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.

What are Composite Materials

A composite material is one in which two or more separate materials have been combined to make a single construct having more desirable properties. What many people don't realize is that composites are probably the most common structural materials in the world, and have always been an essential part of their lives. Concrete, paper, corrugated cardboard, plywood, fiberglass, bamboo, cornstalks, trees, bricks... all are composite materials. Far from being a new invention, composite materials are the main structural elements of nature. Take a close look at the grain and structure of a piece of wood, and you will see how its strength comes from a structure of fibers bound together side by side.

Man's first use of such composite materials was probably the adobe brick. Mud or clay can be shaped and dried into a hard block, but that kind of block has little load bearing strength and can be easily crushed by the weight of other blocks on top of it. At some point in time, it was found that mixing dried grass or straw into the mud produced a brick with superior properties, a brick that could bear much greater loads without being crushed than a brick of plain dried mud could bear.

Plywood is another example. In plywood, thin sheets, or 'plies' of wood are laminated together. In each ply, the wood fibers runs in one particular direction, and each ply is aligned in a different direction than the adjacent plies. This gives the resulting stack of wood plies an optimum strength in all directions, making plywood a very versatile and useful structural material. A third example of a composite material is reinforced concrete, used in the construction of bridges and buildings. Steel rods are encased in a matrix of concrete, producing reinforced concrete, which has much better strength and load-bearing properties than concrete that has not been reinforced.

Go to Next:  Classification of Composites

Redox Reactions or Oxidation Reduction Reactions

Redox Reactions  / Oxidation and reduction reactions


These are reactions where electrons are transferred from one species (atom, molecule or ion) to another. We can write 'half' equations to show only what happens to the species losing electrons or a different 'half' equation to show the species gaining electrons.
The whole equation is put together by making sure that the numbers of electrons are balanced in each half equation and adding them together (when the electrons will cancel out)
Oxidation
This is the name given to removal of electrons from a species - the reagent causing the loss of electrons is called the oxidising agent

Example: Mg(s) Mg2+ + 2e In this (half) equation the magnesium atom loses electrons and becomes an ion.

Reduction
This is the gain of electrons - the species donating the electrons is called the reducing agent

Example Fe3+ + 3e Fe(s) In this (half) equation the iron(III) ion gains three electrons to become an atom.

Redox reactions
Obviously the electrons leave one species and go to another. Consequently reduction has to be accompanied by oxidation and vice versa. For this reason reactions involving transfer of electrons are called reduction and oxidation or redox for short

Example 3Mg(s) + 2Fe3+ 2Fe(s)+ 3Mg2+ The electrons from the magnesium are transferred to the iron(III) ions

Summary

Loss of electrons = Oxidation Gain of electrons = Reduction	Mnemonic (memory aid) OIL-RIG
			Oxidation Is Loss Reduction Is Gain

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10.1.2: Calculate the oxidation number of an element in a compound. Oxidation numbers should be shown by a sign (+ or -) and a number, eg +7 for Mn in KMnO4.

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Oxidation number
This is the apparent valency of an atom within a compound. It is usually considered as if the element were bonded ionically to allow the apparent number of electrons gained or lost to be assessed.
The sum of all the oxidation numbers in a compound must add up to 0. By convention, the oxidation number is written as a Roman numeral in the name, eg. iron II sulphate, sulphur VI oxide.
The oxidation number of an uncombined element is always zero (0)
Calculating the oxidation number
There are some elements that virtually always have the same oxidation number and these can be used to calculate the oxidation numbers of the atoms in question.
Hydrogen, for example always has an oxidation number of -1 when bonded to a metal (more electropositive element) and +1 when bonded to a more electronegative element (non-metal). Oxygen is always -2 (except when in the form of the peroxide ion when it has an O-O bond giving it an oxidation number of -1). Group 1 and 2 metals usually have an oxidation number of 1+ and 2+ respectively.

Example - Calculate the oxidation number of sulphur in sulphuric acid H2SO4 Hydrogen = +1 oxidation number Oxygen = -2 oxidation number Therefore: (2 x H) + S + (4 x O) = 0 2 + S -8 = 0 S = 6

Example - Calculate the oxidation number of nitrogen in calcium nitrate Ca(NO3)2 Calcium is in group 2 = +2 oxidation number Oxygen = -2 oxidation number Therefore: (+2) + [(2 x N) + (6 x -2)] = 0 +2 + 2N -12 = 0 2N = 10 N = +5

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10.1.3: State and explain the relationship between oxidation numbers and the names of compounds. Oxidation numbers in names of compounds are represented by Roman numerals, eg iron(II) oxide, iron(III) oxide.

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Names of compounds
Where there is any doubt about the oxidation state of an element within a compound it is stated using Roman numerals immediately after the ambiguous element. For example Iron compounds may be iron in the oxidation state +2 or +3 - it must therefore be stated as iron II or iron III in the compound name.
In the examples above the full systematic name for sulphuric acid is sulphuric(VI) acid and calcium nitrate is calcium nitrate(V)

Example - Name the following compound - FeSO4 Oxidation state of the oxygen = -2 Oxidation state of the sulphur = +6 Therefore oxidation state of the iron = - (+6 - 8) = +2 The name of the compound FeSO4 is iron(II) sulphate

Example - Name the following compound - TiCl4 Oxidation state of the chloride = -1 Therefore oxidation state of the titanium = - (- 4) = +4 The name of the compound TiCl4 is titanium(IV) chloride

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10.1.4: Identify whether an element is oxidised or reduced in simple redox reactions, using oxidation numbers. Appropriate reactions to illustrate this can be found in topics 3 and 11. Possible examples include: iron(II) and (III), manganese(II) and (VII), chromium(III) and (VI), copper(I) and (II), oxides of sulphur and oxyacids, halogens and halide ions.

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Oxidation and reduction
As stated above, for the purposes of oxidation and reduction the oxidation number can be thought of as the apparent ionic charge of an atom within a compound. For example, in sulphuric acid the sulphur is in the VI (6+) oxidation state. For the purposes of redox we can consider that it has an ionic charge of +6 (even though it is clearly covalently bonded). This makes it easier to follow any transfer of electrons.
If the sulphur changes to an oxidation state of IV during a chemical reaction then it has gone from an apparent ionic charge of +6 to a charge of +4, i.e. it has gained two electrons (negative charges). It has therefore been reduced (gain of electrons) in the process.

Examples 2FeCl2 + Cl2 2FeCl3 The iron changes from 2+ to 3+ and is therefore oxidised (removal of electrons) The chlorine gains an electron to go from 0 to -1 and is therefore reduced (addition of electrons) Zn + CuSO4 Cu + ZnSO4 The zinc changes from oxidation state 0 to +2 (removal of electrons) it is oxidised  The copper changes from 2+ to 0 and is oxidised and is therefore reduced (addition of electrons) Cr2O72- + 3SO2 + 2H+ 2Cr3+ + 3SO42- + H2O The chromium changes from +6 to +3 and is therefore reduced (gain of electrons) The sulphur changes from +4 to +6 and therefore loses electrons = oxidation (loss of electrons) 2KI + Br2 2KBr + I2 The iodide ions (oxidation number = -1) change to iodine (oxidation number = 0) : oxidation Bromine (element, oxidation number = 0) changes to bromide ions (oxidation number = -1) : reduction 5Fe2+ + MnO4- + 8H+ 5Fe3+ + Mn2+ + 4H2O The iron changes from 2+ to 3+ and is oxidised (removal of electrons) The manganese atom changes from +7 to +2 and is therefore reduced (addition of electrons)

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10.1.5: Define the terms oxidising agent and reducing agent.

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Oxidising agents
These are the chemicals that cause the oxidation in a redox reaction. We call the reacting compounds in a reaction the reagents (short form of the words reacting agents).
We consider that the removal of electrons from a species is oxidation and these electrons have to be taken away by another compound or species. This species that attracts the electrons is said to be the oxidising agent i.e. the reagent that causes the oxidation.
Reducing agents
Similarly the reagent that causes reduction in a redox reaction is said to be the reducing agent.
The oxidising agent takes the electron and is itself reduced, the reducing agent loses the electrons and is itself oxidised.

2KI	+ Br2		2KBr	+ I2
Iodide ions get oxidised	Bromine gets reduced
Iodide - reducing agent	Bromine - oxidising agent
Cr2O72-	+ 3SO2 + 2H+		2Cr3+	+ 3SO42- + H2O
Chromium VI gets reduced	Sulphur IV gets oxidised
Chromium VI oxidising agent	Sulphur IV reducing agent