IntroductionKineticsKinetics with UV/VIS spectroscopyExperimental methodData analysisWritten laboratory reportReferencesReferencesExperiment 2: Kinetics of a reversible, first-order, consecutive reaction(Dated: March 3, 2010)I. INTRODUCTIONThe majority of chemical reactions do not take place at a single collision but rather have a reaction mechanism,which involves several elementary reaction steps. Thus the reaction may not follow a simple first- or second-orderrate law. When the rate law is inconsistent with the stoichiometry of the chemical equation, the reaction cannot beelementary. An example is the oxidation of the formate ion by permangate in water, which has been observed to besecond-order experimentally:2MnO−4+ 3HCO−2→ 2MnO2+ HCO2−3+ H2O (1)−d£HCO−2¤dt= k2£HCO−2¤ £MnO−4¤The stoichiometry immediately establishes that the reaction must be more complex than a single bimolecular reactionstep. Although reactions consisting solely of a simple bimolecular encounter process will give second-order kinetics,the converse is not necessarily true.The differential form of the rate law is usually much easier to write than the corresponding integrated rate law.There is no general method of solving complex differential rate equations, since each case usually requires a specifictreatment. In many problmes it is not possible to find analytic solutions and in such cases only numerical methodscan be applied. Special methods exist, for example, for the following three important cases [1–3]:1. Reversible reactions, or reactions proceeding towards equilibrium, wherein measurable concentrations of reac-tants remain.2. Concurrent or side reactions which control product distribution where more than one process is available to areactant.3. Consecutive reactions where the initial products are reactants for subsequent reactions.In this laboratory excercise kinetics of the oxidation of tripeptide gluthatione-γ-L-glutamyl-L-cysteinylglycine (GSH;see Fig. 1) by Cr(VI) is studied at neutral pH, which results in formation of glutathionyl disulfide (“oxidized GSH”,GSSG; see Fig. 2). The overall reaction mechanism for the reaction has been determined as:2CrO2−4+ 6GSH + 10H+→ 2Cr3++ 3GSSG + 8H2O (2)FIG. 1: A two dimensional structure of GSH is shown (“reduced GSH”).This reaction is believed to partly account for the toxicity and carcinogenicity of Cr(VI). GSH and GSSG functionas a redox couple, both in intracellular and plasma environments. GSH provides an important defense mechanismagainst certain toxic compounds, such as some drugs and carcinogens. If the levels of GSH in a tissue such as liver arelowered, then that tissue have been shown to be more susceptible to injury by various chemicals that would normallybe conjugated to GSH.The reaction consists of two consecutive and one reversible step (i.e, categories 1 and 3 above), which can be writtenas:Typeset by REVTEXFIG. 2: A two dimensional structure of GSSG is shown (“oxidized GSH”).(I) CrO−24+ GSH ⇋ CrO2−4– GSH (3)(II) CrO2−4– GSH + GSH → GSSG + Cr3+With excess concentrations of GSH and H+, all three reaction steps (i.e., forward and backward reactions in (I)and forward reaction in (II)) follow effectively first order kinetics. High-order reactions that behave like first orderreactions in presence of excess concentrations of the reactants are called pseudo first order reactions.For more information on kinetics, see Refs. [1–4].II. KINETICSIn this work excess concentrations of GSH and H+are present and thus all of the reactions follow first orderkinetics (i.e., the isolation method). As will be discussed later on, the concentrations of Cr ions are monitored in thisexperiment and therefore the rate equations are written in terms of their concentrations (not GSH or GSSG). Whenwe denote r =£CrO2−4¤, i =£CrO2−4– GSH¤, and p =£Cr3+¤, the reaction mechanism can be written as:rk1⇋k3ik2→p (4)The empirical differential rate equations for this reaction can be written as:dr(t)dt= −k1r(t) + k3i(t) (5)di(t)dt= k1r(t) − (k3+ k2)i(t)dp(t)dt= k2i(t)where functions r(t), i(t) and p(t) are functions that determine concentrations of components r, i and p at given timet. In mathematical terms Eq. (5) represents three coupled ordinary differential equations. Obtaining their analyticsolution is quite laborious and here, instead of solving the problem with a pen and paper, we demonstrate the use ofa modern symbolic algebra package (e.g., Maxima [5]). The following maxima program will compute the unknownfunctions r, i and p, which satisfy Eq. (5):2/** Solve the coupled first order differential equations analytically.**//* Define the equations */eq1: ’diff(r(t),t,1) = -k1 * r(t) + k3 * i(t);eq2: ’diff(i(t),t,1) = k1 * r(t) - (k3 + k2) * i(t);eq3: ’diff(p(t),t,1) = k2 * i(t);/* Form a list of equations and substitute in auxiliary variables */eqs: [eq1, eq2, eq3];eqs: subst(l1 + l2 - k1 - k2, k3, eqs);eqs: subst(l1 * l2 / k1, k2, eqs);eqs: subst((a * l2 + l1) / (a + 1), k1, eqs);/* Solve the coupled differential equations */res: desolve(eqs, [r(t), i(t), p(t)]);/** Substitute in the initial conditions:* r(0) = r0 (constant; initial concentration of the reactant)* i(0) = 0* p(0) = 0**/res: subst(r0, r(0), res);res: subst(0, i(0), res);res: subst(0, p(0), res);/* Simplify and multply through with the exponent function */scanmap(multthru, ratsimp(res));The above program is available as a Maxima batch file on the laboratory course web page. The program can be rununder wxMaxima graphical user interface by choosing “File → Batch file” (see the general section in the laboratorymanual for details). Even though programs like Maxima can perform symbolic calculations, the results should alwaysbe double checked with a pen and paper! The relevant part of the output resides at the very end and looks like:[r (t) =a r0e−l2ta + 1+r0e−l1ta + 1, (6)i (t) = −a l2r0e−l2ta l2+ l2− a l1− l1−l1r0e−l2ta l2+ l2− a l1− l1+a l2r0e−l1ta l2+ l2− a l1− l1+l1r0e−l1ta l2+ l2− a l1− l1, (7)p (t) =l1r0e−l2tl2− l1−l2r0e−l1tl2− l1+ r0] (8)where subscripts have been inserted for clarity and r0denotes the initial concentration of the reactant r. Note thatthe initial concentrations of i and p were assumed to be zero. The following notations were used in order to obtainthe simplified expressions:k1=a × l2+ l1a + 1, k2=l1× l2k1, k3= l1+ l2− k1− k2(9)which relate the variables in Eqs.