Chapter 2 Introduction to Kinetic Processes in Materials 41 Chapter 3 Second-Order and Multistep Reactions 61 Chapter 4 Temperature Dependence of the Reaction Rate Constant 95
2.8 CHAPTER SUMMARY
This chapter introduces several concepts about reaction kinetics. Is the reaction controlled by transport—
diffusion—or an actual chemical reaction? Reactions can occur within single phases, homogeneous reactions, or more common in MSE, heterogeneous reactions involving more than one phase: the depo- sition of Si by CVD and the oxidation of a metal are examples. When discussing actual chemical reaction kinetics, the order of the reaction is important because knowing it can lead to understanding the actual atomistics of the chemical reactions taking place. The dissolution of TiO2 in HF is presented as a pos- sible example of this. Zero-order reactions are the ones that depend neither on the concentrations of the reactants nor those of the products and are physically not very likely. First-order reactions are important because they occur frequently, particularly for heterogeneous reactions. In this case, the reaction rate is proportional to one of the reactants with the reaction rate constant, k, the proportionality constant.
The reaction rate constant follows an Arrhenius equation (Svante Arrhenius), namely, it is exponentially temperature dependent—strongly temperature dependent—and the rate is largely defined by the value of the activation energy, Q. The concepts of fraction reacted, half-life, and relaxation time are introduced for first-order equations. Radioactive decay is used as an example of first-order reactions in radioisotope power sources and radiocarbon dating along with some nuclear chemistry associated with fissionable materials for nuclear reactors. The next chapter goes further into reaction kinetics discussing second- order reactions, multistep reactions including serial and parallel reactions, and reactions that go to equi- librium. In reality, most reactions—even simple gas phase homogeneous reactions such as the formation of water vapor from hydrogen and oxygen—consist of multiple series and parallel reactions that preclude analytic solution and dictate the need for numerical computation of concentrations and reaction rates.
EXERCISES
2.1 The reaction rate constant for a zero-order reaction for the decomposition of another liquid dissolved in water is given by k k= 0exp( Q/RT)− for the reaction
A → B.
a. Write the equation for the reaction rate for the decomposition of A.
b. Give the units of k0 if the units of the reactant concentration [A] are millimol/cm3.
HF HF
HF
HFHF
TiO2 crystal surface
Ti4+ ion Attached
ligands Ligand
“squeezing” in
FIGURE 2.14 Postulated possible mechanism for the first-order rate dependence on [HF] for the dissolu- tion of TiO2 in 0.5 molar HCl solutions at 95°C. The figure shows an isolated Ti4+ ion at the surface of the dissolving TiO2 surrounded by several attached HF ligands (the exact nature of and number of the ligands is not known, possibly F−) and one final HF trying to “squeeze” into this coordination sphere that would provide the necessary bonding for the Ti4+ ligands (TiF62−?) to leave the surface and enter the solution.
(After Bright and Readey 1987.)
c. If Q = 30 kJ/mol and k0 = 10−2, calculate the reaction rate in millimol/liter at 25°F.
d. Calculate how long it takes for the reaction to go to completion if [A]0, the reactant concentration at time zero, is one mole per liter.
e. Neglecting any temperature effects on the density of water, which can be assumed to be 1.0 g/cm3, calculate the number of moles per cm3 of water in pure water.
f. Assuming that the molar volumes of water and the reactant are the same, calculate the m/o, w/o, and v/o of A in water at 25°F at the start of the reaction if the molecular weight of the reactant is 60 g/mol.
2.2 a. The reaction rate constant k (k k= 0exp( Q/RT))− for a first-order reaction is 5.00 × 10−3 s−1 at 500°C and is 10.00 s−1 and 1000°C. Calculate the activation energy for this reaction.
b. Calculate the preexponential term, k0, for the reaction rate constant and give its units.
c. Calculate k at 750°C.
d. Calculate the relaxation time for this reaction.
e. Calculate the half-life of this reaction.
f. Make a plot of log10(rate) versus 103/T (K−1) from 100°C to 1200°C.
g. Calculate how long it takes (hours) at 750°C for the unreacted fraction to be 0.001.
h. If the reaction is A → 2B, plot the concentrations of A and B versus time from time zero up to the time that was calculated in (g).
2.3 A well-known generalization (for back of the envelope calculations) for biochemical reactions near room temperature (300 K) is that their rates double for every 10 K rise in temperature.
a. Determine the activation energy (J/mol) for such reactions implied by this rule.
b. Calculate the value of the activation energy in eV/molecule.
2.4 a. The half-life for a first-order reaction is 1000 s. Calculate the reaction rate constant (s−1) for this reaction.
b. Calculate the relaxation time (s) for this reaction.
2.5 a. The half-life for a first-order reaction is 1200 s. Calculate the reaction rate constant (s−1) for this reaction.
b. Calculate the relaxation time (s) for this reaction.
c. Plot the fraction unreacted versus time for 0–10,000 s. Indicate the half-life and the relaxation time on your plot.
2.6 a. Plutonium-238, 23894Pu, spontaneously decays by alpha particle, 24He, emission with a half- life of 86 years. The nuclear disintegration reaction is
23894
23492
24 5 234 Pu→ U+ He+ . Mev.
This disintegration rate is sufficiently fast to produce enough heat to be used as a power source in satellites. Calculate the first-order reaction rate constant for the nuclear disin- tegration of plutonium-238.
b. Calculate the number of becquerels at time = 0 with one mole of pluonium-238.
c. Calculate the number of curies at time zero in one mole of pure 23894Pu.
d. Calculate the rate of energy generated in the plutonium-238 (watts) for one mole of plutonium-238 at time zero.
e. Calculate how long (years) it will take for the rate of energy generation to drop to 10% of its initial value.
f. If the 23894Pu were in the form of PuO2 rather than pure plutonium, calculate the number of curies in 5 cm3 of PuO2 at time zero if the density of PuO2 is 11.5 g/cm3.
g. Calculate the number of watts that this 5 cm3 of 23894Pu-containing PuO2 generates.
h. If this 5 cm3 of PuO2 were perfectly insulated, calculate how long it takes (hours) for the 5 cm3 of PuO2 to reach its melting point of 2390°C if its molar heat capacity is 75 J/mol K.
2.7 A reaction obeys the following stoichiometric equation:
A + B = Z.
The rates of formation of Z at various concentrations of A and B are given in the following table.
[A] (mol/L) [B] (mol/L) Rate (mol/L-s) 2.00 × 10−2 4.20 × 10−2 3.53 × 10−9 6.50 × 10−2 4.80 × 10−2 4.87 × 10−8 1.10 × 10−1 1.22 × 10−1 9.00 × 10−7
Determine the values of α, β, and k in the rate equation: rate = k [A]α[B]β. Give the units for k as well. You may use programs such as Excel and Mathematica, or by hand, to get the solu- tion. Just clearly indicate what operations were performed, that is, copy that part of the Excel spreadsheet or the Mathematica page that shows the calculations.
2.8 In 2012, what were thought to be the bones of King Richard III of England were found under a parking lot in England and later confirmed by DNA evidence in 2014 that the bones were indeed his. The bones were dated by radiocarbon dating and confirmed that the date of death of the skeleton was 1485, the date of death of the king. Calculate the ratio of carbon-14 remain- ing in the bones.
2.9 If the number of disintegrations per minute per gram of carbon produced by pyrolyzing a piece of recently cut wood is 12.5 dis/min g, calculate the fraction of 14C in carbon of age zero if the half-life of 14C is 5730 years. (The wood is assumed to have an age of zero.)
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