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Unformatted text preview: 5.111
Lecture # 29. REACTION RATES AND RATE LAWS
[Pages 529547 from the “Chemical Principles” textbook, 4‘h edition, by Peter Atkins & Loretta Jones, Freeman, New York, 2008] What thermodynamics does and does not tell us about chemical reactions. Chemical kinetics — investigation of the rates of chemical reactions. Consider reaction
A —> P (1)
By deﬁnition, the rate of this reaction
Rate = d[P]/dt = d[A]/dt (2)
Note (i) the minus sign in reaction (1); (ii) that the rate depends on (typically, declines with) time; and (iii) the units of the reaction rates. One way to experimentally determine the reaction rate is to measure the slope of the tangent of the [A] or [P] versus reaction time curve, e. g., see [Slide 29.1]. Very useful and commonly used approach — determining initial reaction rates (Rateo).
For chemical reactions such as (l), the rate of disappearance of A, whether at t=0 or at a
later time, is often experimentally found to be directly proportional to [A] at that time point. In other words, Rate = k x [A] (3) Amy—wag «Eu.
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9....2: Rod mod low) uonenuaauoa JEOw .l (l Where k is the rate constant for the reaction. The units of k; what k is independent of and dependent on. Equation (3) is an example of a rate law. Each chemical reaction under given
experimental conditions has its characteristic rate law and rate constant; see [Slide 29.2]
for examples. Note in that table that While for some reactions the rate is directly
proportional to the reactant concentration (e.g., for the decompositions of N205, N20, and
C2H6), for others it is directly proportional to the square of the reactant concentration (e.g., for the decompositions of H1 and N02). However, in both of the foregoing instances
.Rate = k x [reactant]" (4) where n is the reaction order. If n = 1, it is a ﬁrst—order reaction; if n = 2, it is a second order reaction. While many chemical reactions are either ﬁrst or second order, other reaction orders are
not uncommon, e. g., n = 0 or even fractional reaction orders. Note that the order of a
reaction M be predicted from the stoichiometry of the chemical equation for the
reaction (such as those in [Slide 29.2]); rather, it is an experimentally determined property of the reaction. One also can see in [Slide 29.2] that some reactions have rate laws that depend on the concentrations of more than one reactants (e.g., the last three). Therefore, in general, $8st “58$:me REEL
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3338 Sam 1v: 85309:“? . k5“— Bmm nouumom mHENHmGOU wand USN mgmwm vumm _..ﬂ_. mdm<h Rate = k[A]“[B]b. .. (5)
where a, [9, etc., are the reaction orders with respect to [A], [1;], etc.; the sum of the
powers a + b + etc.. .. is called the overall reaction order. The units of k differ depending on the overall reaction order, so that the reaction rate
always has the units of concentration/time, usually molL'1 s1 (or Ms1 for short). Thus
for n = 0, the units of k are Ms'l; for n = l, the units of k are s'l; for n = 2, the units of k are M'ls'l; for n = 3, the units of k are M'z's'l; etc. The reaction order, with respect to both the individual reactants and overall, can be
determined experimentally. For example, let’s take the logarithms of both parts of
equation (5) for the initial rate: log(Rate0) = logk + alog[A]o + blog[B]O + (6)
By plotting log(Rateo) as a function of log[A]o, with all the other reactant concentrations
kept constant, we can determine a. The same for b, etc.; and then calculate a + b + etc. . ..
An integrated rate law gives the concentration of reactants or products as a function of time of the reaction. Consider this for different reaction orders. For a zeroorder reaction, d[A]/dt = k or d[A] = — kdt (7)
We can integrate the latter equation between the limits t = 0 (when [A] = [A]0) and the
time of interest, t (when [A] = [A] t): ld[A] =  lkdt to obtain [Ale  [A]: = kt 0r [Alt = [Ale — kt (8)
[Slide 29.3].
Now let’s consider ﬁrstorder integrated rate laws. d[A]/dt = k[A] or d[A]/ [A] = kdt (9)
Integration between the limits t = 0 (when [A] = [A]o) and the time of interest, t (when
[A] = [A] 1:), gives: l(d[Al/[Al) = lkdt (= ~16?)
to obtain mmmmamm m [NFMM“ am Equations (10) represent two forms of the integrated rate laws for a ﬁrstorder reaction. Exponential decay — [Slide 29.4]. Note that if the ﬁrst—order chemical reaction in question is (1), then mﬁwwawwwﬂewneb m) Equation (10) can be used to calculate the reactant and product concentration at any time
point after the reaction started. They also can be used (i) to measure the rate constant and
(ii) to verify that the reaction is indeed ﬁrst—order, by rearranging the ﬁrst equation in
(10) into ln[A]t = ln[A]o  kt (12)
and plotting 1n[A] t as a function of t. The resultant straight line, if obtained, will conﬁrm the ﬁrstorder, and its slope will yield the value of k. (— macaw ,10 uonenueouoo Time > © Bryan Hsu, 2008 Time a 1; (— [V] auepeax ;o uogmnuaauoa mow
E' A very useful parameter in chemical kinetics, especially in the case of ﬁrstorder chemical reactions, is the reactant halflife, t1 /2; by deﬁnition, it is the time required for the reactant’s concentration to drop by half. Therefore, from the ﬁrst form of [equation
(10) above t1/ = (l/k)'ln(0.5[A]o/[A]0) =(1/k)1n2 z 0.69/k (13)
Relationship between rm and k — [Slide 29.5]. Note that for the ﬁrstorder rate law (and
only for it) the halflife is independent of the initial concentration of the reactant; important implications. Finally, let’s consider secondorder integrated rate laws.
d[A]/dt = k[A]2 or d[A]/[A]2 = kdt (14)
Integration between the limits 1‘ = 0 (when [A] = [A]0) and the time of interest, 1‘ (when
[A] = [A] t), following straightforward rearrangements gives: ‘
1/[Alt1/[Alo = kt 01‘ [Alt = [A]o/(1 + MAL) (15)
Plotting [AL as a function of t —— [Slide 29.6]. Experimentally distinguishing between
ﬁrstorder and secondorder reactions and determining the k value for the latter —— [Slide 29.7]. Note that for a secondorder reaction the ﬁrst form of equation (15) gives
rm = 1/k[A]o (16) i.e., the halflife depends on the initial concentration of the reactant. T “.25... .NI
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