Dimension and unit of scale

Before deciding on the initial estimates for the rate or equilibrium constants, we must consider the dimensions and units. Let us consider in turn the dimension, the unit (scale), and the magnitude of rate constants and of equilibrium constants.

Rate constants

In general the dimension of rate constants strictly follows from the molecularity of the elementary reaction which they describe. Rate constants which describe monomolecular reactions have the dimension [1/time], rate constants which describe bimolecular reactions have the dimension [1/concentration $\times$ 1/time], and so on.

Table 4.1: Dimension of rate constants.

reaction type	order	molecularity	dimension of
$\vphantom {\displaystyle \sum_{\stackrel{\scriptstyle M}{M}}^{M}}$ $A \stackrel{k}{\rightarrow}$	0	(constant influx)	${\rm concentration} \times {\rm time}^{-1}$
$\vphantom {\displaystyle \sum_{\stackrel{\scriptstyle M}{M}}^{M}}$ $A \stackrel{k}{\rightarrow} B + \cdots$	1	monomolecular	${\rm time}^{-1}$
$\vphantom {\displaystyle \sum_{\stackrel{\scriptstyle M}{M}}^{M}}$ $A + B \stackrel{k}{\rightarrow} C + \cdots$	2	bimolecular	${\rm concentration}^{-1} \times {\rm time}^{-1}$

Thus in different kinds of rate constants there appear either one or two physical quantities (either time, or time and concentration) for which we must select an appropriate unit. The unit is determined by the experimental data we want to analyze.

$\fbox{%%%{BEGIN_FBOX} \begin{minipage}{3in} The units of time and concentrati... ...ation used to describe the experimental data. \end{minipage} }%%%{END_FBOX}$

Example: An enzyme reaction was followed by monitoring absorbance changes over time. The experimental data are pairs of data values, representing absorbance (dimensionless) vs. time in minutes. Therefore, unless the time axis for the data is first converted to seconds, the unit of time must be ${\rm min}^{-1}$ for all first-order rate constants and ${\rm concentration}^{-1} \times {\rm min}^{-1}$ for all bimolecular rate constants.

The unit of time for rate constants is determined exclusively by the unit of time used in the experimental data. On the other hand, the concentration unit for rate constants is determined by two important factors, namely, the concentration unit for reactants and the molar instrumental responses.

The unit of concentration for all bimolecular rate constants must be the same as the unit in which concentrations or all reactants are also expressed. However, the molar concentrations of reactants (products, substrates, catalysts) are never measured directly. Instead, the measuring device usually provides values of physical quantities linearly related to concentrations, such as absorbance or optical rotation. The proportionality constant is called the molar response coefficient. Thus, the unit of concentration used for all bimolecular rate constants must correspond to the concentration unit obtained when the raw experimental data (in arbitrary instrumental units such as absorbance or fluorescence) are converted to concentrations by using the molar response coefficients.

Example: An enzyme reaction was followed by monitoring absorbance changes over time. The experimental data are pairs of data values, representing absorbance vs. time in minutes. Assume that the concentrations throughout the script file are in the micromolar units ( $\mu$ M). Therefore, unless the time axis for the data is first converted to seconds, the unit must be ${\rm min}^{-1}$ for all first-order rate constants and $\mu{\rm M}^{-1} \times {\rm min}^{-1}$ for all bimolecular rate constants. One mole-per-liter of the reaction product would an increase of absorbance by absorbance units. Therefore, the molar response coefficient (see below) must be expressed in micromolar units also, $\epsilon = 0.01234$ (absorbance units per $\mu$ M of product).

Equilibrium constants

Similar considerations about the dimension the unit, and the magnitude apply for equilibrium constants that appear in the DynaFit script files. The molecularity of forward and backward elementary reactions determine the dimension of each equilibrium constants. Some examples are given in table 4.2.

Table 4.2: Dimension of equilibrium constants.

reaction type	dimension of
$\vphantom {\displaystyle \sum_{\stackrel{\scriptstyle M}{M}}^{M}}$ $A \stackrel{K}{\rightleftharpoons} B$	(none)
$\vphantom {\displaystyle \sum_{\stackrel{\scriptstyle M}{M}}^{M}}$ $A + B \stackrel{K}{\rightleftharpoons} C$	${\rm concentration}^{-1}$
$\vphantom {\displaystyle \sum_{\stackrel{\scriptstyle M}{M}}^{M}}$ $C \stackrel{K}{\rightleftharpoons} A + B$	${\rm concentration}$
$\vphantom {\displaystyle \sum_{\stackrel{\scriptstyle M}{M}}^{M}}$ $A + A + A \stackrel{K}{\rightleftharpoons} A_3$	${\rm concentration}^{-2}$

The scale of each equilibrium constant that appears in the mechanism is strictly dictated by the concentration scale of the experimental data (e.g., mM, $\mu$ M, or nM). Thus, if the data are in the micromolar scale, all binary dissociation constants must have the same scale, while all binary association constants have the scale $\mu{\rm M}^{-1}$ , a trimerization association constant would have the scale $\mu{\rm M}^{-2}$ , and so on.

Next: Initial estimates Up: Kinetic Constants Previous: Formal rules Index

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Petr Kuzmic | Jul 12 2005