relaxation phenomenon

relaxation phenomenon, in physics and chemistry, an effect related to the delay between the application of an external stress to a system—that is, to an aggregation of matter—and its response. It may occur in nuclear, atomic, and molecular systems. Chemists and physicists use relaxation effects to study processes that take only a fraction of a second. When an equilibrated nuclear, atomic, or molecular system is subjected to an abrupt physical change, such as a sudden rise in temperature or pressure, it takes time for the system to re-equilibrate under the new conditions. The relaxation effect may be caused by a redistribution of energy among the nuclear, electronic, vibrational, and rotational energy states of the atoms and molecules that constitute the system, or it may result from a shift in the ratio of the number of product molecules to the number of reactant molecules (those initially taking part) in a chemical reaction. The measurement of relaxation times can provide many insights into atomic and molecular structures and into the rates and mechanisms of chemical reactions.

Historical survey

The word relaxation was originally applied to a molecular process by the English physicist James Clerk Maxwell. In the paper “On the Dynamical Theory of Gases,” which he presented in 1866, Maxwell referred to the time required for the elastic force produced when fluids are distorted to diminish or decay to 1/e (e is the base of the natural logarithm system) times its initial value as the “time of relaxation” of the elastic force. The earliest suggestion of a chemical relaxation effect is contained in a dissertation (Berlin, 1910) based on research directed by the German physical chemist Walther Nernst. Measurements of sound propagation through the gas nitrogen tetroxide—which breaks up, or dissociates, into nitrogen dioxide—led Nernst to suggest that experiments at frequencies at which the dissociation reaction could not keep pace with the temperature and pressure variations that occur within a sound wave would permit evaluation of the dissociation rate. Ten years later, at a meeting of the Prussian Academy of Sciences, Albert Einstein presented a paper in which he described the various theoretical aspects of this relaxation effect.

The detection of the chemical relaxation effect predicted by Nernst and Einstein did not become technically feasible until the last half of the 20th century. In the first half of the century, physicists and chemists studying relaxation concentrated on physical relaxation processes. Peter Debye referred to the time required for dipolar molecules (ones whose charges are unevenly distributed) to orient themselves in an alternating electric field as dielectric relaxation. Sound absorption by gases was used to investigate energy transfer from translational (or displacement in space) to rotational (spinning and tumbling) and vibrational (oscillations within the molecule) degrees of freedom, the three independent forms of motion for a molecule. The former requires only a few molecular collisions, but the transfer of energy between translational and vibrational modes may require thousands of collisions. Because the processes are not instantaneous but time-dependent, relaxation effects are observed. Their measurement provides information about molecular bonding and structure. Chemical relaxation was rediscovered by the German physical chemist Manfred Eigen in 1954. Since then, technological advances have permitted the development of techniques for the measurement of relaxation times covering the entire range of molecular processes and chemical reactivity.

The great variety of relaxation phenomena and of the techniques developed for their study precludes a comprehensive survey. To facilitate a general discussion, the relaxing system, its initial and final states, the nature of the disturbance, and the system’s response are considered separately. Examples are cited that emphasize the important features of relaxation phenomena and illustrate the variety of information that can be obtained from their study. A moderately detailed description of one relaxation technique, the temperature-jump method, is used to summarize the discussion.

Relaxation mechanisms

The chemical relaxation of nitrogen tetroxide is easy to visualize, and it illustrates principles common to all relaxation phenomena. Nitrogen tetroxide (formula N₂O₄; also called dinitrogen tetroxide) actually is a dimer (a molecule formed from two similar constituents called monomers) that dissociates into two molecules of nitrogen dioxide (formula NO₂). The monomer and dimer are easily distinguishable: the former is a brown gas; the latter is a colourless gas. The product and reactants exist in equilibrium, represented by the reversible reaction:

At ambient (room) temperature and atmospheric pressure, approximately 80 percent of the molecules in the mixture are dimers, and the remaining molecules are monomers. The distribution of molecules between the two forms remains unchanged as long as the temperature and pressure are held constant. But when the system is disturbed by a sudden change in temperature or pressure, the gases eventually reach new equilibrium concentrations to suit the new conditions. If the external conditions are altered, then the ratio of monomers to dimers will adjust to a new value. The dependence of the equilibrium on pressure is intuitively understandable as follows: to a good approximation, the volume that a gas occupies at a given pressure and temperature depends directly on the number of gas molecules. The dissociation of one molecule of nitrogen tetroxide into two molecules of nitrogen dioxide entails an expansion of the gas—a doubling of molecules—which is opposed by the external pressure. If the external pressure is increased, the system acts to relieve the stress by reducing its volume—i.e., by combining monomers to form dimers and thus reducing the number of molecules. The equilibrium shifts in favour of dimers under increased pressure and in favour of monomers under reduced pressure. At any steady pressure, the ratio of the two forms eventually becomes constant.

Chemical relaxation results from the inability of systems at equilibria to respond instantaneously to changes in external conditions. The rate of reestablishment of equilibrium, or re-equilibration, is limited by the concentrations of the reactants and their reactivities. At any specified temperature and pressure, there is a definite probability per unit time that a nitrogen tetroxide molecule will dissociate into two nitrogen dioxide molecules and that the latter will recombine to form a dimer. The average lifetime of a nitrogen tetroxide molecule at ambient temperature and atmospheric pressure, for example, is about one-third of a microsecond (one-millionth of a second). The product of the reciprocal of the average lifetime times the concentration of nitrogen tetroxide molecules gives the rate at which they dissociate. At equilibrium there is no net change in the number of nitrogen tetroxide molecules, because their dissociation rate is exactly balanced by the rate at which they are being re-formed through association of nitrogen dioxide molecules. If the external conditions are altered, the reactivities of the monomer and dimer change instantaneously, but their concentrations change at a finite rate until the balance between the association and dissociation rates is reestablished. By determining the relaxation time, it is possible to derive the rate at which nitrogen dioxide combines to form dinitrogen tetroxide, as well as the rate of the reverse reaction.

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Sound propagating through a gas can be pictured as a pressure wave whose alternating increase and falling off of pressure, called a sinusoidal variation of pressure, with time at any point in the medium is accompanied by a corresponding fluctuation in the temperature. The effect of the varying temperature and pressure of a sound wave moving through nitrogen tetroxide gas on the dissociation of nitrogen tetroxide depends on the frequency of that sound wave. When the pressure oscillates slowly enough, the dissociation reaction will remain at equilibrium with the oscillation; that is, the extremes in the monomer-dimer ratio will coincide with the extremes of pressure and temperature. If, on the other hand, the pressure fluctuates too rapidly for the reaction to follow, the ratio of monomers to dimers will remain constant at the equilibrium value for the ambient temperature and pressure; but at intermediate frequencies a relaxation effect may be observed, and a readjustment of the chemical equilibrium will lag behind the pressure variation within the gas.

The relaxing chemical equilibrium results both in the absorption of sound by the gas and in dispersion of, or changes in, the sound velocity. Measurement of either of these effects permits evaluation of the relaxation time. The maximum absorption of sound occurs, for example, when the angular frequency (two π times cycles per second) of the sound wave equals the reciprocal of the relaxation time. The relaxation time can then in turn be related to the mechanism of the chemical reaction and to the reactivities of the reactants.