Rate Laws

By Philip Tsao, Ryan Song, Sherry Gee, Katherine Barbour 

Introduction

A reaction’s progression is largely determined by its rate or the speed at which the reaction occurs. Some reactions may be instantaneous whereas others may be extremely slow. Those that are extremely slow may appear as if there is no reaction at all, for example the rusting of a nail. The field of study investigating the rates and speed at which a reaction occurs is known as Kinetics. In addition, this field also examines the conditions and factors that influence the speed and yield of a reaction, methods of determining reaction rates and - what this exploration will be focusing on - defining rate laws, mathematical descriptions of the rates of reactions.

Origin of Rate Laws

The field of kinetics originated from Waage and Guldberg’s formulation of the law of mass action which stated that the speed of a reaction is associated with quantity of reactants. In a series of largely unnoticed papers, Waage and Guldberg proposed that: 
1. affinity or chemical force between reactants in a substitution reaction is proportional to mass
2. rates of forward and backward reactions must be equal in equilibrium
3. reaction rate is proportional to concentrations
From their investigation of rates of reactions and constants for chemical affinity, the field of kinetics branched off to pursue a deeper understanding of reaction rates and formulate rate laws. Contemporarily, this field of study has become largely centered around defining the orders of reactions and examining which and how certain factors affect the rates and results of reactions. 

The Basics

The rate of a reaction is given by the rate law R=k([A]^x)([B]^y), where R is the rate, k is the equilibrium constant, [A] is the concentration of substance A, and x is the order of reaction for A.

Concentration: The amount of substance per defined space. In kinetics, concentration is typically expressed in molarity, or number of moles of a substance per liter. Equilibrium balances the concentrations of the different substances, driving the reaction forward or backwards.

Rate Constant (k): Each reaction has an equilibrium constant that determines its rate. It gives the fraction of reactant molecules that undergoes the reaction per unit time. For example, if a rate constant of 0.035 sec^(-1) means 3.5% of the reactants will undergo the reaction per second.

Order of Reaction: The magnitude of the effect of all the components of a reaction on its rate. The partial reaction order for each reactant and product is determined by experiment; the higher the reaction order, the greater the effect of the concentration of a substance on the reaction rate. The overall reaction order is the sum of the partial reaction orders. The more components in a reaction, the higher its reaction order.

Order Of Reaction

• Zero Order - The reaction rate is independent of the reactant concentration.
• First Order - The reaction rate is dependent on the concentration of a single reactant. The exponent is one.
• Second Order - The reaction rate is dependent on the concentration of two reactants, or is proportional to the concentration of a single reactant squared.
• nth Order - The sum of the concentrations that affect the reaction rate. 
  

All radioactive decay processes are first order. Many reactions with two reactants are second order; enzyme-driven reactions, however, are often zero order because the rate is far more dependent on the enzyme, making the reactant and product concentrations irrelevant.

Rate Constant

The rate constant is given by the Arrhenius equation, k = Ae^(-Ea/RT), where A is the frequency factor (also known as the pre-exponential or steric factor), Ea is the activation energy, R is the gas constant, and T is the temperature in degrees Kelvin.

Rate Laws

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Reactions of different orders have different rates based on their initial concentrations and rate constants. The differential rate law describes the rate and the integrate rate law, derived from the differential rate law, gives the concentration at a given time.

Thermodynamics versus Kinetics

Thermodynamics is all about the “if” of a reaction, such as whether the process can occur.  Kinetics is all about “how”, such as whether the reaction occurs fast or slow.  The kinetics of the process is how to overcome the energy barrier from the reactants to the products, while the thermodynamics ensures that the reaction is a favorable transformation.
For a process to occur, it must overcome the energy maximum, usually denoted E (activation energy).  The larger the barrier (Ea), which is the difference between the energy maximum and initial minimum, the more difficult the process to occur, which thus results in a slower rate.  For a reaction to occur, not only must the thermodynamically favored with a -ΔG (which could be referred to as the driving force), it must also be fast enough, which results from a small Ea.  
If a reaction is thermodynamically favored, that means that the products are in the lowest possible energy state.  If a reaction is kinetically favored, that means that the reaction went through the easiest possible direction, that is, it went in the direction with the lowest activation energy.  Given inifite amount of time, all reactions would proceed to the thermodynamically favored products.  However, if the reaction was only given a short time to proceed, some kinetically favored products would be in solution, however it is unlikely any thermodyamically favored products would, assuming the activation energy of the thermodynamically favored products is higher than that of the kinetically favored products.  The diagram below describes a possible reaction that is helpful in the understanding of thermodynamics versus kinetics. 

If you started with B and were asked what would be created, C or A, if the reaction was thermodynamically favored, the answer would be C.  Why?  C is at a lower, more stable energy than A, therefore, given infinite amount of time, all of B would eventually form C.  If you started with C and went in the reverse reaction, what would the rate determining step be? As stated before, the rate determining step is the one with the highest activation energy.  Therefore, the rate determining step would be from B to A, because that is the biggest energy “hump” to get over.

Below is another energy diagram that can be studied to help better understand the relationship, and difference, between kinetics and thermodynamics.   

Given the reactant D, what is the kinetically favored product, E or F?  The answer is E, but why?  Does that mean F is the thermodynamically favored product?  To answer the latter, E is the kinetically favored product because it has a lower activation energy i.e. the “hump” is smaller and easier to get over.  To answer the last question, F is not then, by process of elimination, the thermodynamically favored product.  Both products E and F have the same energy, therefore one product is not energetically favored versus the other.  However, if F was lower energy than E, even though the activation energy is higher than that of D to E, then F would be the thermodynamically favored product. 

Handling Multi-step Reactions

Multi-step reactions can be handled by first finding the rate-determining step, which is the slowest elementary step, and then setting it to equilibrium. 

Temperature Increases Reaction Rates

As temperature increases, reaction rates, k, increase extremely rapidly.  Svante Arrhenius suggested that the reaction rate is exponential to the inverse of temperature.

(k = A e^(-Ea/RT).

 Where E_a is a constant with dimensions of energy and A is a constant with the same dimensions as k.  

If E_a is the critical relative collision energy required for a pair of molecules to react, then only a fraction of the molecules will have the amount of energy needed to react. This fraction is represented by the area under the Maxwell-Boltzmann distribution curve.  

The average speed, ū,  is related to temperature by:

Kinetic Energy  = ½ mŪ²= 3/2 k_BT

Thus, as temperature increases, the average speed of molecule increases. Thus, the distribution function spreads out to include molecules with higher speeds or energies that will surpass the E_a and allow more molecules to react. This shows that temperature increases reaction rates.

Pressure Increases Reaction Rates

Along with temperature, pressure can also increase reaction rates. Increasing the pressure, while keeping the other variables (n,T) in the ideal gas law, PV=nRT, constant means that if pressure is increasing, volume is shrinking. This means that there will be more molecules/unit volume, therefore, there is less space for the molecules to spread out, and it would take less time before it collides with the wall or other molecules. This means that more molecules will collide in a given time. Since a reaction occurs by molecules colliding, we can say that as pressure increases, reaction rate increases

Questions:

1. Kinetically favored reactions:

     a. have a comparitively high activiation energy

     b. have a comparitvely low activation energy

     c. are the lowest possible energy product

     d. a and c

     e. b and c

2. The rate constant k = [a_C]^c [a_D]^d / [a_A]^a [a_B]^b

     a. True

     b. False

Answers:

1. b

2. b; the k described in the question is the equilibrium constant, not the rate constant.