CHEMICAL KINETICS Chemical kinetics is the branch of chemistry that is concerned with the study of the rates and mechanisms of chemical reactions. The rate of a reaction is a measure of its speed. Consider the hypothetical reaction The rate of formation of C is E + 2B 2C + D (1) (2) where [C] f and [C] i are the concentrations of C at times t f and t i, respectively. The symbol Δ stands for the change in whatever it is placed before. The rate of formation of C is the change in the concentration of C over the time interval Δt. Similarly, the change in concentration of D over time would be: The rates of consumption of E and B are (3) (4) The negative signs in equations (4) and (5) arise from the fact that although the rates are positive numbers, the concentrations of the reactants decrease with time, and their changes are negative. From the stoichiometry of reaction (1) we see that the consumption of 1 mole of E results in the consumption of 2 moles of B and the formation of 2 moles of C and 1 mole of D. B is consumed twice as fast as E, and C is produced twice as fast as D. Thus, the relationships between the rate expressions in equations (2) - (5) is (5) (6) The rate of the reaction, Rate RXN, can be expressed either in terms of the rate of disappearance of reactants or the rate of appearance of products: (7) In general, the rate of a reaction depends on the concentration of the reactants as follows: Rate RXN = k[e] x [B] y (8)
Equation (8) is the rate law for the reaction, and it shows that the rate is proportional to the product of the concentrations of the reactants each raised to some power x or y. The proportionality constant, k, is the rate constant and depends only on temperature. The exponents x and y determine the reaction order with respect to E and B. The sum of the exponents is the overall order of the reaction. For example, if x = 1, and y = 2, then the reaction is said to be first order in E, second order in B, and third order overall. The values of the exponents in the rate law must be determined by experiment, and cannot be deduced from the stoichiometry of the reaction - the stoichiometric coefficients and the order of the reaction with respect to that reagent are not the same. Rate Laws involving only one reactant: For the reaction The rate law is: E product If the reaction is first order, then x = 1 and the rate law is: Δ[E] Δt = k[e] If we integrate the above equation, the integrated rate law becomes: ln[e] = - kt + ln[e] 0 (10) In the above equation, [E] 0 is the initial concentration of E and [E] is the concentration of E at time t. Thus, the integrated rate law permits us to predict [E] at any time in the course of the reaction, if the temperature does not change. A summary of the rate laws for each possible order is on the next page. Order Rate Law Integrated Rate Law Eq n No. Zero = k [E] = [E] 0 - kt (9) First Δ[E] ln[e] = -kt + ln[e] 0 (10) Δt Second Δ[E] Δ[E] = k[e] = Δt Δt k[e] 2 (11)
The integrated rate laws can be used to determine the order of a reaction. For example, if the reaction is zero order, equation (9) predicts that a plot of [E] versus time is linear with a slope equal to -k. If a reaction is first order, equation (10) predicts that a plot of ln([e]) versus time is linear with a slope equal to -k. Finally, equation (11) predicts that if the reaction is second order, a plot of 1/[E] versus time is linear with a slope equal to k. This is summarized in the figure below. In the experimental situation, we typically will graph the different functions of [E] vs. time, and determine the equation for the line by the least squares method and the correlation coefficient. The graph most nearly linear (highest correlation coefficient) is taken as best describing the reaction, and the order is determined by the form of concentration dependence for that graph. Figure 1. The Method of Isolation (also called pseudo-order kinetics) is an experimental technique for determining the order of a reactant in a reaction separate from the other reactants. In this method, one of the reactants is present in large excess relative to the other. For example, consider the reaction E + B C + D Suppose that the initial concentrations of E and B are 1.000 x 10-4 M and 0.2000 M, respectively. We see that [B] >>[E]. If the reaction goes to completion, all of E is used up since it is the limiting reactant, and the concentrations of both reactants will have decreased by 1.00 x 10-4 M. The final concentration of B at the end of the reaction is 0.2000-1.000 x 10-4 M = 0.1999 M, a decrease of only 0.05 %. The concentration of B remains essentially constant during the reaction, and equation (8) becomes: Rate RXN = k o [E] x (12)
where k o = k[b] y (13) The only variable in equation (12) is the concentration of E, and the order with respect to E can be determined by plotting each of the quantities: [E], ln([e]), and 1/[E] versus time, as in Figure 1, and determining which plot is linear. If [E] versus time is linear then the order with respect to E is zero, etc. As discussed above, k o can be obtained from the slope of the linear plot. The order with respect to B can be determined by using different excess concentrations of B, and observing the effect on k o. For example, if doubling the concentration of B doubles k o then in equation (13), y = 1, and the reaction is first order in B. On the other hand, if doubling the concentration of B causes k o to increase by a factor of 4, then y = 2, and the reaction is second order in B. Why is the rate law important? The rate law of a chemical reaction gives us insight into how the reaction occurs at the molecular level. It helps us to understand what the molecules are doing as the reaction occurs. All reactions occur through a sequence of steps (some only one step long) called a reaction mechanism. Consider the reaction with the observed rate law: Rate = k[e][b]. E + 2B 2C + D A possible mechanism, consistent with this rate law is: E + B X + D X + B 2 C (a) slow, rate-determining (b) fast Each step in a mechanism is considered a molecular event. We say the step is an elementary step. The two possibilities are either something decomposes or a collision between two particles has occurred. Thus each step in a mechanism must be either first order or second order. Also, the sum of the elementary steps must equal the overall reaction. In the above mechanism, E collides with B in a slow reaction that determines the rate of the reaction. Then, very quickly, the intermediate X collides with B. Thus, the reaction order is determined not by the coefficients of the overall reaction, but by the molecularity of the rate-determining step. Purpose You will run a set of experiments to examine how changing the concentration of a reactant affects the rate of the reaction between sodium hypochlorite, NaOCl, and a food dye, FD&C Blue #1. Sodium hypochlorite is the active ingredient in commercial bleaches, such as OClrox and Purex. The reaction is NaOCl + FD&C Blue # 1 Colorless products
FD&C Blue #1 is below. Molecular Formula: C 37 H 34 N 2 Na 2 O 9 S 3 Molecular Weight: 792.84 g/mol The rate law for reaction is Rate = k [NaOCl] x [Blue#1] y (15) The purpose of the experiment is to determine the order with respect to NaOCl and Blue #1. The method of Isolation will be used with an excess of NaOCl. If NaOCl is present in large excess, its concentration is essentially constant, and we define a new rate constant, k o, where k o =k[naocl] x. Equation (15) then becomes Rate = k o [Blue#1] y (16) All of the reactants and products are colorless except Blue #1. The blue color of a solution of blue #1 is due t the dye molecules absorbing a portion of the visible spectrum. That part of the spectrum that is not absorbed is transmitted and gives the solution its blue color. As shown in Figure 2, the dye molecules absorb orange light (wave length of about 630 nm). The concentration of the dye over time can be determined by measuring the amount of light absorbed by the reaction mixture over time. The measurement of light absorption will be
accomplished with a visible spectrometer. In the spectrometer, white light (containing all wavelengths of light) from the light source passes through the sample solution and to reach the light detector. The light detector measures the intensity of the light reaching it. In this case, the only light absorbing substance in solution is the dye. The amount of light absorbed by the solution is a measure of the dye concentration. The higher the concentration of dye in the solution the lower the intensity of the light striking the detector. The amount of light absorbed by the solution is measured in terms of absorbance, A. The absorbance is defined as A = - log(i t /I o ) where I o is the intensity of the light reaching the detector when the cell is filled with pure water, and I t is the intensity observed when the cell contains the reaction mixture at time t. The absorbance of the solution is directly proportional to the concentration of the dye. That is A = ε l [Dye] (17) where ε is the molar absorption coefficient or extinction coefficient, which is a property of the dye at a given wavelength, and l is the path length in centimeters. Our sample cuvettes provide a path length of 1.0 cm. Equation (17) is one expression of Beer s Law (or the Beer-Lambert Law). Beer s Law applies only for dilute solutions. How dilute? Generally, we can use the law without reservation when absorbance is some reasonable range, say between 0.8 and 0.08. Most detectors have a sufficient dynamic range to satisfy a criterion of linearity of response in this range. In other circumstances, where linearity is not assured, rather than rely on Beer s Law, we will use a calibration curve to explicitly relate an observed absorbance to the concentration of the species of interest. To determine the order with respect to Blue #1 you will measure the transmittance (%T) of the reaction mixture versus time manually. We will read off the spectrophotometer %T and then calculate absorbance (see equation 18); %T is easier to read off of the dial, and for use when calibrating the spectrometer. After making a table of absorbance vs. time, a computer will be used for the graphing portion of the data work up. (18) Procedure
Calibration of Spectrophotometer: This will be demonstrated. We make two adjustments to calibrate the instrument: with no sample in the compartment, but a shutter over the detector window, we will adjust the 0 %T that corresponds to the sample absorbing all the incident radiation. This correction is made to allow for thermal noise in the detector (essentially, static in the photocell). This measurement is wavelength independent (black is black!). We then adjust 100 %T by inserting a blank sample: a sample that contains whatever is in the mixture except the species to be measured (we will use a water blank, since the dye is the only colored species). This measurement sets the detector signal strength for a transparent sample: 100 % of the incident photons are transmitted to the detector. Because the sensitivity of the photocell is different at different wavelengths, this measurement must be made every time a new wavelength is selected. Determine the absorbance maximum for the dye: This will be demonstrated. Your instructor may give a presentation on the origin of molecular UV-visible spectra, or assign additional reading covering this topic. The demonstration may include the actual selection of the wavelength with the maximum absorption. This wavelength is usually called lambda-max, symbolized as λ max. Using the diluted solution and a calibrated Spec 20 spectrophotometer, determine the %T for multiple wavelengths from 420 to 700 nm. Record the transmittance every 20 nm throughout this range. At each wavelength, you must reset the 100 %T; the 0 %T (dark current) should be stable, and need not be adjusted. Find the wavelength that has the lowest %T (highest absorbance). This wavelength will be the one used in all subsequent measurements of the kinetics lab. In Excel, make a plot of absorbance on the y axis vs. wavelength on the x axis. You need to convert your %T values to absorbance, using equation (18). Determination of the extinction coefficient of Blue #1: Equation (17) shows that the absorbance of the solution is related to the concentration. We will measure %T of a solution with known concentration in order to calculate A, from which we can derive ε, the extinction coefficient (the path length, l, is 1 cm). We will use this value of ε to convert our absorbance readings to concentration. Mix 10.0 ml working dye solution with 10.0 ml of water, using a volumetric pipette for accuracy. Record the details of the dilution, and calculate the new concentration. The concentration of dye in this solution is the same as the starting dye concentration (time = 0 seconds) in the reaction. Record the details of your dilution, and calculate the new concentration. Calibrate the spectrometer and measure the %T of the solution. Convert your reading to A, and calculate ε.
Measurement of %T versus time for the reaction: Be sure to measure the temperature of the reaction solution! Carefully measure 20 ml of the working dye solution with a graduated cylinder, and pour it into a 150 ml beaker. Prepare 20.0 ml of bleach solution by mixing 18.0 ml of deionized water in a 25 ml graduated cylinder and 2.0 ml of commercial bleach. The bleach should be measured using a volumetric pipet. Mix the bleach solution thoroughly by sealing the top of the cylinder with ParaFilm, and inverting several times. Add the bleach solution to the dye solution in the beaker in a single portion, while simultaneously starting a stopwatch. Quickly stir the mixture by swirling the beaker, and pour it into a clean, dry cuvette. Hold the cuvette by the top edge to avoid getting fingerprints on the light-transmitting walls. Place the cuvette in the colorimeter, cover it, and collect %T values every 30 seconds. Place the beaker containing the remaining reaction mixture next to the colorimeter, and observe the color change as the reaction progresses. Stop data collection when the solution in the beaker is visually colorless, or for 15 minutes, whichever happens first. Repeat this measurement once. 5. Repeat the procedure above using one-half the concentration of bleach, and diluting to a total volume of 20 ml. Make two runs at this concentration. If time permits, you may do the procedure using 50% more or twice the amount of bleach; be prepared to work very quickly! Calculations and Results Please answer each question below clearly. Do the calculations in questions 1 through 3 by hand. 1. Convert the %T value of the blue #1 solution, measured in Part 3 of the procedure, to absorbance, using equation (18). Calculate the constant ε, in equation (17) using your absorbance value and the calculated concentration of the solution. 2. Commercial bleach is a 8.25%, by mass, aqueous solution of sodium hypochlorite, NaOCl. What is the molar concentration of NaOCl? Remember to adjust for density! 3. In Part 4 of the procedure, what are the concentrations of blue #1 and NaOCl in the reaction mixture before reaction occurs? Are we justified in assuming that the concentration of NaOCl is much greater than that of blue #1? 4. Enter your %T versus time data, and your value of ε, into the Excel spreadsheet. Convert the %T values to absorbance. Convert the absorbance to concentration of dye, [E], and calculate ln[e] and 1/[E]. You may simply substitute your observations for the values in the template spreadsheet, but be sure you understand what you are doing! 5. Prepare graphs of each of the quantities [E], ln[e] and 1/[E] versus time. From your graphs determine the order of the reaction with respect to Blue #1. 6. For the linear graph obtained in question 5, determine the slope using the add trendline feature of Excel, and display the equation on chart. The equation of the straight line will be displayed on the graph. The slope will be equal to -k o if the reaction is either zero or first order with respect to blue #1, and it will equal k o if the reaction is second order with respect to blue #1. 7. Compare the values of k o obtained in parts 4 and 5 of the procedure. What can you conclude about the order with respect to NaOCl?
8. What is the value of the rate constant, k, in equation (15)? 9. Rewrite equation (15) with the correct values of k, x and y. Kinetic runs at other than room temperature Our first experiment is sufficient to determine the rate law for the reaction. Chemists are also interested in how the rate of the reaction will change with temperature. The rate law will not change over a reasonable temperature range, but the rate constant is temperature dependent. We know that increasing the temperature always accelerates a reaction, but this is too general an observation to be useful in most situations. The Arrhenius equation describes the temperature dependence of the rate constant: k = A e Ea/RT (19) In order to estimate the energy of activation, we have to determine the rate constant at several temperatures. We can linearize the Arrhenius equation by taking the logarithm: ln(k) = - E a /R(1/T) + ln(a) (20) This puts the equation in the form y = mx + b, where m = E a /R and b = ln(a). Thus graphing ln(k) vs. 1/T gives a line with a slope of E a /R and y-intercept ln(a). Remember, in this discussion, A is the frequency factor, not the absorbance! A useful estimate can be gotten with only two temperatures, but this may be subject to large experimental error. However, sometimes scientists (and students who are In a hurry) are content with a reasonable estimate if the experiment is difficult to perform. So we will do the experiment at one additional temperature, and then decide if a more exact determination is worth the effort. We can graph or data, or more simply solve ln (k 1 /k 2 ) = E a /R (1/T 2 1/T 1 ) (21) What temperature(s) to use? Guess! There is a rule of thumb that says the reaction rate increases by a factor of 2 for every 10 degree increase in temperature. So we need to review the data we collect at room temperature to decide what temperature and concentration combination will allow us enough time to collect useful data, but not so slow as to keep us standing about forever. Since we are interested in the rate constant (we should already know the order from the room temperature data), we are not constrained to using the same concentrations we used in the RT experiment. So we do a bunch of calculations (what if?) on the basis of the RT rate law to determine the best conditions to use in the follow-on experiments. We can use higher temperatures with more dilute reagents, or lower temperatures. A limit on the lower temperature is that we do not want to have water vapor condense on the spectrometer s cuvette. For ease of interpretation, I would choose one of the concentrations from the first experiment.
Procedural Problem(s). In a well-equipped laboratory, the spectrometer would have a thermostated reaction cell, and we would do the entire procedure in this apparatus. But we are faced with the problem of running the reaction at one temperature, then transferring aliquots of the reaction mixture and measuring results while disturbing the temperature as little as possible. Here are some ideas (and no, I don t know if this will work very well - that s why it is an experiment!): All glassware, solutions, spectrometer vials, etc., should be held at the reaction temperature (equilibrated at that temp, as we say) before mixing. It may make sense to mix sufficient reactants to get 4 or 5 time points, each time point requiring filling a spectrometer vial. Time 1 would be done as fast as solution could be transferred, time 2 is 30 seconds to 1 minute (depending on how fast you expect the reaction to go), etc. This will require cooperation and dexterity! Time 0 is when the solutions are mixed, and then times must be recorded when the mixture is introduced into the colorimeter. We need to record a data for each sample as it is introduced into the colorimeter, and see whether the transmittance changes drastically over time (this is because the reaction is still going, but not at the same temperature or rate as when first introduced into the colorimeter). We will possibly need to come up with a method to extrapolate back to measurement time 0 for each sample - what was the transmittance when the sample was introduced into the machine? Hmmmm... confusing? Welcome to life in a chemistry lab! Procedure Incubation of solutions and glassware: After you have selected the temperature at which you wish to do the experiment, assemble glassware, small containers of reagents and spectrometer cuvettes, and place them in a bath at the chosen temperature. Reagents should be stirred or swirled so that they heat or cool uniformly. The bleach and dye solutions and water to dilute them should be at the same temperature when you mix them. 2. Kinetics runs: When you have a firm plan to follow (what concentration to use and how to mix the reagent, how many vials or cuvettes to start reactions in, when you will take samples), determine %T vs. time for the reaction as in step 4 of the room temperature experiment. Because we already know the rate law, you will not need as many time points as in the first experiment, if these are spread out over a reasonable period. Remember, the longer it takes to transfer samples to the cuvette and the longer the cuvette sits in the spectrometer, the less accurate your data will be, since your sample will be warming or cooling. 3. Make two runs at your chosen temperature, if time permits.
Calculations and Results 1. Convert your %T data to absorbance, and make a graph of absorbance vs. time as in Step 6 of the calculations above. The linear form used should not change from the first experiment. What is the slope of the line, and what is its significance? 2. Sanity check: if you did this experiment above room temperature, should your result in question 1 be larger or smaller than what you obtained in the first experiment? If you worked at a lower temperature, how should it change? Does your data meet your expectations? 3. If everything seems on track, substitute your values for temperature and rate constant into equation (21), and calculate the activation energy for this reaction. 4. Plot your two points as described for use of equation (20). Add a trendline to obtain the slope and intercept. Convert these values into E a and A, the frequency factor.