Kinetics IILecture 15

Rates of Complex Reactions

• Complex reactions that involve a series of steps that must occur in sequence are called chain reactions.

• In a chain reaction when one step is much slower than the others, the overall rate will be determined by that step, which is known as the rate-determining step.

• Complex reactions can involve alternate routes or branches (read about H2 combustion in book).o In a branch, the rate of the fastest branch will determine that step.

• Another example of a branched reaction is stratospheric ozone.

Stratospheric Ozone Cycle

• Normally, stratospheric ozone is created by photolysis reactions in the stratosphere where photons of sufficient energy (UV) are abundant:

• Ozone is then destroyed by a similar photolysis reaction:

• Normally, the rates of these reactions are such that it produces a steady-state concentration of ozone. (Ozone is far more likely to be photolyzed (higher rate constant), but it is far less abundant, so the rates balance).

• The ozone hole has developed because Cl in the stratosphere provides an alternate pathway for ozone destruction:

The overall rate is then the sum of the two

rates. But if the rate of one reaction is far

faster, the rate of the slower reaction

becomes irrelevant and the overall rate is

governed by the rate of the fast reaction.

Relating Thermodynamics and

Kinetics

Principle of Detailed Balancing

• Consider a reversible reaction such as A ⇋ B

• The equilibrium condition is k+[A]eq = k–[B]eq

o note typo in book: + and - should be subscripted in eqn 5.40 p172.o (so this, for example would be the case for a steady-state ozone supply).

• where k+ and k– are the forward and reverse rate constants, respectively. This is the principle of detailed balancing.

• Rearranging we have

Kapp?

Relating k and K• We can write the temperature dependence of K

as:

• For constant ∆Sr:

o where C is a constant.

• See anything familiar?

• Then

Barrier Energies• Our equation was

• Its apparent then that:

• We can think of the barrier, or activation energy as an energy hill the reaction must climb to reach the valley on the other side.

• Energy released by the reaction, ∆Hr, is the depth of the energy valley.

Aspects of Transition State Theory

Frequency Factors and Entropies

• A similar sort of derivation yields the following

• A relates to the frequency of opportunity for reaction (beat of as clock).

• This tells us the ratio of frequency factors is exponentially related to the entropy difference, or randomness difference, of the two sides of the reaction.

• It can be shown (but you’ll be glad to note that we won’t) that the frequency factor relates to entropy as:

• where ∆S* is the entropy difference between the initial state and an activated state.

Fundamental Frequency

• The term

• is known as the fundamental frequency• k has units of joules/kelvin, T has units of kelvins,

and h is Plank’s constant and has units of sec-1, so the above has units of time (6.21 x 10-12 sec at 298K, or 2.08 x 10-10 T sec).

Reactive Intermediates

• Transition state theory supposes that a reaction such as • A + BC → AC + B

• proceeds through formation of an activated complex, ABC*, called a reactive intermediate, such that the reaction mechanism is:

A + BC → ABC*ABC* → AC + B

• The reactive intermediate is supposed to be in equilibrium with both reactants and products, e.g.,

• Free energy of reaction for formation of complex is:

Predicting reaction rates

• Combining these relationships, we have:

• Example to the left shows predicted vs. observed reaction rates for the calcite-aragonite transition.

• In this case, the above rate is converted to a velocity by multiplying by lattice spacing.

• Suppose our reaction isA + BC ⇋ AC + B

• i.e., it is reversible. The net rate of reaction isRnet = R+

- R– (1)

• If ∆Gr is the free energy difference between products and reactants and ∆G* is free energy difference between reactants and activated complex, then ∆Gr - ∆G* must be difference between activated complex and products. You can derive (next problem set):

(2)• -∆Gr is often called the affinity of reaction and

sometimes designated Ar (but we won’t).

• Then, substituting (2) into (1) and a little algebra:

∆G and Rates• Provided this is an elementary reaction, then the rate may be

written as:

o Note, activation energy, EA and barrier energy, EB, are the same thing.o In the reaction, the stoichiometric coefficients are 1.

• In a system not far from equilibrium, ∆Gr/RT is small and we may use the approximation ex = 1 + x to obtain:

• ∆G is the chemical energy powering the reaction. At equilibrium, ∆Gr is 0 and the rate of reaction is 0. The further from equilibrium the system is, the more the energy available to power the reaction. Thus the rate will scale with available chemical energy, ∆Gr.

Reaction Quotient (Q)• We’ve seen the rate should depend on ∆Gr, how

can we compute it as reaction progresses?• Consider a reaction

aA +bB ⇋ cC +dD• At equilibrium:

• Under non-equilibrium conditions, this equality does not hold. We define the ratio on the right as the reaction quotient:

Computing the Reaction Affinity (∆G)

• At equilibrium, ∆Gr is 0 (and ln K = –∆G˚/RT)

• Under non-equilibrium conditions,∆Gr = ∆G˚ + RT ln Q

• and ∆Gr = RT ln Q/K

• We expect then that (not far from equilibrium) the reaction should proceed at a rate:

• and also

Note: equation incorrect in book.

Another Approach• Consider now a reaction that depends on

temperature (e.g., α- to β-quartz). Another approach is to remember ∆G = ∆H - T∆S

• At equilibrium this is equal to 0. At some non-equilibrium temperature, T, then

• ∆G = ∆H - ∆Heq - T∆S + Teq∆Seq

• If we are close to the equilibrium temperature, we may consider ∆H and ∆S constant, to that this becomes:

∆G =(T-Teq)∆S = -∆T∆S

• Recall:

• Assume ∆G/RT is small so that e∆G/RT = 1+∆G/RT, then

• We substitute ∆G = -(T-Teq)∆S

o Equation 5.67 should read:

o no negative, no square

Rates of Geochemical Reactions

Wood and Walther used this equation to compute rates. This figure compares compares observed (symbols) with predicted

(line).

∆G & Complex Reactions

• Our equation:

• was derived for and applies only to elementary reactions.

• However, a more general form of this equation also applies to overall reactions:

• where n can be any real number. So a general form would be:

Diffusion

Importance of Diffusion

• As we saw in the example of the N˚ + O2 reaction in a previous lecture, the first step in a reaction is bringing the reactants together.

• In a gas, ave. molecular velocities can be calculated from the Maxwell-Boltzmann equation:

• which works out to ~650 m/sec for the atmosphere• Bottom line: in a gas phase, reactants can come together

easily.• In liquids, and even more so for solids, bringing the

reactants together occurs through diffusion and can be the rate limiting step.

Fick’s First Law• Written for 1 component and 1 dimension, Fick’s

first Law is:

o where J is the diffusion flux (mass or concentration per unit time per unit area)

o ∂c/∂x is the concentration gradient and D is the diffusion coefficient that depends on, among other things, the nature of the medium and the component.

• Fick’s Law says that the diffusion flux is proportional to the concentration gradient. A more general 3-dimensional form (e.g., non-isotropic lattice) is: