Review of Elementary Reaction Kinetics
Review of Elementary Reaction Kinetics
We will first explore the kinetics of noncatalyzed reactions as we did with our study of passive and facilitated diffusion. Before we do that, \(A\) brief review of the two major types of kinetic equations that you studied in general chemistry is in order, using the first order reaction of reactant \(A\) forming product \(P\) with \(A\) rate constant \(k_1\):

initial velocity equation (\(v_0\) as \(A\) function of the concentration of reactants): \(v_0 = k_1[A]\). Initial rate graphs are usually based on measurement of product increase with time (\(\frac{dP}{dt}\)) so \(v_0\) vs. \(A\) plots have positive slopes. This equation is true for any velocity v, not just the initial velocity as v is always directly proportional to \([A]\) at any concentration of \(A\). However it is easiest to measure the initial velocity of the reaction in the linear part of the decay curve when the rate is linear over \(A\) narrow range of \(A\) concentrations.

integrated rate equation which gives concentration as \(A\) function of time. \(A\) differential equation \(v = \frac{d[A]}{dt} = k_1[A]\) that on integration leads to the integrated rate equation: \(A = A_0e^{k_1t}\) giving \(A\) as \(A\) function of time \(t\).
First Order Reactions
Consider the first order reaction
\( A \xrightarrow{k_1} P \)
where \(k_1\) is the first order rate constant. For these reactions, the velocity of the reaction, \(v\), is directly proportional to \([A]\), or
\[ v = \dfrac{d[A]}{dt} = \dfrac{+d[P]}{dT} = k_1[A] \tag{1a} \]
The negative sign in d[A]/dt indicates that the concentration of \(A\) decreases. The equation could also be written as:
\[ v = \dfrac{d[A]}{dt} = k_1[A] \tag{1b}\]
For the rest of the reactions shown below, adopted convention here is to write all rates (velocities) as \(\frac{d[x]}{dt}\) as positive numbers. \(A\) negative sign for \(A\) term on the right hand side of the differential equation (as in 1b) will indicate that the concentration dependency of that term will lead to an decrease in \([x]\) with time. Likewise \(A\) positive sign for the term on the right hand side of the equation will indicate that concentration dependency of that term will lead to an increase in \([x]\) with time. 
Using this nomenclature, the following separable differential equation can be written and solved to find \([A]\) as \(A\) function of \(t\).
\[ \dfrac{d[A]}{dt} = k_1A \]
\[ \int _{[A_o]}^{[A]} \dfrac{d[A]}{dt} = k_1[A] \]
evaluate this integral at the limits assuming \( t_0 = 0 \)
\[ ln\; [A]  ln\; [A_o] =  k_1t  0\]
\[ ln\; [A] = ln\; [A_o]  k_1t \]
or
\[ [A] = [A_o] e^{ k_1t} \tag{2} \]
which is characteristic of firstorder kinetics.
Equation 2 is an example of an integrated rate equation. The following graphs show plots of \(A\) vs. \(t\) and \(\ln\,A\) vs. \(t\) for data from \(A\) first order process. Note that the derivative (\(\frac{dA}{dt}\)) of the graph of \(A\) vs. \(t\) is the velocity of the reaction. The graph of \(\ln\, A\) vs. \(t\) is linear with \(A\) slope of \(k_1\).The velocity of the reaction (slope of the \(A\) vs. \(t\) curve) decreases with decreasing \(A\), which is consistent with equation 1. Again, the initial velocity is determined from data taken in the first part of the decay curve when the rate is linear and little \(A\) has reacted (i.e., \([A] \approx [A_0]\)).
Figure: First Order Reaction kinetics: \( A \rightarrow P\)
Once again, for complete clarification, another way of analyzing the kinetics of \(A\) reaction, in addition to following the concentration of \(A\) reactant or product as \(A\) function of time and fitting the data to an integrated rate equation, is to plot the initial velocity, \(v_o\), of the reaction as \(A\) function of concentration of reactants. The initial velocity is the initial slope of \(A\) graph of the concentration of reactants or products as \(A\) function of time, taken over \(A\) range of times such that only \(A\) small fraction of \(A\) has reacted, so \([A]\) is approximately constant = \(A_o\). From the first order graph of \(A\) vs. \(t\) above, the slope approaches 0 with increasing time as \([A]\) approaches 0, which clearly indicates that the reaction velocity depends on \(A\). For \(A\) first order process, two equivalent equations, 1a and 1b, can be written.
Equation 1a above is written showing the disappearance of \(A\) as
\[v = \dfrac{d[A]}{dt} = k_1[A],\]]
while Equation 1b above is written showing the appearance of \(A\) as
\[v = \dfrac{d[A]}{dt} = k_1[A].\]
Both equations shows that v is directly proportional to \(A\). As \([A]\) is doubled, the initial velocity is doubled.
Velocity graphs used by biochemists often show the initial velocity of product formation (not reactant decrease) as \(A\) function of reactant concentration. Hence, as the concentration of product is increasing, the slopes of initial velocity are positive. \(A\) graph of \(v \, (= \frac{dP}{dt})\) vs. \([A]\) for \(A\) first order process would have \(A\) positive slope and be interpreted as showing that the rate of appearance of \(P\) depends linearly on \([A]\).