29-Reaction Rates and Rate Laws

29-Reaction Rates and Rate Laws - 5.111 Lecture # 29....

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Unformatted text preview: 5.111 Lecture # 29. REACTION RATES AND RATE LAWS [Pages 529-547 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. om ON or o 9395 he p-._._oE c $.wa or . mod 3.32. c 4 .7._._oE mead _..O on A3325 2.5... ON or mv—Ow; NF PUT—OE O 9.33 o 9-....2: Rod mod low) uonenuaauoa JE|Ow .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 .83 wzogficfimfi 059:: 2% SW... Twice; TS x 3 me rmzzommauz 002:6 N I of + :ommsu Twice; TS x MWN me 7:25qu '5 + :ofu Al 1:0 + £65 Twice; :2 x 3 . me I5sz of N TI to + Bf cots—om mucosu< Tm TS X mum mun 7530:5353 “Eugen m ucwmoaofib Tm T2 x 3 m3 1:63 mmu N T £6 TwTEEg Ed mt. N102: No + oz N T Noz N tm or: - Tm one 82 5st No + N2 N T 002 N T2 x N.m wmm TS x .2 wNm TS x S, Mam Tm TS x 5m me 10st No + Noz v T w0N2 N TS x mm 0% TS x 3 2K i: x mm 08 . - _.-m._-_oeg TS x E 8% NEE a + N: Al E N N 8% TS x 3 9K TS x .3 com Twice; TS x 3 o8. @133 E N Al NH + N: 0239 80 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 mol-L'1 s1 (or Ms1 for short). Thus for n = 0, the units of k are M-s'l; for n = l, the units of k are s'l; for n = 2, the units of k are M'l-s'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 zero-order 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 first-order 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 first-order 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 first-order, and its slope will yield the value of k. (— macaw ,10 uonenueouoo Time -> © Bryan Hsu, 2008 Time a 1; (— [V] auepeax ;o uogmnuaauoa m|ow E' A very useful parameter in chemical kinetics, especially in the case of first-order chemical reactions, is the reactant half-life, 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 first-order rate law (and only for it) the half-life is independent of the initial concentration of the reactant; important implications. Finally, let’s consider second-order 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/[Alt-1/[Alo = kt 01‘ [Alt = [A]o/(1 + MAL) (15) Plotting [AL as a function of t —-— [Slide 29.6]. Experimentally distinguishing between first-order and second-order reactions and determining the k value for the latter —— [Slide 29.7]. Note that for a second-order reaction the first form of equation (15) gives rm = 1/k[A]o (16) i.e., the half-life depends on the initial concentration of the reactant. T “.25... .NI tosm 2 N mac. .D =55 « .—? :5 PIN O H < I—l [V] 'JUEJDEBJ 1.0 UOHEJJUBDUO) JEIOW k small k large T 9 [V] 'JUBDEBJ ;o uonenuaauoa Jelow O F! < hi T u 68:. 8: Al 4. .mEt. 3 £5: 6— [VI/l <— [V] uI x n mac—m «I u wnofi as: =_ ...
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29-Reaction Rates and Rate Laws - 5.111 Lecture # 29....

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