John Dolton chemistry

Introduction
John Dalton law was proposed by John Dalton in the year 1801. According to this law at a particular temperature, the total pressure of a mixture of two or more non-interacting gases is equal to the sum of partial pressures of the individual gases.
This law can be mathematically written as:
P_total = P₁ + P₂ + P₃ + ...
Here P₁, P₂, P₃ are the partial pressures of each component in the mixture.

John Daltons Atomic Theory, Laws of multiple proportions, Dalton’s law of partial pressure and for Daltanism.  Dalton concluded that evaporated water exists in air as an independent gas in the course of his studies.  Dalton found that evaporation might be viewed as a mixing of water particles with air particles. He performed a series of experiments on mixtures of gases to determine what effect properties of the individual gases had on the properties of the mixture as a whole and he was the first to associate the ancient idea of atoms with stoichiometry.  Dalton came to know the vital theoretical connection between atomic weights and weight relations in chemical reactions. The core concepts of Dalton’s theory are foundations of modern physical science.


                   John Dolton

Assumptions of John Dolton chemistry

All matter consists of tiny particles
Atoms are indestructible and unchangeable: According to this assumption of John Dalton atoms of an element cannot be created, destroyed, broken into smaller parts or transformed into atoms of another element. So he stated that atoms cannot be created, destroyed or transformed into other atoms in a chemical change.

Elements are characterized by the mass of their atoms: According to this assumption atoms of different elements have different weights.

When elements react their atoms combine in simple, whole number ratios:  In this postulate Dolton explained that compounds contained characteristic atom-to-atom ratios in this postulate he effectively explained the law of definite proportions.

When elements react, their atoms sometimes combine in more than one simple, whole-number ratio:  According to this postulate why the weight ratios of nitrogen to oxygen in various nitrogen oxides were themselves simple multiples of each other.

Atoms in compounds according to John Dalton Chemistry

The law of fixed composition

Conservation of matter and energy

Introduction
The conservation of matter and energy means that the total amount of energy and the total amount of matter is always constant in a given closed, isolated system. In other words, neither energy nor mass can be created or destroyed in any physical or chemical process.

Formation of the law of conservation of matter and energy

Until the discovery of mass - energy equivalence by Albert Einstein in 1905, conservation of matter and conservation of energy were two different conservation laws. Conservation of matter implied that matter can neither be created nor destroyed, and conservation of energy implied that energy can neither be created nor destroyed. The two were not related to each other in any  aspect except that they were both conversations laws.
But, as physics and science increased its parameters with the advance of technology,nuclear reactions were discovered, and it was discovered that in nuclear fusion and fission reactions, the total mass of reactants does not equal to the total mass of the products obtained. For example, in the following nuclear fusion reaction,
4H → 2He,
There is a difference between the mass of the reactants, that is, four atoms of Hydrogen, and the mass of the products, that is, one atom of Helium. This difference can be clearly depicted by the following diagram:-



The difference in the mass could not be explained by any laws, and furthermore, it was a direct failure of the law of conservation of mass. Huge amount of energy was produced in the above reaction, and this "creation" of energy and in was certainly a setback to the law of conservation of energy. But in 1905, Albert Einstein pointed out that in the chemical reactions like above, the difference in mass of the reactants and products is balanced by the release of energy, that is, some mass of the reactants was converted into energy, and since matter and energy are the same thing, conversion of matter into energy cannot be regarded as the creation of energy. He also gave the formula to calculate the amount of energy produced as follows:-
`e = mc^2` .

Conservation of Matter and Energy : Albert Einstein

Albert Einstein pointed out that matter and energy are the same thing, and matter can be converted to energy. Thus, if energy is being "produced" in a nuclear reaction, although the total calculated amount of energy is increasing, but since the calculated amount of mass is decreasing, and matter and energy are the same thing, therefore the total energy of the system can be regarded as constant.
Thus, the law of conservation of energy and the law of conservation of matter were combined, to form the law of conservation of matter and energy. Note that the term "The Law of Conservation of Energy" implies the law of conservation of matter and energy.

Conservation of Mechanical energy

Introduction :
Conservation of mechanical energy states that the whole mechanical power (sum of kinetic energy+ potential energy) of a structure remains stable provided the forces acting on the body are conservative. The standard of conservation of mechanical energy states, the whole mechanical energy of a conservation system leftovers constant.

Proof of Conservation of Mechanical energy

Suppose a particle of mass m moves from position x₁ to another position x₂ under the action of conservation force. As a result its velocity changes from v₁ to v₂. By work-energy theorem, we have

If V₁ and V₂ are the potential energies at positions x₁ and x₂, we have

If effort done by a conservative force is positive, the kinetic energy of body increases and potential energy of body decrease to stay total mechanical energy conserved.

Example of Freely Falling Body

Gravitational force is conservation force. When a body of mass m, initially at rest at height H, above the ground, falls under gravity, then its total mechanical energy remains constant.

At heighest point A:
The body is at rest, therefore kinetic energy,

Potential energy, V=mgh
Total mechanical energy at A=K+V=0+mgH=mgH

At point B:
Let the body reach at intermediate point B at distance x below point.

Potential energy=mg(H-x)

At point C:
Let C be the point on the ground. If v_C is the velocity at point C, then relation
V²=u²+2gs gives v²_C =0 +2gH=2gH
Kinetic energy,

Potential energy = 0
Total mechanical energy at C=mgH=0=mgH
Thus it is clear that for a freely falling body, the total mechanical energy remains constant.

Example of Mass-Spring System in Conservation of Mechanical Energy

Consider a spring-block system located on a horizontal frictionless table. Mass of block is M and force stable of spring is K. when mass is in position A, spring is in its usual length. So that elastic PE of spring is zero. When mass is taken to position B, the spring is stretched by an amount x₀, so elastic potential energy of spring (= (1/2 )kx₀²) and kinetic energy in this position = zero.

Torque Equation of a D.C Motor

Introduction:
The turning or twisting force about an axis is called torque. Consider a wheel of radius R meters acted upon by a circumferential force F Newton’s .Basically the torque is developed in the armature and hence, gross torque produced is denoted as T_a. The mechanical power developed in the armature is transmitted to the load through the shaft of the motor.

Torque Equation of a D.C Motor

Let us see about examples of torque,
The wheel is rotating at a speed of N r.p.m
Then angular speed of the wheel is,



So, work done in one revolution is,
W  =   F * distance travelled in one revolution
      =   F * 2π R Joules
P   =   Power developed = Work done/ Time



P  = T *ω      Watts
T = Torque   in  N  - m
ω = Angular speed in rad / sec.

let T_a be the gross torque developed by the armature of the motor in examples of torque. It is also called armature torque. The gross mechanical power developed in the armature is E_b I_a, as seen from the power equation. So if speed of the motor is N r.p.m. then,
Power in armature = Armature torque * ω



But E_b in a motor is given by,


This is the examples torque equation of a d.c motor.

Example of torque

Let us see about examples of torque,
A 4 pole d.c motor takes a 50 A armature current. The armature has lap connected 480 conductors. The flux per pole is 20 mWb. Calculate the gross torque developed by the armature of the motor.
Solution:

Types of torque in the motor

In examples of torque,the mechanical power developed in the armature is transmitted to the load through the shaft of the motor. It is impossible to transmit the entire power developed by the armature to load.This is because while transmitting the power through the shaft, there is a power loss due to the friction, windage and the iron loss. the torque required to overcome these losses is called lost torque.

Diffraction patterns

A method is described which assigns indices to a set of single-crystal reflections recorded by the rotation-oscillation technique using a fixed X-ray wavelength. It is assumed that the space group and approximate unit-cell parameters are known. The unknown crystal orientation is determined directly from the observed diffraction pattern of one or several oscillation data records. A local indexing procedure is described which tolerates large initial errors in the parameters controlling the diffraction pattern. These parameters are refined subsequently, thereby satisfying the constraints imposed by the space-group symmetry.

Diffraction bending of waves around the edge of an obstacle. When light strikes an opaque body, for instance, a shadow forms on the side of the body that is shielded from the light source. Ordinarily light travels in straight lines through a uniform, transparent medium, but those light waves that just pass the edges of the opaque body are bent, or deflected. This diffraction produces a fuzzy border region between the shadow area and the lighted area. Upon close examination it can be seen that this border region is actually a series of alternate dark and light lines extending both slightly into the shadow area and slightly into the lighted area. If the observer looks for these patterns, he will find that they are not always sharp. However a sharp pattern can be produced if a single, distant light source, or a point light source, is used to cast a shadow behind an opaque body.
The radiation diffraction pattern for amorphous materials requires special analysis for information on atomic groupings. Powdered crystalline material may also give diffuse patterns requiring this type of analysis. Absence of a sharp diffraction pattern does not necessarily indicate absence of the material. Examples are given for chrysotile and powdered chrysotile.

If the diffraction pattern of the hyperbolic umbilical diffraction catastrophe is produced by an optical system of increasing aperture, it passes continuously from the two-dimensional system of Airy rings in the focal plane, made by a very small aperture, to the full three-dimensional pattern corresponding to infinite aperture. The paper studies this transition by examining the truncated diffraction integral and following the evolution of the wave dislocation lines (phase singularities) on which the pattern is based. The seed of the evolution from a two-dimensional to a three-dimensional pattern turns out to be already present asymptotically even for the smallest aperture: namely, a column of small dislocation rings very close to the axis that stream in procession towards the focal plane, and become dislocations lying in the Airy fringe surfaces that run parallel to the main fold caustic, only to disappear ultimately by retreat to infinity. The evolution into the final dislocation pattern takes place by a sequence of primitive local topological events, such as reconnection (hyperbolic interchange) and ring creation.

X Ray diffraction Analysis

Introduction
After the discovery of X-rays, scientist started working on the wave nature of these rays. To test the nature, X-rays have to produce interference and diffraction patterns. For diffraction to occur, the wavelength should be in the order of its slit width. But we know that X-ray have shorter wavelength and it is quite impossible to make such a slit / grating of smaller dimension. So the wave nature was studied in atomic level. Scientist by name Laue suggested that crystal can act like a space grating to observe diffraction. This experiment was later supported by Bragg’s equation.

X Ray diffraction Analysis : Bragg’s law

Consider a plane lattice crystal with inter planar distance d. Suppose a beam of X-rays of wavelength λ  is incident on the crystal at an angle θ , the beam will be reflected in all possible atomic planes. The path difference between any two reflected waves is equal to the integral multiple of wavelength. The ray P gets reflected from the surface while the ray Q has to under go some path difference. The extra distance traveled by the ray Q from the figure is
 (BC +CD). From the diagram either BC or CD is equal to d sin theta. So the path difference is
    d sin θ  + d sin θ  = n λ
    2 d sin θ  = n λ
 Here n is the order = 1,2, 3 …… .This is Braggs law

X Ray diffraction : Analysis of the Pattern

The crystal which is considered as the slit is placed in the Bragg spectrometer for investigation. X-rays are incident on the crystal at different angles and its corresponding ionization current is noted. The below is the plot of ionization current and incident angle
 
  At a certain values of the angle of incidence, the ionization current is increases abruptly or at peak value. Basing on the angle at maximum current, the planar distance can be calculated using Bragg’s law.

 When the angle increases i.e. order of the spectrum, the intensity of the X- rays decreases

Ionization current will never fall to zero.

Thus we can say the diffraction analysis of X-rays helped us to study the crystal structure.

Multiple slit diffraction

Introduction to diffraction:
                The wave nature of the light is further confirmed by the phenomenon of diffraction. The word diffraction is derived from the Latin word diffracts which means break to pieces. When the waves are encountering obstacles they bend round the edges of the obstacles if the dimensions of the obstacles are comparable to the wavelength of the waves. The bending of waves around the edges of an obstacle is called diffraction.

Diffraction

Diffraction in opening:
                The diffraction may be take place in single or multiple slits. Here we are going to see about the diffraction by considering the passage of waves through the opening. When the opening is large compared to the wave length the waves do not bend round the edges which is given as,
                                                                                 
             When the opening is small, the bending effect round the edges is noticeable. When the opening is very small the waves spread over the surface behind the opening. The opening appears to act as an independent source of waves which propagate in all direction behind the opening.

Multiple slit diffraction

Diffraction in multiple slit:
                  Here we are going to discuss about the diffraction at multiple slits. We have seen the narrow slit gives a diffraction pattern considering of a principal maximum flanked by secondary maxima of lower intensity. In case of the multiple slit, each slit produces the similar diffraction effects in the same direction and the observed pattern is crossed by a number of interferences fringes. The regions of first order, second order, etc. maxima contain equally spaced fringes but they will be progressively fainter.
                 The envelope of the intensity variation of the interference fringes is identical to that of the diffraction pattern due to a single slit. In general I_S is the intensity at a point due to interference of light from multiple slits and I_d is the intensity due to diffraction of a single slit, then the resultant intensity I is given by,
                                                                                    I = I_d `xx` I_S
                Hence if I_d=0 at any point, then I=0 at that point irrespective of the values of I_S. The intensity and sharpness of the principal maxima increase and those of the secondary maxima decrease.
                When the slits are large in number, bright narrow lines are visible on a dark background . The greater the number of slits and the closer they are the narrower and brighter are the lines on the screen. Bringing the slits closer results, of course, in an increase in the distance between the lines on the screen.