5/7/07 A quote of the wee (or amel of the wee): Apply yourself. Get all the eduation you an, but then... do something. Don't just stand there, mae it happen. Lee Iaoa Physial Chemistry GTM/5 reation order Reation order is a purely formal quantity, hene, its may be determined only experimentally. In the ase of elementary reations their order is equal to their moleularity (more later). The subjet of the onsiderations to follow may be summarized as: How to find the reation order on the basis of experimental data? It s worthy to notie that the methods presented in next slides are of historial importane rather, beause nowadays at the age of omputerized instrumentation (measuring and analytial devies) it is rather easy to aquire large data sets =f(t), whih subsequently may be fitted to ineti equations of different orders using omputerized regression tehniques. Physial Chemistry GTM/5 reation order () Those regression tehniques, however, are based stritly on the theory presented earlier and the methods presented below. These methods have their origins in times where data olletion was not so easy and omputation methods were more tedious. Therefore, planning of experiments required muh more sophistiated approah. These old methods are also very useful for analysis of ineti data (small data sets) in text problem solving exerises. Hene, their appliation is essential for hemists. It is always useful to inspet the data arefully if any regularities disussed in next setions may be observed. Quite frequently, espeially if the experiment was designed in a way favoring suh observations, one an notie, for example, independene of half-life time of onentration. Suh observations are still important even at the age of omputers, when they may suggest whih model (order) should be tested first. Physial Chemistry GTM/5 3
5/7/07 Equation testing method reation order (3) This is the simplest (most primitive), though most effetive approah (esp. if no regularities might be observed). In this method we use the data in integrated rate laws of several orders and he, whih one yields invariable rate onstant (independent of varying onentrations of reagents). Graphial method In this method, we plot the data in linearized systems of oordinates (suitable for different reation orders) and loo, whih system yields a straight line plot. Its slope is the rate onstant. One an notie that the regression method is a generalization of the both above mentioned methods. Physial Chemistry GTM/5 4 Half-reation time method reation order (4) We saw before that half-reation time id diretly proportional to the initial onentration for 0 th order, independent of the initial onentration for st order, and is inversely proportional for the nd order. This may be generalized by formula: n / 0 After finding the half life-time for several initial onentrations, proportionality may be tested using this formula for different orders n. Integral method by Ostwald-Zawidzi-Noyes This is a variant of the former method, in whih half life-time (or redued time, i.e. time of the same fration of reatants transformed) must be determined for two different initial onentrations. Subsequently, reation order n may be found from the formulae: n t t 0 t or n t 0 0 0 Physial Chemistry GTM/5 5 reation order (5) Differential method by van t Hoff In this method one utilizes no onentrations at given time but rates of reations at given time. It may be employed when reations are not very fast. Most frequently it is utilized for initial rates, when the following formula is valid: v n Isolation method (initial rates) 0 0 v In this method initial rates are measured in several experiments planned in a way permitting elimination of indluene of one or more reatants on the rate observed. An example is shown in the next slides. The method is important beause it permits to find the partial orders versus eah of the reatants!!! 0 0 Physial Chemistry GTM/5 6
5/7/07 Example: reation order (6) X + Y XY + X # Initial onentration mmol/dm 3 X Y Initial rate mmol/(dm 3 min) 450 70 0,8 50 70, 3 450 90 3,6 Physial Chemistry GTM/5 7 reation order (7) When we write rate laws for experiments # and #3 we an eliminate influene of reatant X, whose initial onentrations are the same in these experiments, by dividing both sides of the rate laws: m n m n m m v0 X 0 Y0 v03 X 03 Y03 X 0 X 03 n n v 0 Y0 Y0 v n v 0 v03 n 03 Y03 Y03 Y0 Y03 Even without taing the logarithms, one an see that onentration of Y is 3 times smaller in #3 than in # and the initial rate is 3 times smaller, too. Thus, the reation order vs. Y is. Treating experiments # and # in an analogous manner, eliminating influene of reatant Y, we see that onentration of X is 3 times smaller in # than in #, whereas the initial rate is 9 times smaller, meaning nd order vs. X. Physial Chemistry GTM/5 8 reation order (8) Experimental measurements of onentrations in time may be diffiult. If we measure during the ourse of reation any additive quantity X (e.g. pressure, volume, eletri ondutivity, density, absorption of light), we an alulate onentration after time t using the formula: t X X X 0 X 0 where subsript means measurement after time, when the additive quantity in question does not hange anymore (within the unertainty limits of the used measuring or analytial tehnique). Physial Chemistry GTM/5 t 9 3
5/7/07 A bit of integration or inetis of seond (and higher) order reations at different initial onentrations of reatants: For a seond order reation A=A appliability of formerly derived equations annot be questioned. The same is true when for reation A+B=AB =. What to do, however, if? There are two possible approahes. If >> (or vie versa), then we an treat the seond order reation as a pseudofirst orede reation, inluding onentration of the dominating reatant, whih is pratially onstant, into the rate onstant. v ' A B B Farther integration is no problem at all. Please, note that the onept of half reation time may be applied only to reatant B. Physial Chemistry GTM/5 B 0 A bit of integration() Given is reation A+B=AB, where and both values are ommensurable (the same order of magnitude). Here, we must selet ertain variable x, whih permits us to express onentration of one omponent as a funtion of the seond. da db dab v A B A x; B x; x x AB x We have only one variable x and several onstants. Physial Chemistry GTM/5 A bit of integration (3) Having a differential equation of one variable (and time), we an separate x x the variables and integrate: We must, however, use the integration by parts tehnique. It means we must deompose the integrand in two funtions. It is possible to find two onstants (K, L) giving: K L x x x x K ; L Physial Chemistry GTM/5 4
5/7/07 A bit of integration (4) then we an write x x t onst x the integral as: and finally find the x indefinite integral: From the border onditions: t=0, x=0; we may find the integration onstant: Finally obtaining the formula: Physial Chemistry GTM/5 onst 0 A x t x 3 Bit of integration (5) For a general form of a seond order reation: A B... T when: x A x B T da db dt v T Integrated rate law has the form: 0 A x A B x t x Physial Chemistry GTM/5 4 Elementary reations & omplex reations Chemial reations usually do not our as they are written in hemial equations, whih represent their summarial stoihiometry only. Majority of reations are from the point of view of their inetis omplex reations. It means that their our in several steps (stages), eah of the steps being an elementary reation. All elementary reations onstituting a given omplex reation represent the mehanism of the latter. Elementary (simple) reations do our as indiated by their equations. Therefore, their order may be inferred on the basis of their moleularity, i.e., the number of moleules whih must meet (ollide) to result in the hemial reation. For elementary reations order = number of moleules of the reatants. Physial Chemistry GTM/5 5 5
5/7/07 Reation rates dependene on temperature Rate of majority of hemial reations (and all elementary reations) inreases with temperature. A useful approximate rule (van t Hoff s) is assumption that reation rate inreases twie when temperature is raised by 0 o C (K). Dependene of reation rate on temperature means atually dependene of its rate onstant on temperature. T 0 T More exat relation between temperature and reation rate onstant is given by Arrhenius equation: = Ae E a Physial Chemistry GTM/5 6 Arrhenius equation E A ativation energy A preexponential fator (frequeny fator) = Ae E a Ea = A Results of measurements of as a funtion of T, plotted in a linearized system of oordinates =f(/t), permit determination of the parameters of Arrhenius equation. Slope: Ea tan= Physial Chemistry GTM/5 R 7 Arrhenius equation () E A ativation energy is the lowest energy that the reatants must have to get transformed to produts. It may also be interpreted as the fration of ollisions between moleules of suffiient ineti energy to exeed E a. This is given by Maxwell-Boltzmann distribution. A preexponential fator is frequently nown as the frequeny fator reflets the frequeny of ollisions regardless their energy. The produt of both represents the number of suessful ollisions in time. Physial Chemistry GTM/5 = Ae E a e 8 E a 6
5/7/07 Kinetis of reversible reations So far we have treated all reation as running to ompletion (in stoihiometri sense), though sometimes ompletion was ahieved after time t=. In reality, reations run to the point of equilibrium rather and we must find a suitable desription of suh situations. Equilibrium is not a stati state (altough observation may suggest that the reation does not our at this state nothing happens ). Atually, a dynami equilibrium exists, when two reations our: left to right and right to left, but their rates are the same. A - B A B v A v B v v A K B K is reation (onentration)equilibrium onstant. Physial Chemistry GTM/5 9 Kinetis of reversible reations () da Rate law (differential) may be expressed as: A A0 B0 whih, after integration, assumes t the following form: A B If the equilibrium omposition (onentrations) is nown, one an write: A0 A t A A The ase with two seond order reations is more omplex, beoming very omplex when reations ba and forth are of different orders. B Physial Chemistry GTM/5 0 Parallel reations If, at given onditions, one reatant yields more than one produt, whih may be represented by the sheme: then, assuming that all three reations are st order reations, one an write: da A A 3A obtaining after integration: 3 t A Suh situations an be met in organi hemistry, when three isomers may be obtained (e.g. o-, m-, and p-). Physial Chemistry GTM/5 7
5/7/07 Conseutive reations Conseutive reations are represented by sheme: A B C In whih both reations (A B and B C) are elementary reations. Thus one an say, that from the point of view of the omplete series, substane B is an intermediate produt. One an also write the following rate laws (assuming both reations are first order ones): da A db dc () A B () B (3) We are going to derive the equations permitting us to alulate time dependent onentrations of eah substane: Physial Chemistry GTM/5 Conseutive reations () For substane A we get simple exponential deay: t A e If we put the result into equation (), assuming =0: B e t t e Finally, allowing that for any instant of reation, if =0 and 0C =0, A + B + C = must be true, we get: C e t e t Physial Chemistry GTM/5 3 Conseutive reations (3) The plot of normalized onentrations of A, B and C in time przedstawiono niżej: Conseutive Reaje następze reations. 0.8 [X]/[A 0] 0.6 0.4 0. 0 0 0 0 30 40 50 60 Chem. Fiz. TCH tii/8 4 8
5/7/07 Conseutive reations (4) One an also alulate time, after whih B reahes its maximum: db t t e e This expression is equal to zero, when: e t e t 0 it means that t=t max is equal to t max Physial Chemistry GTM/5 5 9