Geochemical Kinetics

GEOCHEMICAL KINETICS

Kinetics deals with the rate of a chemical reaction, and the reaction mechanism. Thermodynamic, on the other hand, deals with the state and energetics of a reactions. Thermo-dynamic principles give us an idea of whether a reaction is able to occur in a given system, while kinetics tells us how fast. Most natural waters are not at equilibrium!!

Collision Theory

Basic kinetic theory calls for simple collisions between atoms - more or less like billiard balls – so the more collisions in a certain period of time, the faster the reaction. This conceptual model is surprisingly useful for understanding how fast a reaction occurs. Some of the rules are:

With few exceptions (Radioactive Decay), collisions between reacting species are required for a chemical reaction to occur;

Not all collisions are successful. The number of successful collisions determines the rate of the reactions.

a bimolecular collision is far more likely than a trimolecular.

Collisions in solution are more often successful because of the rebounding due to the cage effect.

The number of collisions increases with increasing temperature, or in a gas, with increasing pressure. This is a fundamental observation.

Reactions that require a specific orientation (i.e. SN2, or the precipitation of sodium acetate from a supersaturated solution) are less likely to occur than others (i.e. ionic reactions).

Rate Law

Elementary reaction is a fundamental chemical step.

homogeneous reaction is one with only one phase.

heterogeneous reactions involves more than one phase.

**Most geochemical reactions are heterogeneous, usually liquid-solid

Reaction Order

Reaction order is an expression of the dependence of the elementary reaction rate on the concentrations of the species involved, i.e.

A + B ----> AB

--If the reactions is independent of A, it is Zero order to A

--if the rate is proportional to the concentration of A, the Rx is 1st order to A

--If the reaction is dependent on both, the reaction is 2nd overall, 1st with each.

A + 2B + ....----> P + 2Q + .....

Where d[A]/dt is the disappearance rate of A, k = rate constant, and a, b, p, q, are individual reaction order constants.

From the reaction stoichiometry (1 mole of A reacts for every 2 moles of B, etc) we can generalize all the changing concentrations:

This reaction would be first order with respect to A, second order with respect to B, and third order overall.

Zero Order Reactions:

Consider the irreversible reaction

A -----> Products

This would give the general rate law of:

--to determine the behavior of [A] as a function of time, integrate the rate expression with respect to time, according to the value of n. When n = 0, the reaction is zero order, and

integrating gives us:

[A] = [A]0 - kt

Where [A]0 = the concentration of A at t=0

Half Life:

The t1/2 is the time for 50 percent of the [A]0 to react:

t1/2 =

First Order Reaction

b) When n = 1, the reaction is first order with respect to A and overall:

rearranging and solving the integral we get:
ln [A] = ln[A]0 - kt

or
[A] = [A]0e -kt

and the half life t1/2 = , or 0.693/k

Second Order Reactions

If the reaction order is greater than first order, we can write

integrating to

= -kt

if n = 2, for example, the reaction is second order overall and to A, and

and the half-life is

The case of A + B ----> Products is more complicated, and has the rate law of

when a = 1 and b =1, the reaction is second order overall and first order with respect to both A and B. There are two general cases for 2nd order reactions:

Case 1:

1) When [A]0 = [B]0:

For this reaction, for each mole of A that reacts, one mole of B will react. The concentrations of A and B will therefore decrease at the same rate, and will always be equal. The reaction can therefore be rewritten as:

to be solved as described above

Case 2: Pseudo First order Reactions

2) When the concentrations are not the same, things get messy, except for the case when one reactant is present in such excess that the concentration remains unchanged over the reaction progress, and so the rate is expressed in terms of the changing reactant. This is called a pseudo first-order reaction.

***An example would be the hydrolysis of a solute in aqueous system.

Example 2-1

NH3 + HOCl -----> NH2Cl + H2O

· The rate constant k was found experimentally to be 5.1 x 106 liters/mole sec at 25°C

· the rate order was determined to be:

1. What is the reaction order with respect to each reactant and overall?

2. What percent decrease in reaction rate occurs if the concentration of each reactant is reduced by 50%

a) let the initial concentrations of NH3 and HOCl be x and y, respectively. The rate of reaction becomes

when x and y are each reduced by 50%, the reaction rate becomes

The new reaction rate is 25% of the original

Common Units of Rate

Order:	Units		Differential form
0	mol cm-3 s-1	C t^-1	dC/dt = -k	C=C°- kt
1	s-1	t^-1	dC/dt = kC	C=C°e-kt
2	cm3 mol-1 s-1	C^-1t^-1	dC/dt =-kn	Complex

Reaction Mechanism

Most reactions are complex combinations of elementary reactions that give the reactions a fractional overall order. Each reversible reaction has a forward and reverse rate constant k1 and k-1. For example

H2 + Br2 ---> 2 HBr

The experimental rate law is determined as:

But the reaction is NOT ELEMENTARY!!! Actually

(1) Br2 ---> 2 Br Chain initiation step

(2) Br + H2 ---> HBr + H Chain Propagation

(3) H + Br2 ---> HBr + Br Chain Propagation

(-2) H + HBr ---> H2 + Br

(-1) 2Br ----> Br2 Chain Termination

so:

Effect of Temperature on Reaction Rates

Experimentally, it has been found that reaction rates increase with temperature increase, and that this increase can be formulated by:

This relationship is called the Arhenius equation, where R is the gas constant, A is the pre-exponential frequency factor, and Ea is the activation energy.

The relationship of k to T is:

A straight line should be produced by a plot of ln k vs 1/T, with the slope equal to.

, or

Activation energy is therefore defined as the slope multiplied by –R. The intercept on the 1/T axis is , or on the ln k axis, ln A

Where

A is the pre-exponential factor, or "frequency factor", that is thought to relate to the entropy of the activated complex. A is assumed to be a constant that does not vary with temperature.
Ea is the activation energy, also thought to be constant with respect to temperature.

Emperical Rate Laws

So far we have been dealing with simple (elementary) reaction mechanisms. These kinetic formulations can also be used to provide an empirical mathematical framework to understand data from more complex reactions. The trick, however, is to simplify a complex situation, not to discover a fundamental mechanism.

Example: BOD5 test.

The BOD5 test determines the dissolved oxygen consumed by biological processes, and calculated as:

where the dilution factor is

The test determines only one point on the curve that relates oxygen uptake with time:

Organics + O2 + H2O + oxidized products + ...

and the rate of this reaction in the presence of an excess of dissolved oxygen has often been stated to be first order with respect to organic matter. Empirical! If L = concen-tration of organic matter at any time t (days), then for a first order reaction we can write:

integrating we get:

L = L0e-kt

by going through the process we can estimate the first stage carbonaceous BOD:

K. NUCLEATION:

Nucleation is the formation of a new phase, here the formation of solid phase from a supersaturated solution. The first stage of nucleation is the rapid formation of dimers, trimers, and higher polymers. As the micro-particle forms, however, the surface free energy of the particle becomes an important consideration:

The surface free energy of a very small sphere (<0.1 um) is much higher than for a larger particle. Therefore there is more work associated with overcoming this surface free energy, defined as the interfacial - Free energy liberated

W = 4pr2g - 4/3p r3 DG

Solving for rate, and using W analogous to activation energy (Even though it is thermodynamic in nature...), we get:

This suggests that the rate of nucleation and growth is very sensitive to interfacial energy. At low degrees of supersaturation, nucleation can only take place if there is a way of minimizing interfacial energy. This may occur by nucleating on an established surface.

DISSOLUTION AND GROWTH:

There are five primary steps to a surface reaction:

Diffusion of reactants to a surface
sorption of reactants to the surface
reaction forming a transition state complex
Decomposition of the transition state complex abd desorption
diffusion of reactants away from the surface.

Any of these steps, in theory, can be the rate controlling step. In practice, it is usually either surface control, or diffusion/transport control

Surface Controlled Reactions:

At low degrees of saturation, the rate of dissolution and precipitation is usually surface controlled. Surface control occurs when the rate of dissolution is slower than the rate of diffusion, giving a uniform solution composition with distance from the surface.

The nature of the surface controls bulk rate: kinks, steps, dislocations. During dissolution, the atoms with the highest energy react first. Sorbed compounds may poison a surface controlled reaction, for example PO4.on a calcite surface.
Surface controlled reactions generally have an activation energy greater than that of diffusion, and are often quite high.

Diffusion control

When the rate of reaction is less than the rate of diffusion of products away from the surface, saturation is reached at the surface, and the reaction proceeds only as fast as excess products diffuse away. This assumes no advective process, so a standard experimental observation is that an increase in stirring speed should increase the rate.

Diffusion-controlled rates have a low activation energy (about 5 kcal/mol) compared to surface controlled reactions (12-17 kcal/mol).

Example: Calcite dissolution in Seawater.

This experiment used pH stat to control pH during reaction and maintain a constant degree of undersaturation, as long as the added calcium was small compared to sea water

Dissolution rate is calculated from the rate and amount of acid added to maintain the pH.

SI is calculated as the difference in pH between the actual conditions and the equilibrium pH for that solution composition.

Findings:

Below pH 4, diffusion controlled, no etch pits formed
sensitive to pH, pCO2, possibly reactant sorption controlled (H+)
Surface controlled, dissolving at kinks and dislocations, etch pits formed.

Moral: Even simple calcite dissolution is very complex at the molecular level.

***Kinetics can be used to discern molecular interactions and mechanims that are invisible in a thermodynamic calculation. Thermo is path independent, kinetics is not!!