X eq For the transient process of the system towards equilibrium ∆G of the system is negative, ∆G 0 (5.34) that is larger than zero, and henceΦ is a minimum at the stable stationary state The derivative with respect to time is d dtΦ[{X i , Y i }] i dX i dt {à(X i )−à(X i S )}
The right-hand side of equation (5.34) is negative semidefinite, indicating that the system moves towards the minimum of the function Φ, which corresponds to a stable stationary state Consequently, Φ serves as a Liapunov function for the system Additionally, Φ meets the criteria for the stationary solution of the master equation in the thermodynamic limit These characteristics ensure that Φ provides the necessary and sufficient conditions for the existence and stability of stationary states.
Non-Linear Reaction Mechanisms
This article explores chemical reaction systems featuring autocatalytic steps, where the kinetic reaction terms are non-linear functions of the concentrations of intermediate species We consider a reaction mechanism that allows for changes in the number of X and Y molecules by ±1 or 0 during each elementary reaction step For each set of concentrations (X i, Y i), we can create a thermodynamically and kinetically equivalent system, with a unique mapping from the non-linear system to its linear counterpart The linear equivalent system is defined as presented in (5.26), with coefficients that adhere to the relationships outlined in Table 5.1.
Table 5.1 Relations of the terms in the rate equations of a non-linear system, (4.11) and (4.12), to the kinetically equivalent linear system for each set of variables, see (5.26) From (1)
Linear Non-linear a xi = k 1 AX i m−1 b xi = k 4 X i q−1 Y i r−1 c xi = k 2 X i m−1 +k 3 X i q−1 Y i r−1 a yi = k 6 BY i n−1 b yi = k 3 X i q−1 Y i r−1 c yi = k 5 Y i n−1 +k 4 X i q−1 Y i r−1
The reference state (X i ∗ , Y i ∗ ) is defined by the equations
As the system approaches a stationary state the starred variables approach their values of the stationary state For the Selkov model the stationary state of the linear equivalent system is
, (5.38) where the prime indicates the corresponding value of the linear equivalent system.
The differential excess work for the equivalent linear system is given in (5.29), and the instantaneous differential excess work for the non-linear sys- tem is dΦ˜ i ={à(X i )−à(X i ∗ )}dX i +{à(Y i )−à(Y i ∗ )}dY i
In equation (5.39), the curl on the left-hand side denotes an inexact differential, while the terms in brackets represent the species-specific driving forces for species X and Y towards their reference states (X ∗ i, Y ∗ i) To determine the function Φ for non-linear systems, one can replace the superscript 'S', indicating the stationary state of the linear system, with a '∗' as defined in equations (5.36) and (5.37), with the corresponding relations outlined in Table 5.1 This leads to the calculation of the total excess work.
5.1 Reaction–Diffusion Systems with Two Intermediates 49 Φ i
The function Φ is defined by the chosen integration path, whether it be the deterministic path or its reverse At the stationary state, Φ equals zero, and its first derivatives at this state are also zero.
Since the non-linear system is indistinguishable from the instantaneously equivalent linear system, we have dΦ dt non−linear=dΦ dt linear
The right-hand side of the equation is negative semi-definite, leading to the result that the rate of change of Φ over time is non-linear and less than or equal to zero at every state (X i, Y i), with equality only at the stationary state Consequently, Φ decreases over time and reaches a minimum at the stationary state This establishes Φ as a Lyapunov function, which is essential for determining the necessary and sufficient conditions for the existence and stability of stationary states in the systems being analyzed.
Relative Stability of Two Stable Stationary States
In a reaction system with two stable stationary states, understanding their relative stability under specific external constraints is essential For example, at one atmosphere of pressure and temperatures below 100 °C, liquid water is more stable than its vapor counterpart, as indicated by the lower Gibbs free energy of liquid water When both phases are in contact, the vapor condenses into liquid, demonstrating the stability of the liquid phase Conversely, at temperatures exceeding 100 °C, vapor becomes the more stable state, with its Gibbs free energy being lower than that of liquid water This principle of relative stability applies similarly to reaction diffusion systems with two stable stationary states.
Fig 5.2 Schematic apparatus for determining relative stability of two stable sta- tionary states of a reaction–diffusion system For description see text From [1]
Consider a schematic setup involving two semi-infinite tubes, each containing a different stable stationary state: one designated as state 1 and the other as state 3 Both tubes are subjected to identical external conditions, including temperature, pressure, and species concentration Initially, these tubes are aligned lengthwise with a partition separating them.
After removing the partition, a reaction-diffusion front may develop in the interphase region and advance into a less stable state During this transient period, the front's propagation occurs until external constraints reach equistability, at which point the velocity of the reaction-diffusion front becomes zero.
The concept of excess work, as discussed in previous chapters, serves as a criterion for equistability, similar to how Gibbs free energy functions for equilibrium systems Once a reaction diffusion front is established, we can compute the excess work required to create that front from phase 1 and also from phase 3.
If these two excess works are equal than we expect equistability and zero velocity of front propagation If the excess work to form the front from phase
In order to ensure phase one is more stable than phase three, we observe that the value of 1 is insufficient for forming the front from phase three To conduct this analysis, we divide the interphase region into N boxes of a specified length L, applying boundary conditions for the concentrations on the left side of the interphase.
5.1 Reaction–Diffusion Systems with Two Intermediates 51
The plot of concentration versus position z illustrates the initial concentration profile, with stationary state 1 occupying the negative z region and stationary state 3 in the positive z region The interface is marked by a dotted line, indicating the transition between these two states Initially, the interphase region is set to the concentrations of phases 1 and 3 As diffusion and reaction processes unfold, each segment follows a distinct deterministic path from either stationary state 1 or 3 towards a stable front condition These paths are derived by numerically integrating a set of ordinary differential equations that describe the changes in concentration over time for both states.
Along the calculated trajectories we evaluate (X i ∗ , Y i ∗ ) from (5.35) and (5.36) Then we can obtain the excess work from the last line of (5.40)
We can split the expression on the rhs into two parts
The sign of ∆Φ determines the prediction of the theory of the direction of propagation of the interface: if we have
∆Φ(1→St.Fr.)>∆Φ(3→St.Fr.), (5.47) then 3 is the more stable phase and the interface region moves in the direction which annihilates phase 1 For the opposite case we have
∆Φ(1→St.Fr.)