For completeness, here we review the solution of the unimolecular reversible first-order isomerization equilibrium

$$\begin \textrm\rightleftharpoons & \textrm. \end$$

(49)

The kinetic rate equation for species A is

$$\begin \dot= & -k A+k'B, \end$$

(50)

where k and \(k'\) are the first-order forward and backward rate constants, respectively, both of inverse time units. The rate equation for B is simply \(\dot=-\dot\). The equilibrium constant for this unimolecular transformation is given by \(K_\textrm=k/k'=B_\textrm/A_\textrm\).

At any time, the total amount of species \(M_\textrm\) is conserved \(M_\textrm=A_0+B_0=A+B\). Thus, we can use \(B=M_\textrm-A\) in Eq. (50) to get

$$\begin \dot= & -k A + k'\left( M_\textrm-A\right) =-\left( k+k'\right) \left[ A-A_\textrm\right] , \end$$

(51)

where \(A_\textrm=(k'M_\textrm)/(k+k')=M_\textrm/(K_\textrm+1)\) is the equilibrium concentration of A. From Eq. (51), it is easy to see that when \(A=A_\textrm\) the dynamics on A vanishes. Equation (51) can now be readily integrated

$$\begin \int _^ \fracA})}= & -\left( k+k'\right) t \end$$

(52)

to give the time solution of A

$$\begin A_t= & A_\textrm+\left( A_0-A_\textrm\right) \exp . \end$$

(53)

Using the total number conservation law, we obtain a similar equation for B

$$\begin B_t= & B_\textrm-\left( B_\textrm-B_0\right) \exp . \end$$

(54)

For both solutions, we see that at \(t=0\) we get the expected initial amounts of A and B, while as the limit of \(t\rightarrow \infty\), A and B approach exponentially to \(A_\textrm\) and \(B_\textrm\), respectively. The irreversible solution is easily obtained from Eq. (53) by zeroing one of the rate constant \((k'=0)\) and one equilibrium concentration \((A_\textrm=0)\).

A relevant time quantity is the half-life for equilibration \((t_)\), which is easily obtained by setting \(A_t=(A_0+A_\textrm)/2\) in Eq. (53) to get

$$\begin t_= & \frac. \end$$

(55)

Similarly, the the lifetime \(\tau\) is defined as the time required to get the concentration \(A_\tau =A_\textrm+(A_0-A_\textrm)/e\), where e is the Euler’s number.

Finally, we remark that the solution Eq. (53) resembles the solution for the irreversible decay of A via two first-order parallel reactions (e.g., \(A\rightarrow P\) and \(A\rightarrow P'\)) to yield products P and P′ with rate constants k and \(k',\) respectively

$$\begin A_t= & A_0\exp . \end$$

(56)

This is because the differential equation is very similar to Eq. (56) except for the constant term \(-(k+k')A_\textrm\). In both cases, the overall decay rate is given by the sum of rate constants \((k+k')\) and the half-life is the same as well.

Here we present the analytical solution for the irreversible stepwise (consecutive) homotrimerization of a monomer M to yield a homotrimer T via a reaction intermediate dimer D

$$\begin \textrm + \textrm\Rightarrow & \textrm, \end$$

(57)

$$\begin \textrm + \textrm\Rightarrow & \textrm. \end$$

(58)

Most trimerizations follow this scheme and a classic example is the oligomerization of isocyanates which was recently studied experimentally by Li and coworkers [29]. The system of coupled differential equations to be solved subject to the initial conditions \(M(0)=M_0\) and \(D(0)=D_0\) are

$$\begin \dot= & -2k_\textrm M^2 - k_t D\cdot M \end$$

(59)

$$\begin \dot= & - k_t D\cdot M \end$$

(60)

where the first and second steps are described by the \(k_\textrm\) and \(k_t\) rate constants, respectively, both with units of M\(^\)s\(^\).

Following Croce and coworkers [30], we now divide the first by the second equation to cancel the troublesome coupling term \((k_t D\cdot M)\) that is common in both equations

$$\begin \fracM}D}= & 2\frac}\left( \frac\right) +1. \end$$

(61)

Defining now a new time-dependent dimensionless variable \(u=(M/D)\) in Eq. (61) leads to

$$\begin u+D\fracu}D}= & 2\frac}u+1\Rightarrow \end$$

(62)

$$\begin a\,u+1= & D\fracu}D}, \end$$

(63)

where the unitless parameter \(a=\left[ 2(k_\textrm/k_t)-1\right]\) was defined. Note that for the special case \(k_\textrm=k_t,\) then \(a=1\). Now we must solve Eq. (63) subject to the initial value \(u_0=(M_0/D_0)\). This last equation can be integrated right away via separation of variables u and D.

$$\begin \int _^ \fracD}= & \int _^ \fracu}\Rightarrow \end$$

(64)

$$\begin \ln \right) }= & \frac\ln \right) }. \end$$

(65)

Finally, Eq. (65) gives the exact implicit relation between monomer M and dimer D concentrations at any given time t. Solving for \(M_t\) leads to

$$\begin M_t= & \frac\left( \frac-1\right) D_t, \end$$

(66)

where the positive parameter \(b=D_0^/\left( aM_0+D_0\right)\) with units of M\(^a\) was defined and is given by the initial conditions.

With the relation Eq. (66) we can now go back and substitute it in Eq. (60) to yield an expression in terms of the dimer concentration only

$$\begin \dot= & -c D^2\left( \frac-1\right) , \end$$

(67)

where parameter \(c=k_t/a\) has the same units as \(k_t\) (M\(^\) s\(^\)). We can separate variables in Eq. (67) and integrate to get

$$\begin -c\,t= & b\int _^\fracD}=\left\;\frac;\frac\right) }\right\} _^. \end$$

(68)

This solution is somewhat complicated as it involves the hypergeometric function \(_2F_1(d,e;f;x),\) which can be defined only for non-negative parameter f. However, for the special case that \(k_\textrm=k_t=k\), then \(a=1, c=k\) and we have the simpler implicit solution

$$\begin -k t= & \left( \frac-\frac\right) +\frac\ln \right] }, \end$$

(69)

where \(b=D_0^2/(M_0+D_0)\). An explicit solution should be possible via the Lambert W function. The previous equation can be written in a more symmetrical way as

$$\begin \left( 1-\frac\right) \textrm^}\textrm^= & \left( 1-\frac\right) \textrm^. \end$$

(70)

From Eq. (66) the equation for the monomer concentration reads \(M_t=D_t[(D_t/b)-1]\). Once the concentrations \(M_t\) and \(D_t\) are determined, the trimer concentration at time t is given by the conservation of monomer entities \(T_t=T_0+[2(D_0-D_t)+(M_0-M_t)]/3\).

Using a similar approach, we can solve the kinetics problem corresponding to the irreversible stepwise tetramerization \((M_4)\) from monomer M through the formation of a dimer intermediate D but no trimer

$$\begin \textrm + \textrm\Rightarrow & \textrm\nonumber \\ \textrm + \textrm\Rightarrow & \textrm_4, \end$$

(71)

with the coupled kinetic rate equations

$$\begin \dot= & -2k_\textrmM^2\nonumber \\ \dot= & k_\textrmM^2-2k_tD^2, \end$$

(72)

where \(k_\textrm\) and \(k_t\) are the rate constants of the first and second step in Eq. (71), respectively. We are not going to show the derivation, which is similar as the previous one, and we simply quote our final answer

$$\begin \left( \frac\right) ^}= & \left( \frac\right) ,\,\textrm\nonumber \\ x= & \left( \frac}\right) , \end$$

(73)

\(u=(D/M)\) is the ratio of concentrations of dimer over monomer and \(r=(k_t/k_\textrm)\) is another unitless parameter.

Here we comment on the connection of the expressions developed in the main text with the experimental real-time monitoring of reversible monomer association focusing on UV-Vis spectroscopy. We will assume that we are working in a regime of concentrations where Beer–Lambert’s law is obeyed. We track the time change in absorbance \(A_\lambda (t)\) at some fixed suitably chosen wavelength \(\lambda\) where the monomer M exhibits a maximum in intensity. In this simple approach, it is important to ensure that at the wavelength chosen the contribution of the oligomer is negligible. According to the Beer–Lambert law, there is a linear relation between the monomer absorbance A and its concentration M(t) for dilute enough solutions, \(A_\left( t\right) =\epsilon \left( \lambda \right) l M(t),\) where l is the optical path length (in cm units) and \(\epsilon (\lambda )\) is the molar extinction coefficient (in M\(^\) cm\(^\)) at \(\lambda\). Rather than choosing a particular wavelength \(\lambda\), we can alternatively consider the area under the substrate absorption band and use an integrated extinction coefficient.

The time evolution of the monomer concentration for a generic reversible concerted homo-oligomerization

$$\begin \dot= & k_1M_\textrm-n k_n M^n -k_1M, \end$$

(74)

where \(n=2,3\) is some positive integer indicating the kinetic order relative to the monomer association. Using Beer–Lambert’s law \(M(t)=A(t)/(\epsilon l)\) in Eq. (74) leads to an equation for the time evolution of the absorbance

$$\begin \dot(t)= & k_0-\tildeA^n(t)-k_1 A(t), \end$$

(75)

where the spectroscopically determined rate constants \(k_0=(k_1M_\textrm\epsilon l)\) and \(\tilde_n=n k_n (\epsilon l)^\) were defined. For large association equilibrium constants, \(K_\textrm>>1\), we expect that at short times the first and second right-hand side terms contributed the most, while at long times the middle term is negligible.

In the more general case that at the selected wavelength \(\lambda\) we have the two species (M and oligomer N = M\(_n\), \(n=2, 3\) and with \(M_\textrm=M+nN\) constant at all times) absorbing simultaneously, then it is easy to get

$$\begin A(t)= & \frac\left[ \epsilon _N M_\textrm+\left( n\epsilon _M-\epsilon _N\right) M(t)\right] ,\textrm \end$$

(76)

$$\begin \dot(t)= & \frac\left( n\epsilon _M-\epsilon _N\right) \dot(t). \end$$

(77)

Thus, we see that the rate of change in absorption A is linearly proportional to \(\dot\) and independent of \(M_\textrm\). At the isobestic point \(\lambda _i\) where \(\epsilon _M=\epsilon _N=\epsilon _i,\) we have

$$\begin \dot(t)= & \left( 1-\frac\right) b\epsilon _i\dot(t), \end$$

(78)

which means that the absorption is time-dependent as long as the reaction is out of equilibrium but it eventually becomes stationary once the system reaches equilibrium.

To finish, a brief comment on nuclear magnetic resonance (NMR) spectroscopy. The chemical shift of a distinct signal in the monomer M and oligomer N (either a dimer or trimer) is given by \(\delta _M\) and \(\delta _N\), respectively, with \(\delta _N>\delta _M\) usually. If the equilibrium is fast relative to the NMR timescale, rather than two separate peaks we measure a single peak at the weighted average chemical shift \(\delta _\textrm=\delta _N+(\delta _M-\delta _N)M_\textrm/M_\textrm,\) where \(M_\textrm\) is the total concentration of monomer and expressions for \(M_\textrm\) are given in the main text.

The Python codes and data used to generate the figures in this article are provided in the Supplementary Information. Figures 1, 2, 3, and 4 were generated using scripts homodimer.py, heterodimer.py, homotrimer.py, and heterotrimer.py, respectively. Figure 5 was produced with the code and file data fit_heterodimer.py and fig9.dat, respectively. These programs run on a Linux terminal by simply entering python program-name.