First-Order Reactions

Let's explain First-Order Reaction.

The Differential Form of the Rate Law

Rate law:

 is the reaction rate and  is the reaction rate coefficient. In first order reactions, the units of  are 1/s. However, the units can vary with other order reactions.

The Integrated Form of the Rate Law

First, write the differential form of the rate law.

 

Rearrange

Second, integrate both sides of the equation.

Recall from calculus

 

Upon integration, we get

 

Rearrange to solve for  and we get one form of the rate law

 

We can rearrange the equation above to:

Recall from Algebra y=mx +b is the equation of a straight line, which  demonstrates.

Now that we recall the laws of logarithms we can say that   is at the time t with its final concentration of A and [A]_o is at time 0 and it is at its initial concentration of A and k is  the rate constant. Since, the logarithms of numbers do not have any units, the product of -kt does not have units as well. This concludes that unit of k in a first order of reaction must be time^-1 . Examples of time^-1 would be s^-1 or min^-1. Thus, the equation of a straight line is applicable to represent 

To test if it the reaction is a first order reaction, plot the natural logarithm of a reactant concentration versus time and see whether the graph is linear. If the graph is linear and has a negative slope, the reaction must be a first order reaction.

To create another form of the rate law, raise each side of the previous equation to the exponent, e

Taking the natural log of both sides of the equation, we get the second form of the rate law

The integrated forms of the rate law allow us to find the population of reactant at any time after the start of the reaction. Plotting with respect to time for a first-order reaction gives a straight line with the slope of the line equal to -k. For more information on differential and integrated rate laws.

Graphing First-order Reactions

The following graphs represents concentration of reactants versus time for a first-order reaction.

Plotting   with respect to time for a first-order reaction gives a straight line with the slope of the line equal to -k.

Relationship Between Half-life and First-order reactions

The half-life is a timescale by which the initial population is decreased by half of its orignal value, t_1/2. We can represent the relationship by the following equation.

 

Using the integrated form of the rate law, we can develop a relationship between first-order reactions and the half-life.

 

Substitute

 

Notice that, for first-order reactions, the half-life is independent of the initial concentration of reactant. This is unique to first-order reactions. Using an alternate half life equation of First Order Reaction would be

Thus, if a problem gave initial concentration, final concentration and a time, it would be more applicable to use the alternate half-life equation rather than:   

The practical implication of this is that for A to decrease from 1 M to 0.5 M, it takes just much time as it does for A to decrease from 0.1 M to 0.05 M

Example

If 3.0 g decomposes for 36 min, the mass of A remaining undecomposed is found to be 0.60 g. What is the half life of this reaction?

Solution: Notice there are initial concentrations and final concentrations. This should be a hint to use the alternate form of half life equation.

1) Plug in the values in the appropriate places

into t

into A_o

 into A_t

Note: Don't forget to multiply ln[2] in the equation.

After substituting the values into this equation, the half life is determined:

 

Classification of Composites

Classification of Composites

Composite are materials composed of two or more distinct phases (matrix phase and dispersed phase) and having overall properties significantly different form those of any of the constituents.

Matrix phase

The primary phase, having a continuous character, is called matrix. Matrix is usually more ductile and less hard phase. It holds the dispersed phase and shares a load with it.

Dispersed (reinforcing) phase

The second phase (or phases) is embedded in the matrix in a discontinuous form. This secondary phase is called dispersed phase. Dispersed phase is usually stronger than the matrix, therefore it is sometimes called reinforcing phase.

Many of common materials (metal alloys, doped Ceramics and Polymers mixed with additives) also have a small amount of dispersed phases in their structures, however they are not considered as composite materials since their properties are similar to those of their base constituents (physical properties of steel are similar to those of pure iron).

There are two classification systems of composite materials. One of them is based on the matrix material (metal, ceramic, polymer) and the second is based on the material structure:

Classification of composites I

 (based on matrix material)

Metal Matrix Composites (MMC)

 Metal Matrix Composites are composed of a metallic matrix (aluminum, magnesium, iron, cobalt, copper) and a dispersed ceramic (oxides, carbides) or metallic (lead, tungsten, molybdenum) phase.

Ceramic Matrix Composites (CMC)

Ceramic Matrix Composites are composed of a ceramic matrix and embedded fibers of other ceramic material (dispersed phase).

Polymer Matrix Composites (PMC)

Polymer Matrix Composites are composed of a matrix from thermoset (Unsaturated Polyester (UP), Epoxiy (EP)) or thermoplastic (Polycarbonate (PC), Polyvinylchloride, Nylon, Polysterene) and embedded glass, carbon, steel or Kevlar fibers (dispersed phase).

Classification of composite materials II

 (based on reinforcing material structure)

Particulate Composites

Particulate Composites consist of a matrix reinforced by a dispersed phase in form of particles.

 Composites with random orientation of particles.

Composites with preferred orientation of particles. Dispersed phase of these materials consists of two-dimensional flat platelets (flakes), laid parallel to each other.

Fibrous Composites

Short-fiber reinforced composites. Short-fiber reinforced composites consist of a matrix reinforced by a dispersed phase in form of discontinuous fibers (length < 100*diameter).

        Composites with random orientation of fibers.

        Composites with preferred orientation of fibers.

Long-fiber reinforced composites. Long-fiber reinforced composites consist of a matrix reinforced by a dispersed phase in form of continuous fibers.

        Unidirectional orientation of fibers.

        Bidirectional orientation of fibers (woven).

Laminate Composites

When a fiber reinforced composite consists of several layers with different fiber orientations, it is called multilayer (angle-ply) composite.