Relaxation time first order reactions

Important quantities characteristic of a first-order reaction are the half-life of the reaction, which is the value of t when [A], = [A](,/2, and t, the relaxation time, or mean lifetime, defined as k. ... 

The latter is invariably used in the relaxation or photochemical approach to rate measurement (Sec. 1.8), rmd is the time taken for A to fall to 1/e (1/2.718) of its initial value. Half-lives or relaxation times are eonstants over the complete reaction for first-order or pseudo first-order reactions. The loss of reactant A with time may be described by a single exponential but yet may hide two or more concurrent first-order and/or pseudo first-order reactions. 

This equation resembles (1.26) but includes [A], the concentration of A at equilibrium, which is not now equal to zero. The ratio of rate constants, Atj/A , = K, the so-called equilibrium constant, can be determined independently from equilibrium constant measurements. The value of k, or the relaxation time or half-life for (1.47), will all be independent of the direction from which the equilibrium is approached, that is, of whether one starts with pure A or X or even a nonequilibrium mixture of the two. A first-order reaction that hides concurrent first-order reactions (Sec. 1.4.2) can apply to reversible reactions also. 

A second relaxation time of T = 32-45 ps has been assigned139 to the conversion of the peroxide intermediate P to P. A third relaxation time (t = 100-140 ps) is associated with the oxidation of CuA by a (not shown in Fig. 18-11).143 This electron transfer step limits the rate of step/of Fig. 18-11. Another reduction step with t 1.2 ms is apparently associated with electron transfer in step h. This slowest step still allows a first-order reaction rate of -800 s . 

The half-life of a first order reaction remains constant throughout reaction and is independent of concentration. This applies to any fractional lifetime, though the half-life is the one most commonly used. The relaxation time is the other common fractional lifetime. The relaxation time is relevant only to first order reactions, and is afractional lifetime which bears a very simple relation to the rate constant as a direct consequence of the exponential behaviour of first order reactions. 

The ultraviolet irradiation of halogenonitrobenzenes dissolved in ethyl ether or tetrahy-drofuran leads to an increase in the electrical conductivity of the solution relaxation of the conductivity is observed after the irradiation is stopped384. The kinetics appeared to be complicated the structure of the compound, its concentration, the nature of the solvent, the temperature, the time of irradiation as well as the light intensity had an influence on the effects. The photodegradation of three nitrochlorobenzene isomers in pure water and river water under irradiation follows first-order reaction kinetics the rate constants for the three isomers decrease in the order p-> o-> m-nitrochlorobenzene385. 

This has the same form as the expression derived for a simple first-order reaction with the quantity in the square brackets as the first-order rate constant. The usual practice in relaxation kinetics is to define the reciprocal of this quantity as the relaxation time x for the reaction. Thus,... 

The analysis of the experimental data to obtain the two relaxation times is equivalent to the analysis of two consecutive first-order reactions. If the values of Xl and X2 are quite different, this is not a problem and can be accomplished using standard methods of data analysis [G2]. If xi and X2 are close to one another, separate determination of these quantities is very difficult. Typically, the ratio of the relaxation times must be greater than three to estimate them separately. In this regard, some variation in the relative values of xi and X2 can be achieved by changing the values of CAe and cbb-... 

FIGURE 4.1 First-order reaction of the type A. B. (a) Concentration relative to the original one as a function of time (t) over relaxation time (r). (b) Example of log concentration versus time tan a — k log e — 0.434C. D is the decimal reduction time. 

The existence of segregation in a real reactor system can be investigated by means of three time constants, fr> the relaxation time of a chemical process, fo> the time of micromixing, and i, the mean residence time. The relaxation time (6) is a characteristic parameter for reaction kinetics and it is the time required to reduce the relative concentration c /co from 1 to the value e For instance, for a first-order reaction (r, = —kq) in a BR, the... 

Inspection of Fig. 5.18 shows that the autocorrelation functions for this particular model decay exponentially with time, and that the rate constant for this decay is the sum of the rate constants for forward and backward transitions between the two states (kon + The upper curve in Fig. 5.18B, for example, decays to He (0.368) of its initial value in 16.61 At, which is the reciprocal of (0.05 -t 0.01 )Mt. In classical kinetics, if a system with first-order reactions in the forward and backward directions is perturbed by an abrupt change in the concentration of one of the components, a change in temperature, or some other disturbance, it will relax to equilibrium with a rate constant given by the sum of the rate constants for the forward and backward reactions. The fact that the autocorrelation functions in Fig. 5.18 decay with the relaxation rate constant of the system is a general property of classical time-correlation functions [259-262]. One of the potential strengths of fluorescence correlation spectroscopy is that the relaxation dynamics can be obtained with the system at equilibrium no perturbation is required. 

For a sequence of first order reactions the relaxation times are clearly independent of reactant concentrations and the equations apply equally to the interpretation of large transients. The effects of changing the concentration, for instance of the ligand in the pseudo first order system, will be discussed later. Without such additional diagnostics, which are available in the case of concentration dependent systems, the four rate constants can only be estimated by numerical fitting procedures. If signals in terms of absolute concentrations for A, R and [AR] are available, the equilibrium constants can be evaluated and serve as a useful restriction for the numerical solutions. If the two relaxations are uncoupled, t, T2, then we can simplify from equations (6.2.20) ... 

How does one monitor a chemical reaction tliat occurs on a time scale faster tlian milliseconds The two approaches introduced above, relaxation spectroscopy and flash photolysis, are typically used for fast kinetic studies. Relaxation metliods may be applied to reactions in which finite amounts of botli reactants and products are present at final equilibrium. The time course of relaxation is monitored after application of a rapid perturbation to tire equilibrium mixture. An important feature of relaxation approaches to kinetic studies is that tire changes are always observed as first order kinetics (as long as tire perturbation is relatively small). This linearization of tire observed kinetics means... 

For a reaction of the type shown in (74) with hydroxide ion in excess, the expected variation of the time constant (t-1) for the first-order approach to equilibrium after a temperature perturbation is given by (75). Thus a plot of reciprocal relaxation time (t 1) against hydroxide ion concentration is... 

Kinetic schemes involving sequential and coupled reactions, where the reactions are either first-order or pseudo-first order, lead to expressions for concentration changes with time that can be modeled as a sum of exponential functions where each of the exponential functions has a specific relaxation time. More complex equations have to be derived for bimolecular reactions where the concentrations of reactants are similar.19,20 However, the rate law is always related to the association and dissociation processes, and these processes cannot be uncoupled when measuring a relaxation process. 

Figure 5.2. Grabowski s model of TICT formation in DMABN the locally excited (LE) state with near-planar conformation is a precursor for the TICT state with near perpendicular geometry. The reaction coordinate involves charge transfer from donor D to acceptor A. intramolecular twisting between these subunits, and solvent relaxation around the newly created strong dipole. Decay kinetics of LE and rise kinetics of the TICT state can be followed separately by observing the two bands of the dual fluorescence. For medium polar solvents, well-behaved first-order kinetics are observed, with the rise-time of the product equal to the decay time of the precursor, but for the more complex alcohol solvents, kinetics can strongly deviate from exponentiality, interpretable by time-dependent rate constants. 52 ...

CHEMICAL PROCESSES. In many chemical reactions and in all exchange reactions, the rate of decrease in the concentration of a reactant is directly proportional to the concentration of that reactant i.e., v = k[A]). For a first-order process, the relaxation period (corresponding to the period of time required to reach e or about 0.368, of the original amount) is given as ... 

The steady-state reaction rate and relaxation time are determined by these two constants. In that case their effects are coupled. For the steady state we get in first-order approximation instead of Equation (13) ... 

For reactions between ions of like charge, the term in xrc (1 + kR) 1 should be multiplied by a number 0.6—0.9, whereas for unlike charges, this number is 0.3—0.6 depending on R. Certainly, eqn. (58) is not the appropriate correction term. In eqn. (57), the ionic relaxation time for univalent ions is Tjon = 1/(477[rc Dn), where n is the electrolyte concentration. This is also the characteristic time for reaction (pseudo first-order decay time) of a univalent species reacting with one or other ion of the... 

While it may not be intuitively obvious, if the displacement from equilibrium is small, the rate of return to equilibrium can always be expressed as a first-order process (e.g., see Eq. 9-13). In the event that there is more than one chemical reaction required to reequilibrate the system, each reaction has its own characteristic relaxation time. If these relaxation times are close together, it is difficult to distinguish them however, they often differ by an order of magnitude or more. Therefore, two or more relaxation times can often be evaluated for a given solution. In favorable circumstances these relaxation times can be related directly to rate constants for particular steps. For example, Eigen measured the conductivity of water following a temperature jump18 and observed the rate of combination of H+ and OH for which x at 23°C equals 37 x 10 6 s. From this, the rate constant for combination of OH and H+ (Eq. 9-52) was calculated as follows (Eq. 9-53) ... 

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