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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.
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, t1/2. 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.
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:
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
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?
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
Note: Don't forget to multiply ln in the equation.
After substituting the values into this equation, the half life is determined:
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.
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.
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
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 consist of a matrix reinforced by a
dispersed phase in form of particles.
random orientation of particles.
preferred orientation of particles. Dispersed phase of these materials consists
of two-dimensional flat platelets (flakes), laid parallel to each other.
reinforced composites. Short-fiber reinforced composites consist of a matrix
reinforced by a dispersed phase in form of discontinuous fibers (length <
with random orientation of fibers.
preferred orientation of fibers.
reinforced composites. Long-fiber reinforced composites consist of a matrix
reinforced by a dispersed phase in form of continuous fibers.
orientation of fibers.
orientation of fibers (woven).
When a fiber reinforced composite consists of several layers
with different fiber orientations, it is called multilayer (angle-ply)