An Initiation to Chemical Kinetics

In chemical reactions, we seek to answer two main questions:

  1. How can we know whether a reaction will be spontaneous?
  2. Once started, how fast will the reaction proceed?

The laws of thermodynamics help us answer (1). Chemical kinetics help us answer (2).

The energy difference between products (C+D) and reactants (A+B) pertains to thermodynamics. The energy difference between the transition state and reactants pertains to kinetics.

While diamond is kinetically stable (high activation energy for conversion), graphite is more thermodynamically stable (on a lower energy than diamond).

Now that we have that cleared up, read on as we discover more about reaction rates and equilibrium!


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The study of reaction rates is called chemical kinetics. We study this so we can:

  1. Predict and optimize how quickly a reaction mixture reaches equilibrium
  2. “Find” the mechanisms of reactions.

So what is a reaction rate?

Reaction rate is the rate of change of the concentration of reactant or product with time.

The average reaction rate is the change in concentration over a defined period.

Instantaneous reaction rate is the rate of change in concentration at any particular instant during a reaction – the derivative of the concentration with respect to time.

Initial rate is the instantaneous reaction rate of the reaction when the reactants are mixed.

For the generic reaction aA + bB →cC + dD,screenshot-2016-10-23-at-12-00-20-pmand so rate is the time derivative of concentration divided by the appropriate stoichiometric coefficient and converted to a positive number (by convention).

(The units are always concentration over unit time.)

Factors controlling rate of a reaction

  1. Surface area of the reactants
  2. Concentration of the reactants
  3. Temperature
  4. Presence of a catalyst

Usually increasing these increases reaction rate. Mechanical force and electric field presence can also contribute (but they are not conventional factors).

Rate Equation and the Differentiated Rate Law

The rate law (or rate equation) relates the reaction rate and the concentration of all the chemical species (reactant, product, catalyst, inhibitor) that may be present in the reaction mixture.

For the generic reaction aA + bB →cC + dD,screenshot-2016-10-23-at-12-06-05-pm it is usually inconvenient to express the concentrations of intermediates – so we have what we call the differentiated rate law.

The term k is known as the rate constant.

The rate equation for a reaction can be determined only by experiment and has no general relation to the reaction stoichiometry!!

Reaction Order

The sum of powers to which all reactant concentrations appearing in the rate law are raised is called the overall reaction order. In the generic reaction, the overall reaction order is x+y.

The Integrated Rate Laws (Concentration-Time Relationships)

In addition to knowing how the rates of reactions depend on concentrations, it’s useful to know how concentrations depend on time. The integrated rate law can be obtained from the differentiated rate law by integration.

These laws depend on the order of the reaction, which we will get into below.

Zero-Order Reaction

A → P

In zero-order reactions, the rate of reaction is constant, independent of concentration. The limiting factor is something like solubility, absorption of light, or surface area where the reaction occurs (not concentration).

screenshot-2016-10-23-at-12-16-54-pm

First-Order Reaction

A → P

The graph of ln[A] vs. time is linear in first-order reactions, with slope = -k. We get an exponential decay in concentration versus time graphs.

screenshot-2016-10-23-at-12-21-20-pm
screenshot-2016-10-23-at-12-21-31-pm
(The three equations are identical.)

Second-Order Reaction

A + A → P

The rate of a second order reaction is proportional to the concentration squared.

screenshot-2016-10-23-at-12-26-48-pm


Experimental Determination of Rate Laws

Now, how do we determine k, x, and y in the expression below?screenshot-2016-10-23-at-12-32-40-pm

Experimentally! We have to find the order of a reaction with respect to each reactant.

Method 1: Initial Rates

Carry out several experiments using different initial concentrations. Determine the initial rate for each run.

If when the concentration of a reactant is doubled, the rate of the reaction also doubles (quadruples), the rate law is first (second) order with respect to that reactant.

Method 2: Graphical

Measure the concentration is a function of time. Different functions of concentrations have linear relationships with time depending on the rate law.

Zero order: [A] vs. time
First order: ln[A] vs. time
Second order: 1/[A] vs. time

Method 3: Half-Life

Half life is the time required for the concentration of the reactant to decrease to half its initial concentration. The half-life depends on concentration characteristic of the order of its reaction

Zero order:

screenshot-2016-10-23-at-12-39-44-pm

First order:
The half-life is independent of the concentration of the reactant.

screenshot-2016-10-23-at-12-39-56-pm

Second order:
The half-life is inversely proportional to the initial concentration.

screenshot-2016-10-23-at-12-40-12-pm

So, if we measure the half-life for many different initial concentrations, we should be able to figure out the order of the reaction.

Method 4: Reagents in Excess

If a reactant/reagent is present in great excess then its concentration is virtually constant during the course of the reaction.

If
screenshot-2016-10-23-at-12-44-31-pm, assuming B is in excess, [B] does not change with time. We can form a new constant by lumping it with k forming kobs (the observed/apparent rate constant).

screenshot-2016-10-23-at-12-46-39-pm
screenshot-2016-10-23-at-12-46-48-pm
This reaction is now said to be a pseudo-first-order reaction.


Quick Summary thus far

I think that the following is a very good summary table of zeroth, first, and second order reactions.

5ae8d514bba2df4f1a350329d94c3d8b
Image Source: Lard Bucket

Reaction Mechanisms

Each step in a sequence is called an elementary step. The collection of elementary steps by which an overall reaction occurs is called a reaction mechanism. Intermediates appear in the mechanism of a reaction but not in the overall balanced equation.

The molecularity of a reaction is the number of molecules reacting in an elementary step (unimolecular, biomolecular, and termolecular).

The order of a reaction cannot, in general, be predicted from the stoichiometry of the overall reaction. However, the order of an elementary reaction is predictable – a unimolecular reaction is first order, a bimolecular reaction is always second order, and a termolecular reaction is third order.

Rate Laws

We must determine mechanisms of chemical reactions experimentally.

First, make rate measurements. Then, analyze the data to determine the rate constant and order of the reaction (rate law is known now). Finally, suggest a plausible reaction mechanism.

The elementary steps must:

  1. Be a sum of elementary steps that equal the overall balanced equation
  2. Agree with the experimentally determined rate law.

There can be many reaction mechanisms consistent with a particular rate law, so the rate law by itself is insufficient to determine a unique reaction mechanism.

For a sequence of reaction steps (for a series reaction, opposed to a parallel reaction), products can be formed no faster than the slowest step in the reaction mechanism.

So, the slowest elementary process in a sequence (consecutive reactions) is called the rate-determining step.

Activation Energy and Temperature Dependence of Rate Constants

 

With few exceptions, reaction rates increase with temperature (in addition to concentration, as mentioned before).

The empirical (based on experiment) Arrhenius equation gives a relationship between the rate constant (k) and temperature (T).
screenshot-2016-10-23-at-1-05-18-pm
Screenshot 2016-10-23 at 1.07.08 PM.png

A is the frequency factor (pre-exponential factor), assumed o be independent of temperature, Ea is the activation energy, and R is the universal gas constant (8.314 J/mol/K).

A plot of lnk vs. 1/T (Arrhenius plot) should be a straight line, with slope -Ea/R, and intercept A.

So, we can determine activation energy by measuring k at several temperatures. An Arrhenius plot for a reaction with a high Ea has a steeper slope.


Activation Energy and Temperature Dependence of Rate Constants

In the pre-exponential factor discussed above, we can interpret A as the rate constant in the absence of an energy barrier – or the rate constant at an infinite temperature. According to the Arrhenius equation, a reaction with a zero activation energy does not depend on temperature.

We can calculate the rate constant k2 at T2 if k1 at T1 and Ea are known via the following equation:screenshot-2016-10-23-at-1-16-20-pm

The collision theory of chemical kinetics provides a theoretical explanation of the dependence of the rate constant on temperature. According to the kinetic theory of gases, gas molecules are in constant motion and frequencly collide with one another. In collision theory, the assumption is that chemical reactions occur as a result of collisions between reacting molecules, treated as hard spheres (or billiard balls!).

If collisions were 100% effective in bringing about chemical change, the rate of the reaction would be equal to the collision frequency. But nooooo, it’s way less!

That’s because we have the activation energy (Ea), the minimum energy that two molecules must have in order to react and to initiate a chemical reaction. The collision must occur with enough kinetic energy to break existing bonds.

When molecules collide they form an activated complex (transition state), a temporary species formed by the reactant molecules as a result of the collision before they form the product (really short lifetime).

The activated complex is the highest point in a potential energy profile that shows how the potential energy changes as reactants are converted to products (for example, the veeeeery first image in this post).

The Maxwell-Boltzmann distribution of energy at two temperatures shows that the fraction of effective collisions increases exponentially with temperature.

However, (again!) experiments show that the observed reaction rate is still much smaller than the rate of collision with enough energy to surmount the barrier. This discrepancy is because not all collisions occur with correct molecular orientation needed for the formation of new bonds.

To resolve the discrepancy, we introduce the steric factor, P, which takes into account orientation effects. We can consider P to be the fraction of the collisions that have the right orientation.

So even if two molecules collide with sufficient energy to overcome the barrier to reaction, unless they collide with the correct orientation, no reaction occurs.

Barrierless reactions (rate decreases with temperature)

Ion-molecule and radical-radical reactions may proceed at very low temperature because these reactions involve no activation energy, while other chemical reactions would stop altogether.

Screenshot 2016-10-23 at 2.33.06 PM.png

Image Source: Chris A

There is no barrier for a radical-radical association reaction because no chemical bonds are being broken.

When temperature rises too high, the reaction probability decreases because:

  1. probability of “capture” decreases (will bounce off instead of sticking to each other)
  2. rotational energy of the radicals decreases the probability of combining radicals remaining in a favorable orientation.

Catalysis

A catalyst is a substance that increases the rate of a chemical reaction without itself being consumed. The catalyst may react to form an intermediate but it is regenerated in a subsequent step of the reaction.

They increase the rate by lowering activation energy for the reaction. Note that the total energy of the reactants or the products is unaffected by the catalyst.

Because the activation energy of the reverse reaction is also lowered, a catalyst enhances the rate of the reverse and forward reactions equally.

  1. The first type of catalysis – heterogeneous catalysis – is where the reactants and catalyst are in different phases (usually catalyst is solid and reactants are gas or liquid).
    Very important examples include hydrogenation (of unsaturated fatty acids, for one), the Haber process, and the in catalytic converter found in all cars.
    The greatest advantage is the ease of separation which allows for recycling. However, they are handicapped by limited activity and selectivity.
  2. Homogeneous catalysis is where the reactants and catalyst are dispersed in a single phase (usually liquid).
    The most striking example is enzymes – biological catalysts. They are highly specific for their function.

And that’s it for kinetics! You know have all the information you need to ace your introductory chemistry course. Good luck, and thanks for reading!

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