1 Introduction

The analysis of the dynamics of complex chemical systems does constitute a problem common to many areas of research. Basic reactions, such as proton transfer [1], electron transfer [2, 3], and metal complex formation [4, 5] were the first whose mechanism was elucidated grace to the introduction of relaxation techniques [6]. These achievements opened the way to the investigation of the activation modes of other fundamental reactions, mainly belonging to the biophysical area, such as enzymatic reactions [7], protein structural changes [8,9,10], double and multiple helix formation in nucleic acids [11, 12], and polymer-ligand interactions [13,14,15,16]. More recently, the dynamic methods have been extended for instance to the study of molecular motion [17], and amorphous materials chemistry [18]. The correct description of the mechanism of a multistep reaction requires the evaluation of both the rate constants and the thermodynamic parameters of each step [19]. Relaxation methods and the relevant dynamic analysis have proven to be especially appropriate for this task, in that they allow us to simultaneously obtain the necessary information. These methods are based on small perturbations applied to a chemical system initially at equilibrium. The analysis of the motion of chemical systems is rather similar to that of mechanical systems subject to small oscillations. This analogy of behaviour between systems belonging to different areas of science can appear surprising at first sight. However, it should be remembered that the general mechanical principles were so attractive to the scientists of the nineteenth century that many non-mechanical observations were analysed by the methods of mechanics. Actually, the results appeared to be so successful that the supposition arose that many non-mechanical observations could be analysed by the methods of mechanics. The beautiful scientific achievements by Boltzmann and Gibbs demonstrate, beyond any doubt, that average (probabilistic) properties of non-mechanical nature can be analysed by the (deterministic) methods of rational mechanics. Illuminating for this purpose is a study of J. J. Thomson who derived the equations of chemical equilibrium using the Lagrangian and Hamiltonian functions [20]. The aim of this study is to show the close formal analogy existing between the dynamic behaviour of a mechanical system made of two interacting bodies and that of a chemical system, made by two coupled reactions, in the vicinity of equilibrium. First, the motion equations will be derived; then, the properties associated with the potential energy will be obtained, for both systems, using the “Normal Modes “ analysis [21]. It should be noted that this method has been applied to the analysis of electrical oscillating circuits [22] and molecular vibration [23].

2 The Mechanical System

Let’s consider two bodies of mass m1 and m2 connected to two fixed points by springs of deformation coefficients k1 and k2 respectively and mutually interacting through the spring of deformation coefficient k (Fig. 1). The two bodies are allowed to move only along the horizontal axis.

Fig. 1
figure 1

The linear two-body oscillator

Full size image

If the body of mass m2 is clamped, only the body of mass m1 is allowed to move. If m1, initially in equilibrium, is impressed with a force that produces a displacement q1, the springs will react with a force equal to but opposite to the impressed one, which tends to restore the equilibrium position of m1. This force, here denoted as − Q1i, is expressed by the Hooke law, − Q1i = (k1 + k) × q1, where the minus sign indicates that Q1i and q1 are oriented in opposite directions. If m1 is clamped, a similar action on m2 will produce the displacement q2 and the restoring force − Q2i = (k2 + k) × q2. If clamps are removed, the displacements q1 and q2 will impress the two bodies motions that, owing to the mutual interactions, are, in general, not periodic anymore; the restoring forces will be different from − Q1 and − Q2 and the oscillation frequencies will be the result of the superimposition of two basic frequencies, the so-called “natural frequencies” of the oscillating system [22]. These, however, can be observed separately by impressing the bodies the “right” displacements, as will be shown later.

2.1 Characterization of the Impressed and Actual Forces

Call Ū the potential energy of the system at equilibrium, when the springs are unstressed. The displacements q1 and q2 impressed respectively to m1 and m2 will cause the potential energy to increase up to a value U. Since the displacements are small, the departure from the equilibrium of the system will be small. Under these circumstances, U may be expanded in Taylor’s series giving Eq. (1) [21].

$$U= \overline{U }+ \left[\left(\frac{\partial U}{\partial {q}_{1}}\right){q}_{1}+ \left(\frac{\partial U}{\partial {q}_{2}}\right){q}_{2}\right]+\frac{1}{2}\left[\left(\frac{{\partial }^{2}U}{{\partial {q}_{1}}^{2}}\right){{q}_{1}}^{2}+ \left(\frac{{\partial }^{2}U}{{\partial {q}_{2}}^{2}}\right){{q}_{2}}^{2}+2\left(\frac{{\partial }^{2}U}{\partial {q}_{1}\partial {q}_{2}}\right){q}_{1}{q}_{2}\right]+\dots$$
(1)

Since the term in the first bracket refers to the equilibrium position, where U is minimal, the values of the first derivatives are zero and Eq. (1) reduces to Eq. (2) where derivatives of order higher than 2 are disregarded.

$$U-\overline{U }= \frac{1}{2}\left[\left(\frac{{\partial }^{2}U}{{\partial {q}_{1}}^{2}}\right){{q}_{1}}^{2}+ \left(\frac{{\partial }^{2}U}{{\partial {q}_{2}}^{2}}\right){{q}_{2}}^{2}+2\left(\frac{{\partial }^{2}U}{\partial {q}_{1}\partial {q}_{2}}\right){q}_{1}{q}_{2}\right]$$
(2)

Differentiation of Eq. (2) with respect to q1 and q2 respectively yields the forces F1i and F2i impressed to the system (Eqs. (3–3’)), each of them being a function of q1 and q2

$${-F}_{1}^{i}= {b}_{11}{q}_{1}+ {b}_{12}{q}_{2}$$
(3)
$${-F}_{2}^{i}= {b}_{21}{q}_{1}+ {b}_{22}{q}_{2}$$
(3′)

where b11 = ∂2U/∂q12, b22 = ∂2U/∂q22, b12 = b21 = ∂2U/∂q1∂q2.Footnote 1 These coefficients, denoted as the “spring constants”, are given by linear combinations of the deformation coefficients of the springs [21]. For the system of Fig. 1, b11 = k1 + k, b22 = k2 + k and b12 = b21 = − k. On the other hand, the kinetic energy of the system (T) does not depend on the coordinates, but only on the velocities of the bodies (Eq. 4).

$$T=1/2(m_1{\dot{q}}_{1}^2 +m_2{\dot{q}}_{2}^2)$$
(4)

The derivatives d(∂U/∂\({\dot{q}}_{k}\))/dt = mk \(\ddot{q}\) k yield the accelerating forces F1e and F2e effectively acting on m1 and m2 during the return of the system at equilibrium (Eqs. 5– 5’).

$${F}_{1}^{\mathrm e}= {m}_{1}\ddot{{q}_{1}}$$
(5)
$${F}_{2}^{\mathrm e}= {m}_{2}\ddot{{q}_{2}}$$
(5′)

Therefore, for small displacements from the equilibrium positions, the motions of the two bodies are described by Eqs. (6–6’) commonly known as Lagrange’s equations.

$$- {m}_{1}\ddot{{q}_{1}}={b}_{11}{q}_{1}+ {b}_{12}{q}_{2}$$
(6)
$$- {m}_{2}\ddot{{q}_{2}}={b}_{21}{q}_{1}+ {b}_{22}{q}_{2}$$
(6′)

2.2 The Integration of the Motion Equations

Equations (6–6’) form a system of homogeneous linear second-order differential equations with constant coefficients. Its general solution provides the dependence of the coordinates q1 and q2 on time according to Eqs. (7–7’), where the parameters λI and λII are associated with the fundamental frequencies of the system,Footnote 2 whereas the coefficients Ajk are associated with the amplitudes of the oscillations.

$${q}_{1}= {A}_{11}{e}^{-i{\lambda }_{\mathrm{I} }t}+ {A}_{12}{e}^{-{\mathrm{i}\lambda }_{\mathrm{II}}t}$$
(7)
$${q}_{2}= {A}_{21}{e}^{-i{\lambda }_{\mathrm{I}}t}+ {A}_{22}{e}^{-i{\lambda }_{\mathrm{II}}t}$$
(7′)

The fundamental frequencies are obtained equating to zero the determinant of the square matrix of Eq. (8), with k = 1, 2,

$$\left(\begin{array}{cc}{b}_{11}-{m}_{1}{\lambda }_{k}^{2}& {b}_{12}\\ {b}_{21}& {b}_{22}-{m}_{2}{\lambda }_{k}^{2}\end{array}\right)\left(\begin{array}{c}{A}_{1k}\\ {A}_{2k}\end{array}\right)=0$$
(8)

in the form of equation Eq. (9).

$${\lambda }_{k}^{2}=\frac{1}{2}\left[\left(\frac{{b}_{11}}{{m}_{1}}+\frac{{b}_{22}}{{m}_{2}}\right)\pm \sqrt{{\left(\frac{{b}_{11}}{{m}_{1}}-\frac{{b}_{22}}{{m}_{2}}\right)}^{2}+4\frac{{b}_{12}{b}_{21}}{{m}_{1}{m}_{2}} }\right]$$
(9)

We shall denote as λI the frequency with the positive sign in front of the radical and λII that involving the negative sign.

2.3 The Normal Modes of Oscillation

Moreover, Eq. (8) yields the values of the ratios of the oscillation amplitudes, A2k/A1k (Eq. 10).

$$\frac{{\mathrm{A}}_{2k}}{{\mathrm{A}}_{1k}}=-\frac{\left({b}_{11}- {m}_{1}{\lambda }_{k}^{2}\right)}{{b}_{12}}=- \frac{{b}_{21}}{\left({b}_{22}- {m}_{2}{\lambda }_{k}^{2}\right)}= \frac{{\mathrm{C}}_{2k}}{{\mathrm{C}}_{1k}}$$
(10)

Equations (7–7’) show that the motions of the two bodies are generally aperiodic, being the result of the superimposition of two periodic oscillations with different periods. This complicated behaviour depends on the presence of the mixed second derivative ∂2U/∂q1∂q2 in the expression for the potential energy which reflects the presence of the terms b12q2 and b21q1 in Eqs. (6–6’). However, we can get rid of these terms, thus making much easier the integration of the motion equations and the description of the dynamics of the system, if we can start the motion of the system in the “right way”, i.e. by imparting the bodies particular combinations of the displacements q1 and q2 such that Eq. (10) is obeyed. Under these circumstances, both bodies will display simple harmonic motion and oscillate with the same frequency, λI or λII, while the amplitude ratio will stay constant during motion. If the system is excited in any other way, there will be no permanent ratio between the displacements of the two bodies and the motion will not be periodic. The combinations of the original coordinates are denoted as the “Normal Coordinates” and will be here indicated as YI and YII. This change of coordinates has the great advantage of providing a much easier description of the motion of the bodies in that, within a system of Normal Coordinates, the bodies do oscillate as they were uncoupled. If the ratio A21/A11 of the displacements necessary to start the motion of m1 and m2 in the “right way” is calculated by introducing λI in Eq. (10), both bodies will oscillate with frequency λI and amplitude Y0I, according to Eq. (11). On the other hand, the introduction of λII in Eq. (10), yields a value of the displacement ratio A22/A12 such that, applied to m1 and m2, will make both bodies oscillate with frequency λII and amplitude Y0II, according to Eq. (11′). The two simple modes of oscillation of the system found in this way are called “the Normal Modes of oscillation” [22].

$${Y}_\text{I}= {{Y}_\text{I}^{0}e}^{-{\mathrm{i}\lambda }_{I}t}$$
(11)
$${Y}_\text{II}= {{Y}_\text{II}^{0}e}^{-i{\lambda }_\text{II}t}$$
(11′)

The original coordinates are linked to the normal coordinates according to the relationships (12–12′)

$${q}_{1}= {C}_{11}{Y}_\text{I}+{C}_{12}{Y}_\text{II}$$
(12)
$${q}_{2}= {C}_{21}{Y}_\text{I}+{C}_{22}{Y}_\text{II}$$
(12′)

These equations represent a change of coordinates performed by rotation of the axes. In this case, the coefficients Cjk represent the direction cosines of the angle that measures the inclination of the normal coordinate set (YI, YII) with respect to the original coordinate set (q1, q2) [21]. Introduction of Eqs. (11–11′) in Eqs. (12–12′) and comparison of the results with Eqs. (7–7’) shows that Ajk = Cjk × Y0k. Hence, A2k/A1k = C2k/C1k (Eq. 10). Conversely, the normal coordinates are derived from the original coordinates using the coefficients Cjk’ obtained by inversion of the Cjk matrix (Eqs. (13–13’)).

$${Y}_\text{I}= {C}_\text{11}\mathrm{^{\prime}}{q}_{1}+{C}_{12}\mathrm{^{\prime}}{q}_{2}$$
(13)
$${Y}_\text{II}= {C}_{21}\mathrm{^{\prime}}{q}_{1}+{C}_{22}\mathrm{^{\prime}}{q}_{2}$$
(13′)

How to observe the system oscillating with frequency λI and, separately, with frequency λII, will be shown in the case of the simplified coupled oscillator described below.

2.4 The Simplified Two-Body Oscillator

To appreciate the use of the Normal Mode analysis let’s now analyse the dynamic behaviour of a two-body oscillator simplified by putting m1 = m2 = m and k1 = k2 = K. Under these conditions, Eq. (9) is reduced to Eq. (14).

$$m{\lambda }_{k}^{2}=\left(k+K\right)\pm k$$
(14)

Introduction of Eq. (14) in either of Eqs. (10) yields C21/C11 = A21/A11 = − 1 and C22/C12 = A22/A12 = 1. The remaining equations necessary to assign a value to each of the Cjk coefficients are provided by the normalizing conditions that, for the present system, read:

$${{C}_{11}}^{2}+{{C}_{21}}^{2}=1$$
(15)
$${{C}_{12}}^{2}+{{C}_{22}}^{2}=1$$
(15′)

A combination of Eq. (10) and Eqs. (15–15′) yields C11 = 1/√2, C12 = 1/√2, C21 = − 1/√2 and C22 = 1/√2. The introduction of these values in Eqs. (12–12′) allows one to express the original coordinates as a function of the normal coordinates (Eqs. (16–16′)).

$${q}_{1}= \frac{1}{\sqrt{2}}({Y}_\text{I}+{Y}_\text{II})$$
(16)
$${q}_{2}= \frac{1}{\sqrt{2}}({Y}_\text{I}-{Y}_\text{II})$$
(16′)

The elements of the inverse matrix are: C11’ = 1/√2, C12’ = −1/√2, C21’ = 1/√2 and C22’ = 1/√2. It should be noted that, owing to the symmetry of the bjk matrix, the Cjk matrix is orthonormal. This makes the inverse matrix coincide with its transpose, thus simplifying the job of finding the Cjk’ coefficients in the case of more complex systems. Application of Eqs. (13–13′) to the simplified oscillator yields:

$${Y}_\text{I}= \frac{1}{\sqrt{2}}({q}_{1}-{q}_{2})$$
(17)
$${Y}_\text{II}= \frac{1}{\sqrt{2}}({q}_{1}+{q}_{2})$$
(17′)

Equations (17–17′) illustrate the meaning of the normal coordinates. If the displacements are equal and act in opposite directions (q2 = −q1); then, YII = 0 and the two bodies will oscillate with the same frequency λI and with equal but opposite amplitudes (A21/A11 = −1), i.e. the centre of mass of the system stays at rest while the two masses will oscillate in opposite directions. If the displacements act in the same direction (q2 = q1), then YI = 0 and the centre of mass will perform the same oscillatory motion as the two masses with frequency λII. In conclusion, the Normal Modes describe the motions of the centre of mass of the system.

2.5 Normal Forces and Individual Forces

Whereas each of the impressed forces, F1i and F2i, depends on both individual coordinates, q1 and q2 (Eqs. (4–4’)), the normal mode analysis allows a new set of impressed forces, S1i and S2i, to be found such that the first depends only on YI and the second only on YII. The introduction of Eqs. (11–11′) and Eq. (14) in Eq. (2) reduces the potential energy of the simplified two-body oscillator to a function of the normal coordinates (Eq. 18).

$$U-\overline{U }= \frac{1}{2}\left[\left(K+2k\right){{Y}_{\mathrm{I}}}^{2}+ K{{Y}_\text{II}}^{2}\right]=\frac{1}{2}\left[m{{\lambda }_{1}^{2}{Y}_\text{I}}^{2}+ {m\lambda }_{2}^{2}{{Y}_\text{II}}^{2}\right]$$
(18)

Comparison between Eqs. (2) and (18) shows that the latter is much simpler in that the mixed product of coordinates has disappeared. Equation (2) represents an ellipse, generically lying on the q1q2 plane. If the coordinate system is rotated by an angle α, so that the new axes YI and YII coincide with the principal axes of the ellipse, then the ellipse equation becomes Eq. (18). It can be shown that, concerning the two-body simplified oscillator, this situation does occur for α = π/4 or 3/4 π corresponding to cos α =  ± sin α =  ± \(\frac{1}{\sqrt{2}}\). This finding provides a geometrical demonstration of the meaning of the Cjk coefficients in Eqs. (12–12′). Derivation of Eq. (18) with respect to YI and YII, leads to the expressions for the normal forces

$$-{S}_{\mathrm I}^{i}=\left(K+2k\right){Y}_{\mathrm I}=m{\lambda }_{\mathrm I}^{2}{Y}_{\mathrm I}$$
(19)
$$-{S}_\mathrm {II}^{i}=K{Y}_\mathrm {II}=m{\lambda }_\mathrm {II}^{2}{Y}_\mathrm {II}$$
(19′)

Introduction of Eqs. (17–17′) in Eqs. (19 and 19′) yields Eqs. (20–20′):

$${-S}_\mathrm {I}^{i}=\frac{1}{\sqrt{2}}\frac{(K+2k)}{\left(K+k\right)}({\mathrm{Q}}_{2}-{Q}_{1})$$
(20)
$${-S}_\mathrm {II}^{i}=\frac{1}{\sqrt{2}}\frac{K}{\left(K+k\right)}(- {Q}_{1}-{Q}_{2})$$
(20′)

where – Q1 = b11q1 = (K + k)q1 and – Q2 = b22q2 = (K + k)q2 are the forces that would act on body 1 and body 2, respectively, if the two bodies were uncoupled.

The possibility of obtaining the individual forces from the normal forces, offered by Eqs. (20 and 20′), finds an important application in chemical kinetics, where, as it will be shown in the chemical example, the thermodynamic parameters associated with the individual reactions can be derived from the analysis of the experimentally measured normal parameters.

3 The Chemical System

It could appear surprising, at first sight, that the dynamics of a mechanical oscillator could display analogies with that of a chemical reaction. Actually, in the absence of friction, the two bodies oscillate indefinitely crossing the relevant equilibrium positions at each oscillation. In contrast, two coupled chemical reactions move towards equilibrium, where they stop moving. Differently from the mechanical system, the motion of the chemical system occurs only once; however, it should be noted that such a behaviour could be displayed also by the oscillator, provided it was immersed in a medium of very high viscosity. In such a case, the damping exerted by the medium on the oscillator’s motion would be so efficient that the bodies would stop moving as the relevant equilibrium positions were reached. In this case, the motion of the system takes the form of an exponential decay, exactly like the decay of a system of first-order reactions. The motions of the two systems here analysed can be considered as limiting expressions of the general damped motion described by the equation a\(\ddot{q}\)+ b\(\dot{q}\) + c\(q\) = 0, where, for b\(\dot{q}\) = 0, the system displays the undamped oscillatory behaviour analyzed above, whereas, for a\(\ddot{q}\) = 0, the system displays the exponential decay typical of first-order chemical reactions. Consider now a chemical system consisting of two reactions coupled through the common component B. More complex systems can, not always but often, be reduced to that represented in Fig. 2, as shown in the Supporting Information.

Fig. 2
figure 2

The chemical system: two coupled chemical reactions in equilibrium

Full size image

Denote as k1 and k-1 the rate constants of reaction ① in the forward and reverse direction respectively. Similarly, denote the rate constants of the reaction ② as k2 and k-2. All these parameters have the dimensions of a frequency. This system represents the chemical analogous of the mechanical system described above. If the equilibrium position of the system is slightly perturbed by a small but fast change of an external physical variable, such as temperature, pressure, electric field density, or by a sudden small variation of a chemical variable, as pH, ionic strength or solvent composition, the “extent of reaction’’ of each step belonging to the reacting system will change [24, 25]. The “extent of reaction change” is defined as the concentration change of the reacting species displaying unitary stoichiometric coefficient [26]. Let’s indicate as q1 and q2 the changes in the extent of reaction respectively induced on the steps ① and ② of the chemical system. These changes correspond here to the respective space displacements of the mechanical case. The increase of potential energy following the perturbation is again expressed by Eq. (2), provided that the differences between the two systems are considered. Note that, whereas in the mechanical system variables such as temperature or pressure are of little relevance, in the chemical system case it is necessary to specify the state of the thermodynamic variables. It should be also pointed out that, in this case, the potential energy is an average property. Under conditions of constant temperature and pressure, the potential energy is identified with Gibbs free energy, whereas the first derivatives of the potential energy with respect to the extents of reaction represent the affinities of the respective reactions, i.e. ∂U/∂q1 = A1 and ∂U/∂q2 = A2.Footnote 3 At equilibrium, affinities are zero [26] so the displacement of the energy of the reacting system from the equilibrium position is represented by Eq. (2), as in the mechanical system. Differentiation of Eq. (2) with respect to q1 and q2 yields respectively the forces impressed to the system resulting from the applied perturbation. These are expressed, as in the mechanical system, by Eqs. (21–21’).

$${-F}_{1}^{i}= {b}_{11}{q}_{1}+ {b}_{12}{q}_{2}= \frac{RT}{V}\times ({g}_{11}{q}_{1}+ {g}_{12}{q}_{2})$$
(21)
$${-F}_{2}^{i}= {b}_{21}{q}_{1}+ {b}_{22}{q}_{2}= \frac{RT}{V}\times ({g}_{21}{q}_{1}+ {g}_{22}{q}_{2})$$
(21′)

It has been demonstrated [24] that the coefficients bjk = ∂2U/∂qj∂qk are given by the functions (RT/V)Σ νijνik/[i] = (RT/V)gik, where [i] are the equilibrium concentrations of the ith reacting species, while νij and νik are the relevant stoichiometric coefficients (negative for reactants and positive for products). Since in the system of Fig. 2 all the stoichiometric coefficients are unity, it turns out that g11 = 1/[A] + 1/[B], g12 = g21 = −1/[B], g22 = 1/[B] + 1/[C]. It should be noted that the second derivatives of Eq. (3), i.e. bjk, represent the chemical analogous of the “spring constants” of the mechanical case. On the other hand, the effective forces acting on the elementary reactions ① and ② are the affinities A1 and A2 respectively [26]. According to the thermodynamics of irreversible processes, in the vicinity of equilibrium, the affinity of the jth reaction is related to the corresponding rate of reaction \(\dot{q}\)j through the linear relationship Aj = (RT/V)\(\dot{q}\)j/rj, where rj is the rate of the jth reaction at equilibrium also denoted as the “exchange rate”, i.e., concerning the present system r1 = k1[A] = k-1[B] and r2 = k2[B] = k-2[C] [26]. Hence, for the reacting system ①—② one can write:

$${F}_{1}^{e}={\text{\bf{A}}}_{1}= \frac{RT}{V}\times \frac{\dot{{q}_{1}}}{{r}_{1}}$$
(22)
$${F}_{2}^{e}={\text{\bf{A}}}_{2}= \frac{RT}{V}\times \frac{\dot{{q}_{2}}}{{r}_{2}}$$
(22′)

A combination of Eqs. (21–21′) and  (22–22′) (Fki – Fke = 0), allows the dynamical behaviour of the chemical system to be described by the set of Eqs. (23–23’), known as the “kinetic equations”.

$$-\frac{\dot{{q}_{1}}}{{r}_{1}}= {g}_{11}{q}_{1}+ {g}_{12}{q}_{2}$$
(23)
$$-\frac{\dot{{q}_{2}}}{{r}_{2}}= {g}_{21}{q}_{1}+ {g}_{22}{q}_{2}$$
(23′)

3.1 The Integration of the Kinetic Equations

The general solution of the system of Eqs. (23–23′) is given by Eqs. (24–24′) that correspond to Eqs. (7–7′) of the mechanical system. In this case, λI and λII represent the time constants of the exponential effects that describe the kinetic behaviour of the reacting system (the reciprocals of λk, denoted as τk, are called “relaxation times”), whereas the parameters Ajk are associated to the amplitudes of the corresponding effects.

$${q}_{1}= {A}_{11}{e}^{-{\lambda }_\text{I}t}+ {A}_{12}{e}^{-{\lambda }_\text{II}t}$$
(24)
$${q}_{2}= {A}_{21}{e}^{-{\lambda }_\text{I}t}+ {A}_{22}{e}^{-{\lambda }_\text{II}t}$$
(24′)

Differentiation of Eqs. (24–24′) and introduction of the results in Eqs. (23–23’) yields Eq. (25) which is analogous to Eq. (8) of the mechanical case.

$$\left(\begin{array}{cc}{g}_{11}-{\lambda }_{k}/{r}_{1}& {g}_{12}\\ {g}_{21}& {g}_{22}-{\lambda }_{k}/{r}_{2}\end{array}\right)\left(\begin{array}{c}{A}_{1k}\\ {A}_{2k}\end{array}\right)=0$$
(25)

Equating to zero the determinant of the square matrix of Eq. (25) yields the “determinantal” Eq. (26)

$${\lambda }_{k}=\frac{1}{2}\left[\left({b}_{11}+{b}_{22}\right)\pm \sqrt{{\left({b}_{11}-{b}_{22}\right)}^{2}+4{(b}_{12}\times {b}_{21}) }\right]$$
(26)

where b11 = g11r1, b12 = g12r1, b21 = g21r2 and b22 = g22r2. Equation (26) allows the rate constants of reactions ① and ② to be evaluated from the measured values of λk. A comparison between Eq. (26) and Eq. (9) shows that the exchange rates play the role of the masses in the mechanical system in the sense that the inertia of the two systems depends on the values of these parameters. Although the chemical system could be kept in a state of periodical motion by application of a suitable periodical perturbation as in relaxation techniques based on sound absorption, in most cases the external perturbation (pressure-jump, temperature-jump or electric field-jump) is applied only once [6] and the consequent restoring of the equilibrium, denoted as “chemical relaxation”, manifests itself as a transient phenomenon. From Eqs. (25) one obtains Eqs. (27) where the equality C2k/C1k = A2k/A1k holds, in analogy with the mechanical case.

$$\frac{{C}_{2k}}{{C}_{1k}}=\frac{{A}_{2k}}{{A}_{1k}}=-\frac{\left({g}_{11}-\frac{{\lambda }_{k}}{{r}_{1}}\right)}{{g}_{12}}=- \frac{{g}_{21}}{\left({g}_{22}- {\lambda }_{k}/{r}_{2}\right)}$$
(27)

The scalar normalization criterion used for the analysis of the chemical system could be different from that employed for the mechanical system (Eqs. (15–15′)). For the present system and the more complex system analysed in the Supporting Information, it seems more suitable to proceed as follows: multiply reaction ① by C1k and reaction ② by C2k, then sum the results and divide the stoichiometric coefficients of the resulting reaction by C1k. We are allowed to do that since any scaling of a reaction stoichiometry is arbitrary. The normal reactions, normalized according to this procedure, are shown in Eq. (28).

$${\text{A}}\, + \,\left( {C_{{{{2k}}}} /C_{{{{1k}}}} - 1} \right){\text{B}}\, = \,C_{{{\text{2k}}}} /C_{{{{1k}}}} {\text{C}}$$
(28)

Note that the stoichiometric coefficients of the normal reactions are directly provided by Eq. (27).

3.2 The Normal Modes of Reaction

Following the procedure used to derive Eqs. (11–11′), we obtain the time course of the relaxing system in terms of the normal coordinates Yk (Eqs. 29–29′) which show that

$${Y}_\text{I}= {Y}_\text{I}^{0}{e}^{-{\lambda }_\text{I}t}$$
(29)
$${Y}_\text{II}= {Y}_\text{II}^{0}{e}^{-{\lambda }_\text{II}t}$$
(29′)

while the original reactions are coupled, the normal reactions are uncoupled. As in the mechanical case, the description of the behaviour of the chemical system according to Eqs. (24–24′), where the individual steps exert a mutual influence on the relevant extents of reaction, is replaced by a description according to Eqs. (29–29′) where two independent normal reactions are operative. Introduction of Eqs. (29–29′) in Eqs. (24–24′) and recalling that Ajk = CjkY0k yields Eqs. (12–12′) which shows that, in the chemical system, the original variables are related to the normal variables by the same equations holding for the mechanical system.

3.3 The Simplified Chemical System

Let us now analyse in more detail the chemical system ①–②, making the simplifying assumptions k-1 = k2 = k and k1 = k-2 = K. The former equality will cause the exchange rates to be equal (r1 = r2). Under these circumstances, the behaviour of the chemical system becomes formally analogous to that of the simplified two-body oscillator. Actually, Eq. (26) yields λI = (K + 2k) s−1 and λII = K s−1 which show that the rate constants play the role of the spring constants of the mechanical case. The introduction of λI in Eq. (27) yields C21/C11 = -1, while the introduction of λII yields C22/C12 = 1.

3.4 The Normal Reactions as Linear Combinations of Basic Reactions

Applying these results to reactions (28), it turns out that the first normal reaction, with a time constant λI and stoichiometric coefficients provided by C21/C11 = − 1, reads as follows:

$${\text{A}} + {\text{C}} = {\text{2B}}$$
(I)

which shows that the first normal reaction is made by the basic reaction ① minus the basic reaction ②. The second normal reaction, with a time constant λII and stoichiometric coefficients provided by C22/C12 = 1, reads as follows:

$${\text{A}} = {\text{C}}$$
(II)

which shows that the second normal reaction is made by the basic reaction ① plus the basic reaction ②. The importance of this result becomes now evident: the normal mode analysis provides a powerful tool to simultaneously derive the kinetic and the thermodynamic properties of basic reactions from the measured parameters of the normal reactions. On this basis, if the system here analysed is investigated using the temperature-jump technique [6], one can write:

$$\Delta H_{{\text{I}}} = \Delta H_{{1}}^{0} - \Delta H_{{2}}^{0}$$
(30)

and, concerning the normal reaction (II)

$$\Delta H_{{{\text{II}}}} = \Delta H_{{1}}^{0} + \Delta H_{{2}}^{0}$$
(30′)

A detailed analysis of the kinetic and thermodynamic behaviour of a system investigated by the T-jump method is provided in the Supporting Information. The pressure–jump technique, applied to the system would provide the reaction volumes of the normal reactions, which, in turn, will provide the reaction volumes of the individual reactions. On the other hand, the analysis of λI and λII, according to Eq. (26), allows us to find the values of the rate constants, whose ratios k1/k-1 and k2/k-2, provide respectively the values of the free energies, ΔGo1 and ΔGo2 of the original reactions at equilibrium. These, combined with the relevant reaction enthalpies, allow the reaction entropies of the basic reactions to be evaluated.

3.5 The Cyclic System

Consider now the cyclic system shown in Fig. 3.

Fig. 3
figure 3

Cyclic system of three chemical reactions

Full size image

The forces Fji impressed to the three reactions are again obtained by derivation of Eq. (2) and read as \(-F_j^i = (RT/V)\times\sum_{i=1}^{3}{g}_{ij}{q}_{i}\) for j = 1,3. These are balanced by the effective forces Fje expressed by the affinities of the three reactions \(\text{\bf{A}}_j = \frac{RT}{V}\times \frac{\dot{{q}_{j}}}{{r}_{j}}\) for j = 1,3. Hence, the kinetic equations read:

$$-\frac{\dot{{q}_{1}}}{{r}_{1}}= {b}_{11}{q}_{1}+{b}_{12}{q}_{2}+{b}_{13}{q}_{3}=({g}_{11}{q}_{1}+{g}_{12}{q}_{2}+{g}_{13}{q}_{3})$$
(31)
$$-\frac{\dot{{q}_{2}}}{{r}_{2}} = {b}_{21}{q}_{1}+{b}_{22}{q}_{2}+{b}_{23}{q}_{3}=({g}_{21}{q}_{1}+{g}_{22}{q}_{2}+{g}_{23}{q}_{3})$$
(31′)
$$-\frac{\dot{{q}_{3}}}{{r}_{3}} = {b}_{31}{q}_{1}+{b}_{32}{q}_{2}+{b}_{33}{q}_{3}=({g}_{31}{q}_{1}+{g}_{32}{q}_{2}+{g}_{33}{q}_{3})$$
(31″)

where g11, g12 = g21, g22, r1, r2 have been already defined, while g13 = g31 = −1/[A], g23 = g32 = −1/[C], g33 = 1/[C] + 1/[A] and r3 = k3[C] = k−3[A]. The determinantal equation for a three-reaction system would be a cubic one, being derived from the determinant of a 3 × 3 matrix. However, since the present system is cyclic, each reaction is thermodynamically dependent on the remaining ones. Assuming that the dependent reaction is reaction ③ (C = A), the thermodynamic properties of this reaction can be expressed as functions of the properties of the independent reactions ① and ②. Under these circumstances the three-reaction system (31–31’’) can be reduced to a system of two equations [24, 27]. The reduction is performed by summing columns 1 and 2 to column 3 and then subtracting row 3 from rows 2 and 1 of the determinant of the 3 × 3 matrix derived from Eqs. (31–31’’). The 2 × 2 matrix so obtained corresponds to a system formally identical to the system (23–23’), with the difference that now b11 = (g11r1 – g13r3), b12 = g12r1 – g13r3, b21 = g21r2 – g23r3 and b22 = g21r2 – g23r3. The relaxation problem is then solved by evaluating λk for k = 1,2 using Eq. (26) and the ratios C2k/C1k using Eq. (27). This method was employed to analyse a proton transfer system [28] and more complex systems involving proton transfer coupled with complex formation reactions [11, 29, 30]. The Normal Mode analysis of a system made by coupled reactions is shown in the Supporting Information section where the complex formation of Ni(II) by 3,5-dinitrosalycilate ion in the presence of the cacodylate/cacodylic acid buffer is studied. The Normal Mode analysis can be easily extended to more complex systems displaying independent modes of equilibration. In such a case, the system behaviour will be described by a n × n matrix leading to a determinantal equation of the nth order. A paper from H.J.G. Hayman [27] shows that the characteristic n × n matrix can be reduced to a symmetric form which, in turn, provides a set of orthonormal coordinates.

4 Conclusion

The present paper shows that the kinetic behaviour of coupled chemical reactions and the dynamic behaviour of coupled mechanical bodies can be interpreted on a common scientific ground. This aspect, which connects apparently separated areas of Science, is not well highlighted, even in more recent and currently used books. The dynamic analysis of small displacements from the equilibrium position enables to obtain simultaneous information about the time-dependent and potential energy-dependent behaviours of mechanical and chemical systems using the same mathematical description. Focusing on systems of coupled chemical reactions, it should be noted that, if these are fast-reacting, the classical chemical techniques are not suitable for a study of the thermodynamics of the individual steps because these techniques can provide only average (and therefore scarcely useful) values of the relevant parameters. In contrast, the relaxation techniques, being based on small perturbations of equilibria, allow the dynamic and thermodynamic behaviour of the individual steps to be elucidated (as shown in the Supporting Information) grace to the application of the Normal Mode analysis, which was originally devised for the study of mechanical systems.