• Rate law introduction
• Writing rate laws examples
• Collision model and Boltzmann Distribution Curve
• Integrated rate laws and graphs
• Half life
• How to change reaction rate

## Rate Laws #### Introduction

Kinetics is the study of how fast a chemical reaction progresses. The rate of a chemical reaction is measured by the disappearance of reactants or the appearance of products. These rates of appearance and disappearance are consistent with the stoichiometric ratio present in the reaction’s chemical equation. For example, if a reactant has a coefficient of 1 but a product has a coefficient of 2, the rate of appearance of the product will be double the rate of disappearance of the reactant.

#### What is a Rate Law?

A rate law is an equation that represents the rate of a reaction. The reaction A + B → C, for example, can be represented by rate = k*[A]^x*[B]^y, which is visualized more effectively to the left. The rate law is based on the concentrations of reactants raised to a power. These exponents are not necessarily the coefficients in front of the reactant in the balanced reaction; the power can only be determined experimentally. The exponent along with each term is the “order” with respect to the reactant, and the sum of the exponents is the overall reaction order. The “k” term is the rate constant. The value of this constant changes with the temperature of the reaction and the units of k depend on the overall reaction order. The formula 1/(M^(overall order-1)xtime) can be used to find the units of k if you don't want to memorize them.

#### Mechanisms and Elementary Steps

A reaction mechanism or elementary steps are a series of reactions that when added together result in the original reaction. The steps in a mechanism can be slow or fast, and this helps us determine rate laws as shown in the examples below. This process relies on being able to identify intermediates and catalysts. Intermediates are substances that are formed in one step but used up in another. Catalysts are not used up in the reaction so they should be present on both the reactants and products sides of a given pair of elementary steps. When added together, matching substances on the produces and reactants sides of the reaction cancel out and should leave the original reaction.

## Collision Model and BoLtzmann Distribution Curve #### Temperature and Kinetic Energy

Temperature is proportional to kinetic energy. Given the equation KE=1/2mv^2, the kinetic energy of a particle is dependent on the particle’s mass and its velocity. Since a particle’s mass remains constant, a higher temperature causes particles to move faster, while a lower temperature causes a particle to move slower. In a container of constant temperature, all the particles have the same kinetic energy. However, since the mass of these particles can vary, smaller particles will move faster while the larger particles will move faster.

#### The Collision Model

These observations of kinetic energy are important in kinetics because based on the Collision Model Theory, a reaction occurs when two reactants collide with enough velocity and in the correct orientation to form products.

#### Boltzmann Distribution Curve

The effects of the collision model can be visualized in the Boltzmann Distribution Curve, pictured on the left. The blue line represents a warmer sample, while the red line represents a colder sample. The vertical line represents the reaction’s activation energy. The kinetic energy of the particles is plotted on the horizontal axis, and the number of particles is plotted on the vertical axis. At a higher temperature, the graph shifts to the right as more particles have a higher kinetic energy. However, the peak becomes lower as the area under the curve is the total number of particles and must remain constant. As a result of this shift, more particles collide with enough energy to exceed the activation energy which causes the reaction to progress more quickly. ## Integrated Rate Laws

The rate law discussed above is often known as the differentiated rate law and compares reaction rate to time. The integrated rate laws (which are integrated from the differentiated rate law using calculus) compare concentration to time

The overall order of a reaction can be determined by looking at a graph of concentration vs. time, or the reaction’s integrated rate law. As depicted in the image, a zero order reaction will have a linear relationship when concentration is plotted on the vertical axis. A first order reaction is linear when the natural log (ln) of concentration is plotted on the vertical axis. A second order reaction is linear when 1/concentration (the reciprocal) is plotted on the vertical axis.

In these graphs, the intercept with the vertical axis can be used to calculate the initial concentration depending on the order of reaction. In a zero order reaction, this intercept is the initial concentration, and in first and second order reactions, the intercept is ln(concentration) or 1/concentration respectively. The slope of these graphs corresponds to k, the rate constant.

## Half Life Half life is a special example of a first order reaction. The equation t(½) = 0.693/k can be used with the first order graph to find k, since t(½) is the time for the concentration to halve. More commonly, the equation pictured can be used to complete calculations with half life.

## How to Change REaction Rate

The following factors influence the kinetics of a reaction: concentrations, temperature, surface area, and catalysts.

• Concentrations: increased concentrations of reactants increases the rate of a reaction as the rate law is dependent on concentrations as seen above.
• Temperature: As seen in the section on collision theory above, particles must have sufficient energy to collide and react with other particles. When temperature is increased, the kinetic energy of the particles increases as well because temperature is proportional to kinetic energy. As a result, more particles have enough energy to exceed the reaction’s activation energy when they collide, thus increasing the rate of reaction.
• Surface area: when a solid reacts with a liquid, the solid with a larger surface area, such as a powder compared to a solid block, will react faster. This is because collisions with sufficient energy between reactants allow the reaction to progress. With increased surface area, more of these collisions occur which causes the reaction to progress more quickly
• Catalysts lower the activation energy of a reaction which causes the reaction to progress faster as less energy is required to convert reactants into products