Reactions—First Order

Reactions displaying first-order reaction kinetics are extremely common. Fortunately, the mathematics needed to describe first-order reactions are also quite straightforward. In a first-order reaction, the reaction rate is directly proportional to the concentration of one of the reactant concentrations. Thus, increasing the concentration of this reactant will speed up the rate of the reaction proportionally. This behavior reflects the fundamental kinetic principle that the speed of most 

FIGURE 3.2 Variation in reactant concentration as a function of time for a zero-order reaction that consumes species A. The concentration of species A decreases linearly with time from an initial concentration c. The slope is given by the rate constant k. 

It is important to note that a first-order reaction depends on the concentration of only one reactant. Other reactants can be present, but they will have zero order. 

As we did with the zero-order rate law, this differential equation can be integrated to obtain an equation that directly expresses how the concentration of the reactant varies as a function of time during the reaction process. For a general first-order reaction involving the consumption of a reacting species A, this integration yields. 

Most nuclear decay processes obey first-order reaction kinetics. An example is the radioactive decay of carbon-14 (an unstable radioactive isotope of carbon) to nitrogen-14 (which is the stable isotope of nitrogen) via the emission of an electron and an antineutrino (Vg)  

A first-order reaction is one in which the rate of the reaction is proportional to the concentration of only one of the reacting substances. Algebraically, 

By using decadic instead of natural logarithms this may be written as 

The integrated form of the first-order equation (II.4.3) provides us with a simple graphical method of representation as shown in Fig. II.la, in which 

In Ca is plotted as a function of time L For purposes of comparison Fig. II.lb shows the appearance of the data when the concentration Ca is plotted directly against the time. The logarithmic plot of Fig. II. la is 

It is convenient to define two other quantities related to the specific rate constant. The first is called the mean life or decay time of the reaction and is represented by the symbol r. It is defined as the time for the concentration of A to fall to l/e of its initial value, where e is the natural number 2.718. By substitution in Eq. (II.4.3), we see that (7a will reach the value Chje after a time r = I/Aa. That is, the mean life t of the reaction is equal to the reciprocal of the rate constant. 

A first-order reaction is one whose rate depends on the concentration of a single reactant raised to the first power. For a reaction of the type A — products, the rate law may be first order  

This form of a rate law, which expresses how rate depends on concentration, is called the differential rate law. Using an operation from calculus called integration, this relationship can be transformed into an equation that relates the initial concentration of A, [A]q, to its concentration at any other time t, [AJ  

This form of the rate law is called the integrated rate law. The function In in Equation 14.12 is the natural Ic arithm (Appendix A.2). Equation 14.12 can also be rearranged to 

Equations 14.12 and 14.13 can be used with any concentration units as long as the units are the same for both [A]( and [A]q. 

For a first-order reaction. Equation 14.12 or 14.13 can be used in several ways. Given any three of the following quantities, we can solve for the fourth k, t, [A]q, and [A],. Thus, you can use these equations to determine (1) the concentration of a reactant remaining at any time after the reaction has started, (2) the time interval required for a given fraction of a sample to react, or (3) the time interval required for a reactant concentration to fall to a certain level. 

It IS not necessary for you to be able to do the calculus required to arrive at Equation 14.3, but it is very important that you know how to use Equation 14.3. 

Setting these two expressions of the rate equal to each other we get 

Applying calculus to the preceding equation, we can show that Equation 14.3 

In Sample Problem 14.4 we apply Equation 14.3 to a specific reaction. 

The decomposition of hydrogen peroxide is first order in H202- 

A plot of In [A] versus time t gives a straight line with a slope of —k. 

The half life of a first-order reaction is independent of the starting concentration and is given by the equation  

2 First order reactions The elementary reaction A ential equation 

In this case the rate constant k has the dimension [s ]. By integration within the limits f = 0, a = a, jc = 0 and t, respectively, one obtains 

First order reactions show no dependence of the half-life on the initial concentration. According to eq. (2.23), the half-life yields 

The slope at the beginning of the reaction is Oq = —ka. The definition of the general coordinates 

Furthermore the initial slope of a with respect to r becomes 

An overall first-order reaction has a rate law in which the sum of the exponents, m + n + , is equal to 1. A particularly common type of first-order reaction, and the only type we will consider, is one in which a single reactant 

The rate of reaction depends on the concentration of H2O2 raised to the first power, that is. 

It is easy to establish that the reaction is first order by the method of initial rates, but there are also other ways of recognizing a first-order reaction. 

Let us begin our discussion of firsf-order reactions as we did zero-order reactions, by examining a hypothetical reaction 

We can obtain the integrated rate law for this first-order reaction by applying the calculus technique of integration to equation (20.12). The result of fhis derivation (shown in Are You Wondering 20-4) is 

Often a rate depends only on the first power of one concentration. Consider the conversion A — P that proceeds to completion according to the rate law 

After separating the variables, this equation can be solved by integration between the limits (fo = 0, [A]0) and (t, [A],). The integrated expression is 

If other reagents affect the rate, then setting their concentrations much higher may sometimes be desirable. They would remain effectively constant during the course of 

The rate of a second-order reaction may be proportional to two concentrations, v = [A][B] with [B]0 s [A]o it follows first-order kinetics. Some authors refer to these as the order with respect to concentration and the order with respect to time. 

For example, imagine that the reaction between BrOj and Br in acidic solutions, Eq. (1-11), is conducted with [H+]0 = 910 X [BrOj ]0 and [Br-]0 = 280 x [BrOj ]0. The effective concentrations of H+ and Br- being nearly constant, the rate equation would become 

An irreversible first-order reaction involves only one reactant  

Since 0 has units of moles per volume per time and a has units of moles per volume, the rate constant for a first-order reaction has units of reciprocal time, for example, reciprocal seconds. The best example of a truly first-order reaction is radioactive decay, for example. 

The rate of first-order reactions is determined by one concentration term and may be written using equation (4.5) as 

Reproduced from K. C. James, J. Pharm. Pharmacol., 10, 363 (1958) with permission. 

According to equation (4.11), a plot of the logarithm of the amount of dmg remaining (as ordinate) as a function of time (as abscissa) is linear if the decomposition follows first-order kinetics. The first-order rate constant may be obtained from the slope of the plot (slope = -kj/2.303). kj has the dimensions of time.  

The time taken for half of the reactant to decompose is referred to as the half-life of the reaction, fg j. An expression for 1q j for a first-order reaction may be derived from equation (4.10), noting that when t= Iq j, x = a/2  

The half-life is therefore independent of the initial concentration of reactants. 

in a mixture of A and B, these components react by parallel first-order processes to give a common product C, and A and B do not interconvert, then a first-order plot of the rate of appearance of P will be curved, having a fast and a slow component. 

This situation is commonly encountered in the measurement of radioactive decay of a mixture of radioisotopes. 

The unstable czs-octahedral isomer of the nickel(II) complex of the macrocyclic ligand 13aneN4 (1,4,7,10-tetraazacyclotridecane) isomerizes to an intermediate planar isomer, which then converts to the stable planar isomer 

Anne M. Martin, Kenneth J. Grant and E. Joseph Billo, Inorg. Chem. 1986, 25,4904. 

Equation (2-4) is the stoichiometric equation for an elementary first-order reaction, and Eq. (2-5) is the corresponding differential rate equation. 

Separating the variables and integrating between the limits shown below yields Eqs. (2-6), (2-7), and (2-8) as equivalent forms of the integrated first-order rate equation. 

For a first-order reaction, therefore, a plot of In Ca (or log Ca) vs. / is linear, and the first-order rate constant can be obtained from the slope. A first-order rate constant has the dimension time , the usual unit being second.  

The half-life tvi is defined to be the time required for the reactant concentration to decay to one-half its initial value. To find tvi for a first-order reaction we use Eq. (2-6) with the substitutions Ca = c°/2 and t = finding 

Note that t is independent of concentration for a first-order reaction. 

The treatment here, due to Wei and Kuo , lumps first-order reactions with f = -Kc. It projects the system onto a lower dimensional space via a linear transformation c = Me where M is an xn lumping matrix. Thus M transforms the -tuple vector c into an h -tuple vector c of a lower rank h n n). the system is exactly lumpable by M, then one finds m h xn matrix K such that 

The consequence of the lumpability on the eigenvalues and eigenvectors of the system is that the vector Mx, either vanishes or is an eigenvector of K with the same eigenvalue Xi, that is, 

Hence the matrix K = XAX has only h eigenvectors X = Mx,. Of the original n eigenvectors, n - n eigenvectors vanish after the projective transformation. 

If M is known in advance, K can be found from Eq.(9). To construct M, one rewrites Eq.(9) (the superscript T denotes transpose) 

Viewing as a mapping, Eq.(ll) says that the action of on is just to 

The most basic type of rate equation is the first-order decay and we will give complete details of the mathematics here. There are a number of spontaneous reactions in nuclear chemistry and organic chemistry. A basic characteristic of any reaction is that the more reactant there is, the more the reaction will proceed but as the amount of reactant decreases the reaction will be slower. Thus, the rate of the reaction is proportional to the concentration of the reactant. 

The method we want to teach here is to identify the extent of reaction x relative to the initial concentrations. An increase of B corresponds to the appearance of x. Let a be [A] at time f = 0 and for simplicity, let [B] = 0 at t = 0. Then, ki has units of (1/f) and where x is now the concentration of [B] for times greater than zero (zero is whenever you start your clock ). 

Perhaps it is a good idea to write the variables under the chemical reaction as follows  

we use the key concept that if there is a proportionality such as C oc D we can immediately write C = fcD using the basic idea that a proportionality symbol oc can be replaced by = k, where k is called the proportionality constant. In kinetics, the proportionality constant is called the rate constant. Here we add a subscript to the rate constant to indicate the order of the reaction as k for a first-order reaction. We will try to do this for higher orders but eventually in complicated cases we may abandon this simpler convention. Next, we can rearrange the kinetic equation to separate x and t variables. 

If we recall our joke in the math review chapter --—— = In (cabin) + C, we can integrate the 

Since substance A is the only reactant, we choose to balance the equation with the coefficient of A equal to unity. Suppose that the reaction is first-order with respect to A and that the rate does not depend on the concentrations of any products then the rate law, Eq. (32.11), becomes 

To integrate this equation we must either express c as a function of i/V or /V as a function of c. In either case, we obtain the relation by dividing Eq. (32.2) by V, 

Thus for a first-order decomposition, the concentration of A decreases exponentially with time. After measuring c as a function of time we can test whether the reaction is first order in A by plotting In (c/cq) versus t. According to Eq. (32.16) this plot should be a straight line if the reaction is first order in A If we find that our experimental points lie on a straight line we conclude that the reaction is first order in A. The slope of this line is equal to — k. 

The half-life, t, of the reaction is the time required for the concentration of A to reach one-half of its initial value. Therefore, when t = t, c = jCq. Putting these values into 

One way to evaluate the rate constant of a reaction is to determine the half-life for various initial concentrations of the reactant A. If the half-life is independent of the initial concentration, then the reaction is first order, and the rate constant is calculated using Eq. (32.18). It is only for first-order reactions that the half-life is independent of the initial concentration. 

As well as the obvious example involving one reactant (even with this solvent may be involved) a number of reactions between A and B that might have been expected to be second-order, first-order in A and in B, turn out to be first-order only (say in A). Obviously some feature of A, not directly connected with the main reaction with B, must be determining the rate. The product of this rds, A, must react more readily with B than A does. It is possible to check the correctness of this idea by independent study of the A — A, interconversion. An isomerization within a complex may limit the rate of its reaction with another reagent. 

The reaction of planar Ni ([14]aneN4) + represented as shown in (2.10) with a number of bidentate ligands (XY) to produce c -octahedral Ni ([14]aneN4) XY + is first-order in nickel complex and [OH ] and independent of the concentration of XY.In the preferred mechanism, the folding of the macrocycle (base-catalyzed tmns — cis isomerization) is rate determining, and this is followed by rapid coordination of XY  

In an alternative mechanism a monodentate intermediate (1) is in rapid equilibrium with reactants and it undergoes at high [XY] rate-determining ring closure. Such a type of mechanism is believed to operate for Ni(trien) + interacting with XY. Reasons for the preferred mechanisms are given. The isomerization may take the form of a conformational change in a metalloprotein. 

The reactivity of a dimer may be limited by its fragmentation. The rates of a number of reactions of cobalt (III) peroxo species (any charges omitted) are limited by their breakdown (rds) 

The scavenging of Co(II)L5 or O2 by added reagent follows rapidly. The rate law does not therefore include the concentration of the added reagent,except in certain instances. Finally, the first-order dominance of a reaction between A and B may only become apparent at higher concentrations of B (Sec. 1.6.3). 

It is assumed that at the beginning of the reaction (t = 0) the concentration of A is a and that of P (products) is zero. If after time t the concentration of P is x, that of A would be (a - x). The rate of formation of P is d%/dt. A first-order reaction can be expressed as  

This is the integrated rate equation for a first-order reaction. When dealing with first-order reactions it is customary to use not only the rate constant, k for the reaction but also the related quantity half-life of the reaction. The half-life of a reaction refers to the time required for the concentration of the reactant to decrease to half of its initial value. For the first-order reaction under consideration, the relation between the rate constant k and the half life t0 5 can be obtained as follows  

There can arise two possibilities in the case of reactions of the second order the rate may be proportional to the product of two equal initial concentrations or to the product of two different initial concentrations. The first case corresponds to a situation where a single reactant is involved, the process being represented as  

It may also be pertinent to a reaction between two different reactants, 

This equation is put into a form so that a standard integral results take all terms in one variable, i.e. [A], to one side and all terms in the other variable, i.e. t, to the other side. 

These can be integrated. Because k is a constant it can be taken outside the integral sign  

In kinetics the change in concentration with time is followed from the start of the reaction, [A]0 at t = 0 to [A], at time t. These are the limits between which the integral is taken. 

If the experimental data fit first order kinetics, then a plot of loge[A]r versus t should be linear. If the plot is curved the data do not fit first order kinetics. The slope is negative and equals —k, hence k is found. 

Equation (3.23) is also the equation of the experimental curve of [reactant] versus time. This can be confirmed by calculating values of loge[Ah and thence [Ah from equation (3.23), and plotting [A], versus t. This should regenerate the experimental curve. 

The way we name things inevitably affects how we perceive those things. 

Reaction rates depend on concentrations of reactants that change during the reaction. Thus, the rate of a reaction will usually be greatest at the beginning of a reaction and will decrease with time until equilibrium is established. One common type of chemical reaction is the first-order reaction, in which the rate of disappearance of a reactant is proportional to the reactant concentration. This leads to an exponentially decreasing reaction rate as time proceeds (see Section 4.3). Other types of kinetics are also possible. 

The solvolysis of 2-chloro-2-methylpropane in water occurs by the mechanism 

The first step is the rate-determining one for this reaction, and we will follow the velocity of this reaction. In this step, the haloalkane ionises to produce an intermediate species, in this case a carbocation. Subsequently, the carbocation can be attacked by water to give either 2-methylpropanol or 2-methylpropene. The overall reaction is known as unimolecu-lar nucleophilic substitution, S l, when the nucleophilic attack predominates. Alternatively, if after initially forming the same carbocation elimination of a proton occurs to give 2-methylpropene, the process is termed unimolecular elimination, El. At room temperature, the predominant reaction is S l, but the rate-determining step is the same for both reactions, and both lead to formation of the same amount of HCI. It is not uncommon in chemical kinetics for different pathways to occur via the same intermediate. As the only ionic species formed in either case is HCI, the progress of the reaction can be followed by the increase of the conductance of the solution. If Aq is the conductance of the solvent. A, the conductance of the solution at time t and A the final conductance of the solution (r = t ), then 

The proportionality constant can be obtained experimentally using the same conductivity cell by measuring the increase in conductivity in the same solvent or in a mixture of solvents owing to the formation of HCI. 

Experiment 3.1. Experimental determination of the rate constant and reaction order for the solvolysis of 2-chloro-2-methylpropane in ethanol-water mixture [2-4] 

Material. Stopwatch, thermostat bath, conductivity cell, 1 cm pipette (or syringe), 25 cm pipette, 100 cm measuring cylinder, 250 cm Erlenmeyer flask. 

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An example of a two-stage hydrolysis is that of the sequence shown in Eq. IV-69. The Idnetics, illustrated in Fig. IV-29, is approximately that of successive first-order reactions but complicated by the fact that the intermediate II is ionic [301]... 

A monolayer undergoes a first-order reaction to give products that also form monolayers. An equation that has been used under conditions of constant total area is (t - K°°)/(ifi - t") = exp(-)kr). Discuss what special circumstances are implied if this equation holds. 

Let us compare computations of the effectiveness factor, using each of the three approximations we have described, with exact values from the complete dusty gas model. The calculations are performed for a first order reaction of the form A lOB in a spherical pellet. The stoichiometric coefficient 10 for the product is unrealistically large, but is chosen to emphasize any differences between the different approaches. 

The most crucial observation concerning the effects of added species is that nitrate ion anticatalyses nitration without changing the kinetic form of the reaction. This shows that nitrate does not exert its effect by consuming a proportion of the nitronium ion, for, as outlined above, this would tend to bring about a kinetically first-order reaction. Nitrate ions must be affecting the concentration of a precursor of the nitronium... 

The effect of nitrous acid on the nitration of mesitylene in acetic acid was also investigated. In solutions containing 5-7 mol 1 of nitric acid and < c. 0-014 mol of nitrous acid, the rate was independent of the concentration of the aromatic. As the concentration of nitrous acid was increased, the catalysed reaction intervened, and superimposed a first-order reaction on the zeroth-order one. The catalysed reaction could not be made sufficiently dominant to impose a truly first-order rate. Because the kinetic order was intermediate the importance of the catalysed reaction was gauged by following initial rates, and it was shown that in a solution containing 5-7 mol 1 of nitric acid and 0-5 mol 1 of nitrous acid, the catalysed reaction was initially twice as important as the general nitronium ion mechanism. 

II [Anisole] = 2 x lo mol i" first-order reactions. For the experiment using pure nitric acid the half-life was about i min, but for that using fuming nitric acid reaction was complete in < 30 s. 

Fixed-time integral methods are advantageous for systems in which the signal is a linear function of concentration. In this case it is not necessary to determine the concentration of the analyte or product at times ti or f2, because the relevant concentration terms can be replaced by the appropriate signal. For example, when a pseudo-first-order reaction is followed spectrophotometrically, when Beer s law... 

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

We know from equation 13.6 that for a pseudo-first-order reaction, the concentration of picrate at time t is... 

For a first-order reaction we can write, from equations 13.2, 13.21, and 13.22... 

Equation 13.14 shows how [A]o is determined for a two-point fixed-time integral method in which the concentration of A for the pseudo-first-order reaction... 

First-Order Reactions The simplest case is a first-order reaction in which the rate depends on the concentration of only one species. The best example of a first-order reaction is an irreversible thermal decomposition, which we can represent as... 

Reaction A5.5 is not the only possible form of a first-order reaction. Eor example, the reaction... 

Proceeding in the same manner as for a first-order reaction, the integrated form of the rate law is derived as follows... 

To determine the reaction order we plot ln(%p-methoxyphenylacetylene) versus time for a first-order reaction, and (%p-methoxyphenylacetylene) versus time for a second-order reaction (Figure A5.1). Because the straight-line for the first-order plot fits the data nicely, we conclude that the reaction is first-order in p-methoxyphenylacetylene. Note that when plotted using the equation for a second-order reaction, the data show curvature that does not fit the straight-line model. 

Chemical Properties. Vacuum thermal degradation of PTFE results in monomer formation. The degradation is a first-order reaction (82). Mass spectroscopic analysis shows that degradation begins at ca 440°C, peaks at 540°C, and continues until 590°C (83). 

In the normal process ( ), step (J) occurs very rapidly and step (/) is the rate-determining step, whereas in the inhibition process (B), step (3) occurs very slowly, generally over a matter of days, so that it is rate determining. Thus it has been demonstrated with AChE that insecticides, eg, tetraethyl pyrophosphate and mevinphos, engage in first-order reactions with the enzyme the inhibited enzyme is a relatively stable phosphorylated compound containing one mole of phosphoms per mole of enzyme and as a result of the reaction, an equimolar quantity of alcohoHc or acidic product HX is hberated. 

The reaction/mass-transfer technique is based on Danckwerts theory of mass transfer accompanied by a fast pseudo first-order reaction (10) ... 

Numerous kinetic mechanisms have been proposed for oil shale pyrolysis reactions (11—14). It has been generally accepted that the kinetics of the oil shale pyrolysis could be represented by a simple first-order reaction (kerogen — bitumen — oil), or... 

For weU-defined reaction zones and irreversible, first-order reactions, the relative reaction and transport rates are expressed as the Hatta number, Ha (16). Ha equals (k- / l ) where k- = reaction rate constant, = molecular diffusivity of reactant, and k- = mass-transfer coefficient. Reaction... 

Fig. 15. Temperature vs heat generation or removal in estabHshing stationary states. The heavy line (—) shows the effect of reaction temperature on heat-generation rates for an exothermic first-order reaction. Curve A represents a high rate of heat removal resulting in the reactor operating at a low temperature with low conversion, ie, stationary state at a B represents a low rate of heat removal and consequently both a high temperature and high conversion at its stationary state, b and at intermediate heat removal rates, ie, C, multiple stationary states are attainable, c and The stationary state at c ...

As a reactant molecule from the fluid phase surrounding the particle enters the pore stmcture, it can either react on the surface or continue diffusing toward the center of the particle. A quantitative model of the process is developed by writing a differential equation for the conservation of mass of the reactant diffusing into the particle. At steady state, the rate of diffusion of the reactant into a shell of infinitesimal thickness minus the rate of diffusion out of the shell is equal to the rate of consumption of the reactant in the shell by chemical reaction. Solving the equation leads to a result that shows how the rate of the catalytic reaction is influenced by the interplay of the transport, which is characterized by the effective diffusion coefficient of the reactant in the pores, and the reaction, which is characterized by the first-order reaction rate constant. 

Even when there is a transport disguise, the reaction order remains one for a first-order reaction. But for reactions that are not intrinsically first order, the transport disguise changes the observed reaction order for an intrinsically zero-order reaction, the observed order becomes 1/2 and for an intrinsically second-order reaction it becomes 3/2 when 0 10. For all reaction orders the apparent activation energy is approximately half the intrinsic... 

A model for coal fluidity based on a macromolecular network pyrolysis model has been developed (33). In that model, bond breaking is described as a first-order reaction having a range of activation energies. A variety of lattices have also been used to describe the bonding in coal. In turn these stmctures... 

For combustion of simple hydrocarbons, the oxidation reactions appear to foUow classical first-order reaction kinetics sufficiently closely that practical designs can be estabUshed by appHcation of the empirical theory (8). For example, the general reaction for a hydrocarbon ... 

The stabilization of chloromethoxycarbene (234) was intensively studied. It is formed from diazirine (233) in a first order reaction with fi/2 = 34h at 20 C. It reacts either as a nucleophile, adding to electron poor alkenes like acrylonitrile with cyclopropanation, or as an electrophile, giving diphenylcyclopropenone with the electron rich diphenylacetylene. In the absence of reaction partners (234) decomposes to carbon monoxide and methyl chloride (78TL1931, 1935). 

Sets of first-order rate equations are solvable by Laplace transform (Rodiguin and Rodiguina, Consecutive Chemical Reactions, Van Nostrand, 1964). The methods of linear algebra are applied to large sets of coupled first-order reactions by Wei and Prater Adv. Catal., 1.3, 203 [1962]). Reactions of petroleum fractions are examples of this type. 

Daily Yield Say the downtime for filhng and emptying a reactor is and no reaction occurs during these periods. The reaction time of a first-order reaction, for instance, is given by = —In (1 — x). The daily yield with n batches per day will be... 

The simplest proWem is when all of the stages have the same kt-, then one of the three variables (kt, n, or Ca ) can be found when the others are specified. For first-order reactions,... 

Different Sizes Ordinarily, it is most economical to make all stages of a CSTR battery the same size. For a first-order reaction the resulting total volume is a minimum for a specified performance, but not so for other orders. Take a two-stage battery ... 

Example 10 Reactor Size with Recycle For first-order reaction with Cq/C<) = 10 and i = 5, C ICo = 2.5. The relative reactor sizes with recycle and without are... 

When the RTD of a vessel is known, its performance as a reaclor for a first-order reaction, and the range within which its performance will fall for other orders, can be predicted. 

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