First order rate constants dimensions

Equation (2.14) is exactly of the same shape as Eq. (2.5) for a first-order reaction. This type of reaction is known as pseudo-first-order reaction, and fcBo is a pseudo-first-order rate constant (dimension s 0-... 

For a first-order reaction, therefore, a plot of In Ca (or log Ca) vs. / is linear, and the first-order rate constant can be obtained from the slope. A first-order rate constant has the dimension time , the usual unit being second. ... 

The Arrhenius equation relates the rate constant k of an elementary reaction to the absolute temperature T R is the gas constant. The parameter is the activation energy, with dimensions of energy per mole, and A is the preexponential factor, which has the units of k. If A is a first-order rate constant, A has the units seconds, so it is sometimes called the frequency factor. 

A first-order rate constant has the dimension time, but all other rate constants include a concentration unit. It follows that a change of concentration scale results in a change in the magnitude of such a rate constant. From the equilibrium assumption of transition state theory we developed these equations in Chapter 5 ... 

One other point should be noted. The dimensions of the right-hand side of Eq. (7-57) are time-. That is, only first-order rate constants appear to be permitted, when in fact, the derivation assumed a bimolecular mechanism. The problem is entirely artificial, arising from the different ways in which concentration units are ordinarily dealt. 

Here y denotes the time-dependent or dynamic variable (usually a concentration), J is the so-called inhomogeneous or input term, and k is the overall first-order rate constant, which can be a sum of several first-order rate constants each describing a different process. Ify has the dimension of a concentration [ML-3], then J has [ML-3T-1]. Note that J as well as k can be time-dependent. We discuss the solution of Eq. 1 by starting with the simplest case and then move to the more complex ones. 

As shown above, the concentration of a compound that is transported in a river is affected by various mixing and elimination processes. Their relative importance for reducing the riverbome mass flow and the maximum concentration depends on the characteristics of the river as well as on the compound under consideration. This section gives a summary of the relevant rate constants by emphasizing the simplest descriptions. In order to compare their relative importance, all processes will be approximated by first-order rate constants, either in time (fc-rates, dimension T-1) or in space along the river (e-rates, dimension L 1). 

Significance of the Michaelis Constant, Km. The Michae-lis constant Km has the dimensions of a concentration (molarity), because k x and k2, the two rate constants in the numerator of equation (23), are first-order rate constants with units expressed per second (s 1), whereas the denominator fc is a second-order rate constant with units of m-is-1. To appreciate the meaning of Km, suppose that [S] = Km. The denominator in equation (25) then is equal to 2[S], which makes the velocity v = VmaJ2. Thus, the Km is the substrate concentration at which the velocity is half maximal (fig. 7.6). 

Note that both the quantum yield and the photocurrent density are a function of the modulation frequency, co. The ks in Eq. 24a are first-order rate constants (with dimensions of s ) for electron transfer and carrier recombination (/free), respectively. The prime in A , distinguishes it from its bimolecular (second-order) counterpart discussed earlier (Section 1.5.1). 

According to equation (4.11), a plot of the logarithm of the amount of dmg remaining (as ordinate) as a function of time (as abscissa) is linear if the decomposition follows first-order kinetics. The first-order rate constant may be obtained from the slope of the plot (slope = -kj/2.303). kj has the dimensions of time. ... 

Here it has been assumed that the first step (Si(0) Si(I)) occurs via the valence band where the three other subsequent steps proceed either under hole consumption or electron injection. The /cjS are first-order rate constants having a dimension of s". The best fit was obtained with k = 2- lO s , /cb = 500 s and = 0.5 s . These values of the first-order rate constants are very small. They were interpreted in terms of relatively large activation energies in the order of about 0.5 eV. 

These dimensions (bold) are what we expect from mass action kinetics for a second-order and for a first-order rate constant. 

This is an expression for the rate of decay of the concentration of species A. (It should remind us of the expression we derived for the change in level of the draining tank for which we used a linear constitutive relationship between level and rate of flow.) The dimensions of k in this case are reciprocal time, that is, sec or min etc. The reason for this is that the rate of reaction is given in dimensions of voi n,e- Therefore to be dimensionally consistent the first-order rate constant must be in dimensions of inverse time. 

The concentration of A leaving the reactor is the reciprocal of the sum of one plus the product of the first-order rate constant and the holding time. The first-order rate constant, we recall, has dimensions of reciprocal time, and the holding time is just time, so their product is dimensionless. In fact this product is actually the ratio of the holding time to the characteristic time required for the chemistry to occur. If the rate constant is taken to be of order unity, then we will see how the concentrations of A and D change with holding time. 

What is a pseudo-first-order rate constant How do its dimensions differ from those of a second-order rate constant ... 

The first-order rate constants have the dimensions whereas the second-order constant has the dimension conc Thus, we can carry out the following dimensional analysis ... 

The catalytic constant k at is quite another matter it is a first-order rate constant the dimension ([time] ) of which does not contain concentration. Therefore, this parameter in reverse micelles is not complicated by the distribution effects of the substances and may be regarded as an objective parameter, reflecting a true reactivity of the enzyme solubilized in the system of reverse micelles. 

FIGURE 13 The dependence of the first order rate constant fcj on reaction product fractal dimension for transesterification reaction in logarithmic coordinates. Type of kinetic curves Q ty. 1 - linear, 2 - autoaccelerated, 3 - autodecelerated. Vertical shaded fine indicates the value D... 

In chemistry, the quantity of matter is usually expressed in moles (mole) and the concentration of matter is usually expressed in mole/liter (M). Reaction rates are expressed in mole/Uter/second (Ms 0- The first-order rate constants have the dimension oftime (s" ) and the second-order rate constants have the dimension of concentration" x time" (M s" ) zero-order rate constants have the dimension of concentration" x time" (M" s" ). 

All of the quantities required in steady-state kinetics are either concentrations, normally measured in mol litre" or a submultiple, rates, measured in mol litre" s", or rate constants, with units that vary according to the type of rate constant a first-order rate constant has dimensions of reciprocal time, and is typically measured therefore in s", and a second-order rate constant has dimensions of reciprocal time multiplied by reciprocal concentration, and is typically measured therefore in mol" litre s" . It is obvious from elementary dimensional considerations that a pseudo-first-order rate constant has the same dimensions and units as a first-order rate constant, and that constants of different order caimot be meaningfully compared. 

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