Use our NMR service to measure dynamics by NMR.
Dynamic NMR is the NMR spectroscopy of samples that undergo physical or chemical changes with time. The timescales studied can be from picoseconds to centuries and the techniques used for their study depend on the timescale. NMR can be used to determine the equilibrium and the rate constants, which can be used to calculate the thermodynamic parameters of the system.
Here we will only discuss an equilibrium between two species, which is quite a common occurrence in NMR spectra. For more complex systems, see the relevant textbooks and other material.
Given an equilibrium between two species A⇋B, the equilibrium constant is defined as . (The concentration of a species is indicated by square brackets .)
The free energy, ΔG
o, of a reaction is the amount of mechanical work that can be
extracted from the reaction. The enthalpy, ΔH o, of a reaction is the amount of heat
that can be extracted from the reaction. The entropy, ΔS o, of a reaction is the
amount of disorder produced by the reaction, or more precisely, "the amount of thermal energy not available to do
work." The standard free energy, enthalpy and entropy are related to the equilibrium constant as follows where
R is the gas constant (8.3143 J K mol-1):
The rate constant in such a reaction is defined by where t is time. Where the concentration can be followed directly such as in slow exchange, the concentration difference is:
and the rate constant is:
The free energy of activation, ΔG‡, is directly related to the rate constant according to the following equation where kB is the Boltzmann constant (1.381 × 10-23 J K-1) and h is Plank's constant (6.626 × 10-34 J s).
The enthalpy and entropy of activation (ΔH‡ and ΔS‡, respectively) can be calculated from the rate constants at two or more temperatures.
Fast submillisecond processes are completely averaged out on the NMR timescale (around a second) and yield a
normal spectrum that is an average of the exchanging states. However, the individual states have an effect on the
NMR spectrum. The exchange equilibrium is temperature dependent. If each of the exchanging entities has a
different chemical shift (δA and δB) and the difference in standard enthalpy
o) is similar to the standard entropy difference (ΔS o) times an
accessible temperature (T in K), i. e., ΔH o ≈ TΔS o, then the
chemical shift will vary with temperature according to the equations below. If the chemical shift is measured at
many temperatures, the values of the chemical shifts of each state, enthalpy difference and entropy difference
can be determined by a fitting function.
Given an equilibrium A⇋B, the averaged chemical shift, δ, is given by:
Solving for the equilibrium constant, one obtains:
Measuring the chemical shift over a range of temperatures results in an S-curve (fig. 1) if
o = TΔS o somewhere in the middle of the
temperature range. The values of δA, δB, ΔH o and
ΔS o can be obtained using a non-linear fit of the S-curve to the rather complex
Fig. 1. Chemical shift as a function of temperature of H2 of monosodium 9-phenylanthracene ionized at position 9. At low temperature, it is in a contact ion pair and at high temperature it is a solvent separated ion pair.
For the above example (fig. 1), the fitted values are as follows: δA = 5.71 ppm, δB
= 6.02 ppm, ΔH
o = 32.6 kJ mol-1 and ΔS o =
29.4 J K-1 mol-1. Alternatively, a simpler iterative method may be used as described below.
If the lowest temperature measured corresponds to pure A and the highest temperature to pure B then a plot of
the Rln(Keq) against
(fig. 2) should yield a straight line
intersecting the y axis at ΔS
o with a slope of
-ΔH o. However, this is rarely the case and iteration is required as described in the
Fig. 2. Plot of Rln(Keq) against 1/T for H2 of monosodium
9-phenylanthracene ionized at position 9. The y intersect corresponds to
o and the slope to -ΔH o.
Medium fast (from the millisecond to second range) exchanges cause line broadening. At the fast end of the
range, a single spectrum is broadened. For the simplest case of two exchanging singlets of equal intensity
(i. e., ΔG
o = 0 and Keq = 1), the line-width, w½ in Hz, for relatively
fast exchange is
where Δν is the resonance frequency difference between the species in Hz, k is
the rate constant in s-1 and T2* is the rate of dephasing or effective
transverse relaxation time in seconds. As the exchange slows there is a deviation from this value and the
spectrum splits into two at a value of k=2.22Δν
which is called the coalescence point. As the exchange slows further, the spectrum starts narrowing again with a
line-width soon becoming
and two sharp spectra are observed for slow exchange. The rate constant can therefore be determined at the
coalescence point or at a moderate distance from the coalescence using the following equations.
for fast exchange
k=2.22Δν at coalescence
for slow exchange
Varying the temperature changes the exchange rate allowing the determination of thermodynamic constants of the transition state. At low temperatures, the exchange rate is slower and may be slow enough that allowing the measurement of T2*. At high temperatures, the exchange rate is faster and may be fast enough that . Ideally, the coalescence point should occur at an accessible temperature, Tc.
Medium slow (up to about a minute) exchanges yield sharp separate spectra but also yield exchange peaks in an EXSY/NOESY spectrum. If small mixing times and long relaxation delays are used, the EXSY/NOESY spectra are quantitative and the results at varying temperature can be used to calculate thermodynamic parameters of the transition state as in the case for medium fast exchange. The relative concentrations may be used to determine the thermodynamic properties of the equilibrium. For a simple exchange between two equal intensity signals the cross-peak integral, [AB], is related to the diagonal peak integral, [AA] as follows and can be used to determine the rate constant, where τ is the mixing time.
If the cross-peak is small relative to the diagonal peak, the value of k is small and the above equation approximates to the linear relationship below that can be solved for k.
Where there are more than two species interchanging, the above linear equation can be used for each exchange only when the cross-peaks are small. When the cross-peaks are more intense then iterative matrix analysis is required.
For slow exchange (over a minute), one can start with a mixture not at equilibrium and observe the change in the concentration of each species over time at a fixed temperature. This is enough to determine the difference in free energy and can be repeated at different temperatures to yield the enthalpy and entropy differences. If the exchange is too slow to observe at room temperature then the sample may be heated. As a general rule of thumb, the rate of reaction doubles with every 10°C of heating although the precise value depends on the entropy of the excited state. So a reaction that takes an hour (k = 0.00028) at 160°C would likely take on the order of a year at room temperature. For even slower reactions, the reaction mixture can be kept in an oven for a few weeks then measured by NMR. Such experiments can determine the order of magnitude of the expected life-spans of centuries for materials. If such measurements are carried out at two or more temperatures, then the entropy of activation can be measured yielding a more accurate rate of reaction at room temperature.