Chemistry · Ch 7 — Chemical Kinetics

Integrated Rate Law for a First Order Reaction

7.5.1

Integrated Rate Law for a First Order Reaction

Setting up the integral. A first order reaction has $\text{Rate}=k[A]^1$ for $A\rightarrow\text{product}$ , so

-\frac{d[A]}{dt}=k[A]\quad\Rightarrow\quad-\frac{d[A]}{[A]}=k\,dt

Integrating both sides between $t=0$ (concentration $[A]_0$ ) and $t=t$ (concentration $[A]$ ):

-\int_{[A]_0}^{[A]}\frac{d[A]}{[A]}=k\int_0^t dt\quad\Rightarrow\quad-\big(\ln[A]-\ln[A]_0\big)=kt\quad\Rightarrow\quad\ln\frac{[A]_0}{[A]}=kt

Converting the natural log to a base-10 log (multiply by 2.303, since $\ln x=2.303\log x$ ):

k=\frac{2.303}{t}\log\frac{[A]_0}{[A]}

This is the working formula used in virtually every first order numerical problem in this chapter.

The graphical form. Rearranging $\ln[A]_0-\ln[A]=kt$ as $\ln[A]=-kt+\ln[A]_0$ matches the straight-line equation $y=mx+c$ exactly, with $y=\ln[A]$ , $x=t$ , slope $m=-k$ , and intercept $c=\ln[A]_0$ . So plotting $\ln[A]$ (y-axis) against t (x-axis) gives a STRAIGHT LINE with a NEGATIVE slope for any genuinely first order reaction (Fig 7.3) -- and this is the standard experimental route to confirming a reaction is first order (the plot comes out straight) and to reading off k (from the slope). …

Figure 7.3A plot of ln[A] vs t for a first order reaction

What this figure shows. For $A\rightarrow\text{product}$ with $[A]_0=1.00$ M and $k=2.5\times10^{-2}\ \text{min}^{-1}$ : a straight line plotted with $\ln[A]$ on the y-axis (0 down to $-2.5$ ) against time in minutes on the x-axis (0 to 60), starting at $\ln[A]=0$ (since $\ln1=0$ ) and falling steadily as t increases -- the line's constant negative slope IS $-k$ , and it never curves, confirming the $\ln[A]$ vs t relationship is exactly linear for …

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Pseudo First Order Reaction

The problem pseudo first order kinetics solves. A reaction that is genuinely second (or higher) order, involving two DIFFERENT reactants, is awkward to follow kinetically -- you would need to measure the concentration of BOTH reactants changing simultaneously, which is experimentally difficult and error-prone. The workaround: deliberately take ONE of the two reactants in a huge excess relative to the other. Because its concentration barely budges over the whole course of the reaction (a small absolute amount reacted is a tiny fraction of a huge starting amount), it can be treated as effectively CONSTANT -- collapsing the apparent kinetics down to first order in the other reactant alone. This is called a pseudo first order reaction. …

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