# Structural Biochemistry/Enzyme/Rate constant

## Introduction

The rate constant is a proportionality constant where the rate of reaction that is directly correlated to the concentration of the reactant. In first order reactions, the reaction rate is directly proportional to the reactant concentration and the units of first order rate constants are 1/sec. In bimolecular reactions with two reactants, the second order rate constants have units of 1/M*sec. Second order reactions can be made to appear as first order reactions, such reactions are called pseudo-first order since adding one reactant in excess will make the reaction first order with respect to the other reactant. There are also zero order reactions in which the reaction is independent of the reactant concentrations where the units of the rate constant are mol/L*sec.

For a general chemical reaction of the form: aA + bB --> products

The expression of the reaction rate would be: ${\frac {d[C]}{dt}}=k[A]^{a}[B]^{b}$ where: k is the rate constant of the reaction, [A] and [B] are the concentrations of the reactants, and a and b are the order of the reaction respect to A and B respectively. The overall order of the reaction is the sum of m and n. Keep in mind here that the above rate equation refers to the disappearance of A and B, thus the rate will be negative (indicating that the reactants were consumed). Products, on the other hand, have a positive value for the rate, since they are being generated. To account for this, many texts list the equation as $rate=k[A]^{a}[B]^{a}$ .

It is important to note that not every reactant will appear in the rate constant since a reaction may be zeroth order in a given reactant. Furthermore, except in very limited cases, the order of the reaction cannot be determined from the stoichiometric equation, but rather must be experimentally calculated.

Although this is generally not seen in general chemistry courses, the reaction order can be negative and/or not a whole number. Rate=K(Concentration)

## Basic Kinetics

For reference purposes the following rate laws are listed without detail into their derivation:

 rate law integrated rate law Rate Constant Units (k) 0 Order $r=-{\frac {d[A]}{dt}}=k$ $\ [A]_{t}=-kt+[A]_{0}$ $\ Ms^{-1}$ 1st Order $r=-{\frac {d[A]}{dt}}=k[A]$ $\ \ln {[A]}=-kt+\ln {[A]_{0}}$ $\ s^{-1}$ 2nd Order $r=-{\frac {d[A]}{dt}}=k[A]^{2}$ ${\frac {1}{[A]}}={\frac {1}{[A]_{0}}}+kt$ $\ M^{-1}s^{-1}$ ### Pseudo 1st Order Rate Laws

In some instances it proves difficult to near impossible to monitor concentrations of each reactant. A rate law for a reaction can be written in the "pseudo" 1st order for reactions involving multiple species, for instance species X and Y. By holding concentrations of one species constant, for instance X, there is essentially no net change in X vs time. As a result a new rate law can be defined to incorporate the X into the preexisting rate constant.

$\ k'$ is defined as $\ k[X]$ , thus our new rate equation is written as $\ r=k'[Y]$ Varying experimental conditions and the resulting data can be used to determine k.

A steady state approximation is useful in systems where it proves ultimately difficult to measure the concentration of one reactant (or one of its intermediates). However, if it is assumed that the concentrations of the species in question remains in a constant steady state, an equation can be written in terms of other species in which we can measure.

Lets examine a system that involves the following reactions:

 reaction rate equation A + B $\rightarrow$ X k1[A][B] X + C $\rightarrow$ D + E k2[X][C] X + E $\rightarrow$ F k3[X][E]

First we are going to assume that species X is in steady state.

$\ {d[X]}{dt}=0$ As part of out assumption X remains constant thus:

$\ {d[X]}{dt}=0=$ production terms - loss terms

$\ {d[X]}{dt}=0=$ k1[A][B] - k2[X][C] - k3[X][E]

Simplifying :

k1[A][B] = k2[X][C] - k3[X][E]

Solving for [X] in terms of reactants in which the concentrations can be experimentally measured, we obtain the equation:

${\frac {k_{1}[A][B]}{k_{2}k_{3}[C][E]}}=[X]$ 