Electrolytic Methods

Here mth term is A_m = n

and nth term is A_n = m

Let first term = a

and common difference = d

Thus Am = n = a + (m -1) d ...(1)

An = m = a + (n-1) d ...(2)

Solving equations (1) and (2) simultaneously we obtain

d = -1

and a = n + m - 1

Thus sum of (m + n) terms is

S_{(m + n)} = [(n + m) / 2] * [2a + ( n + m - 1)d]

or S_{(m + n)} = [(n + m) / 2] * [2(n + m - 1) - ( n + m - 1)]

or S_{(m + n)} = (n + m) * (n + m - 1) / 2

Using the expression

$N_A = N_{A0}e^{-{\lambda}t} \,\!$

λ = 0.693 / t1/2

or λ = 0.693 / (1.37 X 10⁹)

or λ = 5.05 x 10^-10

N_A / N_A0 = 1/8 = exp (-5.05 x 10^-10t)

so t = 4.11 x 10^-9 yrs

Basicity of Amines, Acidity of Ammonium Ions

N lone pair relatively easily protonated

note that K_b x K_a = [H⁺] [OH^-] = Kw = 10^-14
or pK_a + pK_b = 14
recall that when pH = pK_a , there are equal concentrations of the conjugate acid and conjugate base present (i.e., RNH₂ and RNH₃⁺ )
for typical aliphatic amines, pK_b = 3 - 4
so pK_a = 10 - 11 for their ammonium ions
so at around pH 10 - 11 , RNH₂ and RNH₃⁺ are both present
for typical aromatic amines, pK_b = 9 - 10
so pK_a = 4 - 5 for their ammonium ions
so at around pH 4 - 5 , ArNH₂ and ArNH₃⁺ are both present
water solubility of amines can be easily changed with pH
aromatic amines are water-soluble (protonated) below pH 4
aliphatic amines are water-soluble (protonated) below pH 9

Basicity Trends

aromatic amines are less basic due to resonance delocalization of the N lone pair

amides are nonbasic due to strong delocalization of the N lone pair

electron withdrawing effects decrease basicity
because the N lone pair is less available for bonding to a proton

Electrolytic Methods

Introduction

Unlike potentiometry, where the free energy contained within the system generates the analytical signal, electrolytic methods are an area of electroanalytical chemistry in which an external source of energy is supplied to drive an electrochemical reaction which would not normally occur. The externally applied driving force is either an applied potential or current. When potential is applied, the resultant current is the analytical signal; and when current is applied, the resultant potential is the analytical signal. Techniques which utilize applied potential are typically referred to as voltammetric methods while those with applied current are referred to as galvanostatic methods.

Voltammetric Methods

Voltammetry refers to the measurementof current that results from the application of potential. Unlike potentiometry measurements, which employ only two electrodes, voltammetric measurements utilize a three electrode electrochemical cell. The use of the three electrodes (working, auxillary, and reference) along with the potentiostat instrument allow accurate application of potential functions and the measurement of the resultant current. The different voltammetric techniques that are used are distinguished from each other primarily by the potential function that is applied to the working electrode to drive the reaction, and by the material used as the working electrode. Common techniques to be discussed here include:

Hydrodynamic Voltammetry

Polarography
Normal-pulse polarography (NPP)
Differential-pulse polarography (DPP)
Cyclic voltammetry
Anodic-sttripping Voltammetry

Time Based Techniques

Chronoamperometry
Chronocoulometry

Pair Production

Every known particle has an antiparticle; if they encounter one another, they will annihilate with the production of two gamma-rays. The quantum energies of the gamma rays is equal to the sum of the mass energies of the two particles (including their kinetic energies). It is also possible for a photon to give up its quantum energy to the formation of a particle-antiparticle pair in its interaction with matter.

The rest mass energy of an electron is 0.511 MeV, so the threshold for electron-positron pair production is 1.02 MeV. For x-ray and gamma-ray energies well above 1 MeV, this pair production becomes one of the most important kinds of interactions with matter. At even higher energies, many types of particle-antiparticle pairs are produced.

Electron-Positron Pair Production

When a photon has quantum energy higher than the rest mass energy of an electron plus a positron, one of the ways that such a photon interacts with matter is by producing and electron-positron pair.

The rest mass energy of the electron is 0.511 MeV, so for photon energy above 1.022MeV, pair production is possible. For photon energies far above this threshold, pair production becomes the dominant mode for the interaction of x-rays and gamma-rays with matter.

Weightlessness is simply due to the absence of reaction force.

Operating principles in case of parabolic flight path

Trajectory for zero gravity maneuver

The aircraft gives its occupants the sensation of weightlessness by following an (approximately parabolic) elliptic flight path relative to the center of the Earth. While following this path, the aircraft and its payload are in free fall at certain points of its flight path. The aircraft is used in this way to demonstrate to astronauts what it is like to orbit the Earth. During this time the aircraft does not exert any ground reaction force on its contents, causing the sensation of weightlessness.

Initially the aircraft climbs with a pitch angle of 45 degrees. The sensation of weightlessness is achieved by reducing thrust and lowering the nose to maintain a zero-lift angle of attack. Weightlessness begins while ascending and lasts all the way "up-and-over the hump", until the craft reaches a declined angle of 30 degrees. At this point, the craft is pointed downward at high speed, and must begin to pull back into the nose-up attitude to repeat the maneuver. The forces are then roughly twice that of gravity on the way down, at the bottom, and up again. This lasts all the way until the aircraft is again halfway up its upward trajectory, and the pilot again initiates the zero-g flight path.

This aircraft is used to train astronauts in zero-g maneuvers, giving them about 25 seconds of weightlessness out of 65 seconds of flight in each parabola. During such training the airplane typically flies between 40-60 parabolic maneuvers. In about two thirds of these flights, this motion produces nausea due to airsickness, especially in novices, giving the plane its nickname.

More abstractly, a charge is any generator of a continuous symmetry of the physical system under study. When a physical system has a symmetry of some sort, Noether's theorem implies the existence of a conserved current. The thing that "flows" in the current is the "charge", the charge is the generator of the (local) symmetry group. This charge is sometimes called the Noether charge.

Thus, for example, the electric charge is the generator of the U(1) symmetry of electromagnetism. The conserved current is the electric current.

In the case of local, dynamical symmetries, associated with every charge is a gauge field; when quantized, the gauge field becomes a gauge boson. The charges of the theory "radiate" the gauge field. Thus, for example, the gauge field of electromagnetism is the electromagnetic field; and the gauge boson is the photon.

Sometimes, the word "charge" is used as a synonym for "generator" in referring to the generator of the symmetry. More precisely, when the symmetry group is a Lie group, then the charges are understood to correspond to the root system of the Lie group; the discreteness of the root system accounting for the quantization of the charge.

Examples

* Various charge quantum numbers have been introduced by theories of particle physics. These include the charges of the Standard Model:

* The color charge of quarks. The color charge generates the SU(3) color symmetry of quantum chromodynamics.

* The weak isospin quantum numbers of the electroweak interaction. It generates the SU(2) part of the electroweak SU(2) × U(1) symmetry.

* Weak isospin is a local symmetry, whose gauge bosons are the W and Z bosons.

* The electric charge for electromagnetic interactions.

Charges of approximate symmetries:

* The strong isospin charges. The symmetry groups is SU(2) flavor symmetry; the gauge bosons are the pions. The pions are not fundamental particles, and the symmetry is only approximate. It is a special case of flavor symmetry.

* Particle flavor charges, such as strangeness or charm. These generate the global SU(6) flavor symmetry of the fundamental particles; this symmetry is badly broken by the masses of the heavy quarks.

* Hypothetical charges of extensions to the Standard Model:

The magnetic charge, another charge in the theory of electromagnetism. Magnetic charges are not seen experimentally in laboratory experiments, but would be present for theories including magnetic monopoles.

A white dwarf, also called a degenerate dwarf, is a small star composed mostly of electron-degenerate matter. They are very dense; a white dwarf's mass is comparable to that of the Sun and its volume is comparable to that of the Earth. Its faint luminosity comes from the emission of stored thermal energy.

Electron degeneracy is a stellar application of the Pauli Exclusion Principle, as is neutron degeneracy. No two electrons can occupy identical states, even under the pressure of a collapsing star of several solar masses. For stellar masses less than about 1.44 solar masses, the energy from the gravitational collapse is not sufficient to produce the neutrons of a neutron star, so the collapse is halted by electron degeneracy to form white dwarfs. This maximum mass for a white dwarf is called the Chandrasekhar limit. As the star contracts, all the lowest electron energy levels are filled and the electrons are forced into higher and higher energy levels, filling the lowest unoccupied energy levels. This creates an effective pressure which prevents further gravitational collapse.

Sirius-B gives an example of the size of a white dwarf. Electron degeneracy halts the collapse of this star at the white dwarf stage. Though comparable in mass to the Sun, its white dwarf stage is smaller than the Earth.

http://en.wikipedia.org/wiki/Wireless_energy_transfer

If the number of people are 'n' then

number of handshakes = (n-1) + (n-2) + (n-3)+ ...+1

or number of handshakes = n(n-1)/2

here number of handshakes = 28

So, n(n-1) / 2 = 28 on solving gives

n = 7

In this circuit the vertices E and D (F and C) are equivalent, they are on the same potential. The resistance of the whole cube is not changed by merging these vertices into one. Let us merge the vertices E and D (F and C) into one junction, redraw the circuit into the plane and supplement each cube edge with a resistor. The resistance of each edge is R.

The plotting of the circuit in the plane

Let us consider each loop consisting of two resistors R in parallel connection as a single resistor whose resistance R₁ is:

$\frac{1}{R_1}\,=\,\frac{1}{R}+\frac{1}{R}$ $R_1\,=\,\frac{R}{2}$

Let us simplify the sketch of the circuit:

The expression for the resistance R₂ between the junctions (ED) and (CF) reads:

$\frac{1}{R_2}\,=\,\frac{1}{R_1}+\frac{1}{\left(R_1+R+R_1\right)}$ $\frac{1}{R_2}\,=\,\frac{1}{\frac{R}{2}}+\frac{1}{2\frac{R}{2}+R}\,=\,\frac{2}{R}+\frac{1}{2R}\,=\,\frac{5}{2R}$ $R_2\,=\,\frac{2}{5}R$

The expression for resistance of the whole cube reads:

$\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{1}{R_1+R_2+R_1}$

Inserting $R_1\,=\,\frac{R}{2}$ , $R_2\,=\,\frac{2}{5}R$ we obtain R_AB

$\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{1}{2\frac{R}{2}+\frac{2}{5}R}\,=\,\frac{1}{R}+\frac{1}{R+\frac{2}{5}R}\,=\,\frac{1}{R}+\frac{1}{\frac{7}{5}R}$ $\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{5}{7R}\,=\,\frac{12}{7R}$

Thus the total resistance of the cube between the vertices A and B is

$R_{AB}\,=\,\frac{7}{12}R$

In this circuit the vertices E and D (F and C) are equivalent, they are on the same potential. The resistance of the whole cube is not changed by merging these vertices into one. Let us merge the vertices E and D (F and C) into one junction, redraw the circuit into the plane and supplement each cube edge with a resistor. The resistance of each edge is R.

Let us consider each loop consisting of two resistors R in parallel connection as a single resistor whose resistance R₁ is:

$\frac{1}{R_1}\,=\,\frac{1}{R}+\frac{1}{R}$ $R_1\,=\,\frac{R}{2}$

Let us simplify the sketch of the circuit:

The expression for the resistance R₂ between the junctions (ED) and (CF) reads:

$\frac{1}{R_2}\,=\,\frac{1}{R_1}+\frac{1}{\left(R_1+R+R_1\right)}$ $\frac{1}{R_2}\,=\,\frac{1}{\frac{R}{2}}+\frac{1}{2\frac{R}{2}+R}\,=\,\frac{2}{R}+\frac{1}{2R}\,=\,\frac{5}{2R}$ $R_2\,=\,\frac{2}{5}R$

The expression for resistance of the whole cube reads:

$\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{1}{R_1+R_2+R_1}$

Inserting $R_1\,=\,\frac{R}{2}$ , $R_2\,=\,\frac{2}{5}R$ we obtain R_AB

$\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{1}{2\frac{R}{2}+\frac{2}{5}R}\,=\,\frac{1}{R}+\frac{1}{R+\frac{2}{5}R}\,=\,\frac{1}{R}+\frac{1}{\frac{7}{5}R}$ $\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{5}{7R}\,=\,\frac{12}{7R}$

Thus the total resistance of the cube between the vertices A and B is

$R_{AB}\,=\,\frac{7}{12}R$

In this circuit the vertices E and D (F and C) are equivalent, they are on the same potential. The resistance of the whole cube is not changed by merging these vertices into one. Let us merge the vertices E and D (F and C) into one junction, redraw the circuit into the plane and supplement each cube edge with a resistor. The resistance of each edge is R.

Let us consider each loop consisting of two resistors R in parallel connection as a single resistor whose resistance R₁ is:

$\frac{1}{R_1}\,=\,\frac{1}{R}+\frac{1}{R}$ $R_1\,=\,\frac{R}{2}$

Let us simplify the sketch of the circuit:

The expression for the resistance R₂ between the junctions (ED) and (CF) reads:

$\frac{1}{R_2}\,=\,\frac{1}{R_1}+\frac{1}{\left(R_1+R+R_1\right)}$ $\frac{1}{R_2}\,=\,\frac{1}{\frac{R}{2}}+\frac{1}{2\frac{R}{2}+R}\,=\,\frac{2}{R}+\frac{1}{2R}\,=\,\frac{5}{2R}$ $R_2\,=\,\frac{2}{5}R$

The expression for resistance of the whole cube reads:

$\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{1}{R_1+R_2+R_1}$

Inserting $R_1\,=\,\frac{R}{2}$ , $R_2\,=\,\frac{2}{5}R$ we obtain R_AB

$\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{1}{2\frac{R}{2}+\frac{2}{5}R}\,=\,\frac{1}{R}+\frac{1}{R+\frac{2}{5}R}\,=\,\frac{1}{R}+\frac{1}{\frac{7}{5}R}$ $\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{5}{7R}\,=\,\frac{12}{7R}$

Thus the total resistance of the cube between the vertices A and B is

$R_{AB}\,=\,\frac{7}{12}R$

In this circuit the vertices E and D (F and C) are equivalent, they are on the same potential. The resistance of the whole cube is not changed by merging these vertices into one. Let us merge the vertices E and D (F and C) into one junction, redraw the circuit into the plane and supplement each cube edge with a resistor. The resistance of each edge is R.

Let us consider each loop consisting of two resistors R in parallel connection as a single resistor whose resistance R₁ is:

$\frac{1}{R_1}\,=\,\frac{1}{R}+\frac{1}{R}$ $R_1\,=\,\frac{R}{2}$

Let us simplify the sketch of the circuit:

The expression for the resistance R₂ between the junctions (ED) and (CF) reads:

$\frac{1}{R_2}\,=\,\frac{1}{R_1}+\frac{1}{\left(R_1+R+R_1\right)}$ $\frac{1}{R_2}\,=\,\frac{1}{\frac{R}{2}}+\frac{1}{2\frac{R}{2}+R}\,=\,\frac{2}{R}+\frac{1}{2R}\,=\,\frac{5}{2R}$ $R_2\,=\,\frac{2}{5}R$

The expression for resistance of the whole cube reads:

$\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{1}{R_1+R_2+R_1}$

Inserting $R_1\,=\,\frac{R}{2}$ , $R_2\,=\,\frac{2}{5}R$ we obtain R_AB

$\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{1}{2\frac{R}{2}+\frac{2}{5}R}\,=\,\frac{1}{R}+\frac{1}{R+\frac{2}{5}R}\,=\,\frac{1}{R}+\frac{1}{\frac{7}{5}R}$ $\frac{1}{R_{AB}}\,=\,\frac{1}{R}+\frac{5}{7R}\,=\,\frac{12}{7R}$

Thus the total resistance of the cube between the vertices A and B is

$R_{AB}\,=\,\frac{7}{12}R$

Using the properties

ω³ = 1 and 1 + ω + ω² = 0

1 + ω = - ω² ...(1)

Therefore, (1 + ω)⁷ = A + Bω can be expressed as

( - ω² )⁷ = A + Bω (from 1)

or - ω¹⁴ = A + Bω

or - (ω³)⁴ω² = A + Bω

or - ω² = A + Bω

or 1+ ω = A + Bω (from 1)

Thus A = 1 and B = 1

Pressure is an intensive property.

Work and volume are both extensive properties; but pressure is intensive.

Basic Vector Operations

Both a magnitude and a direction must be specified for a vector quantity, in contrast to a scalar quantity which can be quantified with just a number. Any number of vector quantities of the same type (i.e., same units) can be combined by basic vector operations.

Graphical Vector Addition

Adding two vectors A and B graphically can be visualized like two successive walks, with the vector sum being the vector distance from the beginning to the end point. Representing the vectors by arrows drawn to scale, the beginning of vector B is placed at the end of vector A. The vector sum R can be drawn as the vector from the beginning to the end point.

The process can be donemathematically by finding thecomponents of A and B,combining to form the components of R, and then converting to polar form.

Derivations of Equations of Motion (Graphically)

First Equation of Motion

Graphical Derivation of First Equation

Consider an object moving with a uniform velocity u in a straight line. Let it be given a uniform acceleration a at time t = 0 when its initial velocity is u. As a result of the acceleration, its velocity increases to v (final velocity) in time t and S is the distance covered by the object in time t.

The figure shows the velocity-time graph of the motion of the object.

Slope of the v - t graph gives the acceleration of the moving object.

Thus, acceleration = slope = AB =

v - u = at

v = u + at I equation of motion

Graphical Derivation of Second Equation

Distance travelled S = area of the trapezium ABDO

= area of rectangle ACDO + area of DABC

(v = u + at I eqn of motion; v - u = at)

Graphical Derivation of Third Equation

S = area of the trapezium OABD.

Substituting the value of t in equation (1) we get,

2aS = (v + u) (v - u)

(v + u)(v - u) = 2aS [using the identity a² - b² = (a+b) (a-b)]

v² - u² = 2aS III Equation of Motion

http://www.reuk.co.uk/Smoothing-Capacitors.htm

Thermionic emission is the heat-induced flow of charge carriers from a surface or over a potential-energy barrier. This occurs because the thermal energy given to the carrier overcomes the binding potential, also known as work function of the metal. The charge carriers can be electrons or ions, and in older literature are sometimes referred to as "thermions". After emission, a charge will initially be left behind in the emitting region that is equal in magnitude and opposite in sign to the total charge emitted. But if the emitter is connected to a battery, then this charge left behind will be neutralized by charge supplied by the battery, as the emitted charge carriers move away from the emitter, and finally the emitter will be in the same state as it was before emission. The thermionic emission of electrons is also known as thermal electron emission.

The themionic emitter is not depleted by loss of electrons as electron is essentially a point particle and contributes not even fraction of a percent towards the contitution of atom and thus the depletion of material is essentially nil even if its all the electrons (hypothetical case) bound as well as free are emitted.

However,its depletion do takes place due to loss of matter owing to the vaporzation of external surface at elevated temperature over a long period of time.