Introduction and Examples

Yaar, m a new-comer to this site----the day i joined i ws extremely happy but later on i find at mny instances there is "cold war" goin on btwn the goiitians---------wich is definitely nt expected----------------cm on guys & gals we r nt here to quarell with each other---

Well, regarding our concerned topic, i mst say Christiano Ronaldo hs gvn a lot to dis site but at the same time i think we shud consider the upcoming goiitians who r working hard as well-------------so i want to live a msg 4 them

Encouragement

Is of constant need

Both in the inner

World of progress

And in the outer

World of success.

U all hve my best wishes & encouragement ..........................................

GOIIT ROCKXXXXXXXXXXXXXXXXXXXXXXX

srrrrrrrrrryyyyyyyyy ji late repli ke liye

Laser

In physics, a laser is a device that emits light through a specific mechanism for which the term laser is an acronym: light amplification by stimulated emission of radiation. This is a combined quantum-mechanical and thermodynamical process discussed in more detail below. As a light source, a laser can have various properties, depending on the purpose for which it is designed. A typical laser emits light in a narrow, low-divergence beam and with a well-defined wavelength (or color, when the laser is operating in the visible spectrum). This is in contrast to a light source such as the incandescent light bulb, which emits into a large solid angle and over a wide spectrum of wavelength. These properties can be summarized in the term coherence.

Physics behind it

A laser is composed of an active laser medium, or gain medium, and a resonant optical cavity. The gain medium transfers external energy into the laser beam. It is a material of controlled purity, size, concentration, and shape, which amplifies the beam by the quantum mechanical process of stimulated emission, predicted by Albert Einstein while he studied the photoelectric effect. The gain medium is energized, or pumped, by an external energy source. Examples of pump sources include electricity and light, for example from a flash lamp or from another laser. The pump energy is absorbed by the laser medium, placing some of its particles into high-energy ("excited") quantum states. Particles can interact with light both by absorbing photons or by emitting photons. Emission can be spontaneous or stimulated. In the latter case, the photon is emitted in the same direction as the light that is passing by. When the number of particles in one excited state exceeds the number of particles in some lower-energy state, population inversion is achieved and the amount of spontaneous emission due to light that passes through is larger than the amount of absorption. Hence, the light is amplified. Strictly speaking, these are the essential ingredients of a laser. However, usually the term laser is used for devices where the light that is amplified is produced as spontaneous emission from the same gain medium as where the amplification takes place. Devices where light from an external source is amplified are normally called optical amplifiers.

The light generated by stimulated emission is very similar to the input signal in terms of wavelength, phase, and polarization. This gives laser light its characteristic coherence, and allows it to maintain the uniform polarization and often monochromaticity established by the optical cavity design.

The optical cavity, a type of cavity resonator, contains a coherent beam of light between reflective surfaces so that the light passes through the gain medium more than once before it is emitted from the output aperture or lost to diffraction or absorption. As light circulates through the cavity, passing through the gain medium, if the gain (amplification) in the medium is stronger than the resonator losses, the power of the circulating light can rise exponentially. But each stimulated emission event returns a particle from its excited state to the ground state, reducing the capacity of the gain medium for further amplification. When this effect becomes strong, the gain is said to be saturated. The balance of pump power against gain saturation and cavity losses produces an equilibrium value of the laser power inside the cavity; this equilibrium determines the operating point of the laser. If the chosen pump power is too small, the gain is not sufficient to overcome the resonator losses, and the laser will emit only very small light powers. The minimum pump power needed to begin laser action is called the lasing threshold. The gain medium will amplify any photons passing through it, regardless of direction; but only the photons aligned with the cavity manage to pass more than once through the medium and so have significant amplification.

The beam in the cavity and the output beam of the laser, if they occur in free space rather than waveguides (as in an optical fiber laser), are often Gaussian beams. If the beam is not a pure Gaussian shape, the transverse modes of the beam can be described as a superposition of Hermite-Gaussian or Laguerre-Gaussian beams. The beam may be highly collimated, that is being parallel without diverging. However, a perfectly collimated beam cannot be created, due to diffraction. The beam remains collimated over a distance which varies with the square of the beam diameter, and eventually diverges at an angle which varies inversely with the beam diameter. Thus, a beam generated by a small laboratory laser such as a helium-neon laser spreads to about 1.6 kilometers (1 mile) diameter if shone from the Earth to the Moon. By comparison, the output of a typical semiconductor laser, due to its small diameter, diverges almost as soon as it leaves the aperture, at an angle of anything up to 50°. However, such a divergent beam can be transformed into a collimated beam by means of a lens. In contrast, the light from non-laser light sources cannot be collimated by optics as well or much.

The output of a laser may be a continuous constant-amplitude output (known as CW or continuous wave); or pulsed, by using the techniques of Q-switching, modelocking, or gain-switching. In pulsed operation, much higher peak powers can be achieved.

Some types of lasers, such as dye lasers and vibronic solid-state lasers can produce light over a broad range of wavelengths; this property makes them suitable for generating extremely short pulses of light, on the order of a few femtoseconds (10^-15 s).

Although the laser phenomenon was discovered with the help of quantum physics, it is not essentially more quantum mechanical than other light sources. The operation of a free electron laser can be explained without reference to quantum mechanics.

It is understood that the word light in the acronym Light Amplification by Stimulated Emission of Radiation is typically used in the expansive sense, as photons of any energy; it is not limited to photons in the visible spectrum. Hence there are infrared lasers, ultraviolet lasers, X-ray lasers, etc. For example, a source of atoms in a coherent state can be called an atom laser.

Uses

When lasers were invented in 1960, they were called "a solution looking for a problem". Since then, they have become ubiquitous, finding utility in thousands of highly varied applications in every section of modern society, including consumer electronics, information technology, science, medicine, industry, law enforcement, entertainment, and the military.

The first application of lasers visible in the daily lives of the general population was the supermarket barcode scanner, introduced in 1974. The laserdisc player, introduced in 1978, was the first successful consumer product to include a laser, but the compact disc player was the first laser-equipped device to become truly common in consumers' homes, beginning in 1982, followed shortly by laser printers.

Laser safety

Even the first laser was recognized as being potentially dangerous. Theodore Maiman characterized the first laser as one "Gillette"; as it could burn through one Gillette razor blade. Today, it is accepted that even low-power lasers with only a few milliwatts of output power can be hazardous to human eyesight.

At wavelengths which the cornea and the lens can focus well, the coherence and low divergence of laser light means that it can be focused by the eye into an extremely small spot on the retina, resulting in localized burning and permanent damage in seconds or even less time. Lasers are classified into safety classes numbered I (inherently safe) to IV (even scattered light can cause eye and/or skin damage). Laser products available for consumers, such as CD players and laser pointers are usually in class I, II, or III. Certain infrared lasers with wavelengths beyond about 1.4 micrometres are often referred to as being "eye-safe". This is because the intrinsic molecular vibrations of water molecules very strongly absorb light in this part of the spectrum, and thus a laser beam at these wavelengths is attenuated so completely as it passes through the eye's cornea that no light remains to be focused by the lens onto the retina. The label "eye-safe" can be misleading, however, as it only applies to relatively low power continuous wave beams and any high power or q-switched laser at these wavelengths can burn the cornea, causing severe eye damage.

The Optical Damage Threshold test station at NASA Langley Research Center has three lasers: a high-energy pulsed ND:Yag laser, a Ti:sapphire laser and an alignment HeNe laser.

Lord Lens

In analogy with optical lasers, a device which produces any particles or electromagnetic radiation in a coherent state is also called a "laser", usually with indication of type of particle as prefix (for example, atom laser.) In most cases, "laser" refers to a source of coherent light or other electromagnetic radiation.

Introduction and Examples

DEFINITION: A matrix is defined as an ordered rectangular array of numbers. They can be used to represent systems of linear equations, as will be explained below

Here are a couple of examples of different types of matrices:

Symmetric	Diagonal	Upper Triangular	Lower Triangular	Zero	Identity

And a fully expanded mxn matrix A, would look like this:

nxn matrix

or in a more compact form: simplified

Matrix Addition and Subtraction

DEFINITION: Two matrices A and B can be added or subtracted if and only if their dimensions are the same (i.e. both matrices have the identical amount of rows and columns. Take:

Addition

If A and B above are matrices of the same type then the sum is found by adding the corresponding elements a_ij+b_ij

Here is an example of adding A and B together

Subtraction
If A and B are matrices of the same type then the subtraction is found by subtracting the corresponding elements a_ij-b_ij

Here is an example of subtracting matrices

Matrix Multiplication

DEFINITION: When the number of columns of the first matrix is the same as the number of rows in the second matrix then matrix multiplication can be performed.

Here is an example of matrix multiplication for two 2x2 matrices

Here is an example of matrices multiplication for a 3x3 matrix

Now lets look at the nxn matrix case, Where A has dimensions mxn, B has dimensions nxp. Then the product of A and B is the matrix C, which has dimensions mxp. The ij^th element of matrix C is found by multiplying the entries of the i^th row of A with the corresponding entries in the j^th column of B and summing the n terms. The elements of C are:

Note: That AxB is not the same as BxA

Transpose of Matrices

DEFINITION: The transpose of a matrix is found by exchanging rows for columns i.e. Matrix A = (a_ij) and the transpose of A is:
A^T=(a_ij) Where i is the row number and j is the column number.

For example, The transpose of a matrix would be:

In the case of a square matrix (m=n), the transpose can be used to check if a matrix is symmetric. For a symmetric matrix A = A^T

The Determinant of a Matrix

DEFINITION: Determinants play an important role in finding the inverse of a matrix and also in solving systems of linear equations. In the following we assume we have a square matrix (m=n). The determinant of a matrix A will be denoted by det(A) or |A|. Firstly the determinant of a 2x2 and 3x3 matrix will be introduced then the nxn case will be shown.

Determinant of a 2x2 matrix

Assuming A is an arbitrary 2x2 matrix A, where the elements are given by:

then the determinant of a this matrix is as follows:

Determinant of a 3x3 matrix
The determinant of a 3x3 matrix is a little more tricky and is found as follows ( for this case assume A is an arbitrary 3x3 matrix A, where the elements are given below)

then the determinant of a this matrix is as follows:

Determinant of a nxn matrix

For the general case, where A is an nxn matrix the determinant is given by:

Where the coefficients are given by the relation

where is the determinant of the (n-1) x (n-1) matrix that is obtained by deleting row i and column j. This coefficient is also called the cofactor of a_ij.

The Inverse of a Matrix

DEFINITION: Assuming we have a square matrix A, which is non-singular ( i.e. det(A) does not equal zero ), then there exists an nxn matrix A^-1 which is called the inverse of A, such that this property holds:

AA^-1= A^-1A = I where I is the identity matrix.

The inverse of a 2x2 matrix
Take for example a arbitury 2x2 Matrix A whose determinant (ad-bc) is not equal to zero

Inverse of 2x2 matrix

where a,b,c,d are numbers, The inverse is:

The inverse of a nxn matrix
The inverse of a general nxn matrix A can be found by using the following equation:

Where the adj(A) denotes the adjoint (or adjugate) of a matrix. It can be calculated by the following method

Given the nxn matrix A, define

to be the matrix whose coefficients are found by taking the determinant of the (n-1) x (n-1) matrix obtained by deleting the i^th row and j^th column of A. The terms of B (i.e. B = b_ij) are known as the cofactors of A.
And define the matrix C, where
.
The transpose of C (i.e C^T) is called the adjoint of matrix A.

Lastly to find the inverse of A divide the matrix C^T by the determinant of A to give its inverse.

Solving Systems of Equations using Matrices

DEFINITION: A system of linear equations is a set of equations with n equations and n unknowns, is of the form of

The unknowns are denoted by x₁,x₂,...x_n and the coefficients (a's and b's above) are assumed to be given. In matrix form the system of equations above can be written as:

nxn Systems of equations

A simplified way of writing above is like this ; Ax = b

Inverse Matrix Method

DEFINITION: The inverse matrix method uses the inverse of a matrix to help solve a system of equations, such like the above Ax = b. By pre-multiplying both sides of this equation by A^-1 gives:

Ax=b derivation

or alternatively this gives

Ax=b derivation

So by calculating the inverse of the matrix and multiplying this by the vector b we can find the solution to the system of equations directly. And from earlier we found that the inverse is given by

Inverse

From the above it is clear that the existence of a solution depends on the value of the determinant of A. There are three cases:

If the det(A) does not equal zero then solutions exist using
If the det(A) is zero and b=0 then the solution will be not be unique or does not exist.
If the det(A) is zero and b=0 then the solution can be x = 0 but as in 2. is not unique or does not exist.

Looking at two equations we might have that

Inverse

Written in matrix form would look like

Inverse

and by rearranging we would get that the solution would look like

Inverse

Similarly for three simultaneous equations we would have:

Written in matrix form would look like

and by rearranging we would get that the solution would look like

Inverse

Cramer's Rule

DEFINITION: Cramer's rules uses a method of determinants to solve systems of equations. Starting with equation below,

nxn Systems of equations

The first term x₁ above can be found by replacing the first column of A by bxn

. Doing this we obtain:
nxn Systems of equations

Similarly for the general case for solving x_r we replace the r^th column of A by bxn

and expand the determinant.
This method of using determinants can be applied to solve systems of linear equations. We will illustrate this for solving two simultaneous equations in x and y and three equations with 3 unknowns x, y and z.

Two simultaneous equations in x and y

2x2 equations

To solve use the following:

and

or simplified:

and

Three simultaneous equations in x, y and z

ax+by+cz = p
dx+ey+fz = q
gx+hy+iz = r

To solve use the following:

,

and

cooooooooooooooooollllllllllllllllllllllllllllllllll

ammmmmmmmmmmmmmmmmmmmmmaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaazzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzziiiiiiiiiiiiiinnnnnnnnnnnnnnnnngggggggggggggg yaaaaaaaaarrrrrrrrrrr !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

frnz the chart in the appelet as mntioned here is nothing but the periodic table itself.

Evidence for the Structure of Atoms

How do we know what the structure of atoms is like?

Our current picture of the atom wasn't created all at once; it was built up and improved step by step based on experimental evidence and some extremely clever insights.

Some ancient Greek philosophers speculated that everything might be made of little chunks they called "atoms." The name comes from a Greek word meaning "uncuttable"; atoms were supposed to be unbreakable, the smallest possible units of anything.

In later centuries, scientists more or less left the atom idea on the shelf, until the early nineteenth century, when the chemist John Dalton came out with an updated atomic theory of his own.

This is the same John Dalton I mentioned in our discussion of atomic weight. As I said then, he and others had noticed that elements in chemical reactions combine in certain definite proportions; this, Dalton guessed, had to mean that the elements were made of tiny, unbreakable chunks that always stick together in the same ways--two hydrogen chunks plus an oxygen chunk always makes water, for example.

So Dalton's guess was right.

It was, and it wasn't. The scientific world soon accepted that atoms did exist...but were they really unbreakable?

Meanwhile, a lot of apparently unrelated experiments about electricity had been going on. By the nineteenth century, scientists knew a fair amount about how electricity behaved--but what was it, exactly? Some kind of fluid? Waves? A bunch of little particles?

In 1897, J. J. Thomson answered this question. He found that cathode rays were bent in certain directions by electric and magnetic fields, and therefore, he thought, must be made up of negatively charged particles of some sort. Later, those particles were named electrons.

Atomic Structure and Periodic Properties

After Mendeleev's time, scientists discovered what you already know: an atom consists of a positively charged nucleus, made of neutrons and protons, and some negatively charged electrons swarming around it.

But what exactly is the configuration of those electrons? That's the key to understanding why each element behaves the way it does.

"Configuration"? I'm not sure I understand what that means. Does it have something to do with that chart in the applet, the one that says "s p d" at the top?

Yes; that chart shows how the electrons are arranged in the selected element. I'd be happy to explain in detail how the electrons organize themselves; if you'd prefer, I can also give you a short crash course in interpreting the chart.

Now that we've talked about the structure of atoms, can you answer my question about their sizes?

There are two patterns to be explained: atoms get bigger as you go down a group, and smaller as you go to the right across a period. The reason for the first one shouldn't be so hard to see now; look again down the column of alkali metals in the applet.

Each time you move down, you add another primary level--lithium's highest electron is in a 2s state, for sodium it's 3s, and so on.

Exactly. And the higher an electron's energy, the farther from the nucleus it is.

So the atoms get bigger as you add electrons to higher energy levels--that makes sense. But why do they get smaller as you move to the right?

Well, you'll notice that within a period, the outermost electrons are all in the same primary level--that is, at (roughly) the same distance from the nucleus. But as you move to the right, the elements increase in atomic number; each element has one more proton than its left-hand neighbor. The more protons in the nucleus, the more strongly the valence electrons are pulled in...

...and so the atoms shrink! Also, I can see from the chart that the ionization energies get larger as you go to the right; that must be for the same reason.

Very good! Similarly, the ionization energies decrease as you move down a group.

"Everything that has a beginning, has an end"

In Matrix Revolutions, Agent Smith said that "Everything that has a beginning has an end.".And technically speaking, college has truely come to an end. But there may be certain things that won't end for certain people like friendship and love. And there may be things that didn't start but still sadly ended.

"Man is a gregarious animal" was the first line I learnt while learning Sociology in Class 9. This means that most people prefer being in groups and form social relationships rather than remaining alone.

3 years being together has provided a good platform for us. The only problem arises when things we get used to and usually take for granted, comes to an end. It's only then that we get to know the real extent of friendship and love.

But, we have to accept the fact. Most of us have different paths at this point of time. And we all have to move forward in life, but we should never leave our friends behind. Surely, life, time and commitments will change us in life, but we should never forget those 3 wonderful years in our lives. We should in fact learn to include in us those 3 years and not to classify them as "memories", keep them aside and remember them occasionally. That would be too unfair. It would be those same 3 years that will make us what we would be later. That would be too unfair to forget everything, and most importantly, everyone.

Bunking classes, getting kicked out of classes, trying to cheat in exams, laughing on good jokes, pretending to laugh on Tokas' jokes, Abhishek taking pictures of AM and AK, Anand's verbal fight with Dr GLA, Dr GLA himself, crushes, infatuations, true love, friendship, Sahil's unbiased sincere flirting towards every girl, Abhinav's cricket, football, dancing and wooing "skills", sleeping or hard rock mp3 music in AM's class, hanging out in LP, Purnima's flirtatious adventures with Dhan Singh, Amit's ajeeb aur gareeb harkatein, Rohit's special fascination for my jeans and me and Charanya, dosa and idli sambhar, college festivals, Euphoria, 10 Feb, winning cricket matches, a few matches of football, dance parties, Abhishek's transformation to being called the "Seductive Ape", bike rides, good days, bad days, some unforgettable days, unforgettable moments, unforgettable words, unforgettable persons, missed opportunities, regrets, late night calls, words of comfort, getting close with people, gazing at girls during classes, enjoying life with friends, sometimes just showing a fake smile, being a support for others, being a support for oneself, tears of sadness, tears of joy, tears for love, making fun of others, others making fun of us, gradually understanding life, growing mature, fun trips, fun floor, fun and food trip, Chowkidhani trip, Vaibhav's blog, my first one week girlfriend, my first one-day agreement valentine date, broken hearts, made couples, my waterpark story, SRK, Samriti's craze for SRK, Tokas' achievement (or rather the vodka shots) in Chowkidhani, swimming practices, my missed father-in-law Mr M.P Sharma, water polo matches, arguments, small fights, ke baat hogi??, Prateek's devotion to cigarettes and booze, our booze party and all that followed, sundays at my place, omelettes and tea at India Gate, India Gate, Sahil's maggi and cooking skills, my fish curry, metro trips with Sonali, Raghav's lost love, Nonu's acting and coffee and momos, volleyball matches, Ankit's unimitable hairstyle, months of hardwork and toil for exams, Priyanka's practical jokes, PVR Priya (my second home), hard times when I missed home, trips to Anand's place and cricket matches, Charanya's so much talked about lungi, entertaining sms-chatting, non enternaining sms-chatting with Vaibhav, my childhood friend cum bhabhi Shikha, inspired from Rani in "Black" is khamosh Kanika aunty, masti, mazaak, mahabharat play and my mauritian friend Dhavishkar aka George who helped us and everything all which I forgot. How can I forget all these. These are those precious moments that will always be etched in my heart forever.

Truely, it was a nice experience to know you people of a different culture, though of a same religion. Interacting with you all made me discover a whole new world. This is a hell of an opportunity no one should ever miss in life. Gradually, I had reached a point where I was no more discovering, but unknowingly became a a part of your ( I mean 'our' ) own world. Except for a few mispronounciations (I pronounciate BC and MC better now, kyun Tokas guru??), I guess it becomes difficult to distinguish me from a Delhiite, especially when I use my jat accent (Ke baat hogi, BC??) and punjabi accent. And more then an experience, I got to know all you people and many of you are really good friends of mine. But, I believe that destiny had it planned that we had to part ways one day. No probs, I'll be waiting for you guys in Mauritius.

I have never said it, but you all mean loads to me. And those who mean mush to me, know it and it's no use telling them. They all understand my unspoken words. Maybe that's what we mean by friendship.........

(Not self-written but found it

interesting)

ya, hve gvn u a salute !!!!!!!!!!!!!!!!

Electrostatic Potential Energy

Consider the following situation:

a) Particle with mass m in gravitational field

b) Charged particle with charge q in an electric field

Both gravitational and electrical forces are conservative. The work done by a conservative force: can be expressed as difference between initial and final values of potential energy, it is reversible, it is independent of the path taken, for a path around a closed loop, the work done is zero.

In each case the particle accelerates uniformly: it gains kinetic energy K, as it looses potential energy U.

Work done on the body by the field = increase in kinetic energy

W = - DU = DK

This relationship can be used to define the difference in potential energy between two points. To define the absolute value of U, we need to choose a zero point. We choose the potential energy to be zero when the charge q is infinitely far from any other charges.

Example: A charge q = 1.0 C moves distance s of 1.5 m in the direction of a uniform electric field E of magnitude 2.0 N/C. Find its change in electrostatic potential energy.

Work W = -DU = F.s = (qE).s = qEs (as s and E are parallel and cos 0^o is 1)

DU = -qEs = -(1.0 C)(2.0N/C)(1.5 m) = -3.0 Nm = -3.0 Joule (units of work and energy)

Electrostatic potential energy decreased by 3.0 J

Potential energy of a pair of point charges

Fix charge q₁ at the origin and move charge q₂ from distance A to distance B

What is the change in potential energy of the system?

Work done in this case is more difficult to calculate, as the force between the charges depends on their distance. We have to calculate W at each point and add all together: integration.

(Note that we have renamed distance s as r, since q₂ moves radially from charge q₁)

We can show mathematically (from the rules of the scalar product), that any non-radial displacement dl is equivalent to the radial component dr (= dl cos f).

Now take B as infinity, where U = 0 and A as r:

The potential energy of two point charges q₁ and q₂ at a distance r apart.

As you would expect for a scalar product: potential energy is a scalar.

The potential energy decreases as (1/r)

Compare with electric field (1/r²)

Potential energy for a group of charges

For a charge q_o in the neighbourhood of several point charges q₁, q₂ ?

Work done is additive, potential energy is additive.

If we add up the energy of all the pairs of charges, we get the total potential energy

The sum extends over all the pairs of charges, but do not let i = j, as there is no self-interaction. Also, need to count each charge pair only once.

Electric Potential

A test charge q in an electric field experiences force proportional to q, therefore potential energy is proportional to q.

Define "electric potential" V as the potential energy per unit charge.

V = U/q

This is independent of the test charge.

Change in potential between two points

From the definition unit is Joule/Coulomb = 1Volt (V)

The electron-volt

On the microscopic scale, we define a different unit of energy:

One "electron-volt" is the energy gained by electron in falling through a potential difference of 1 V. Denoted 1 eV

W = -qDV = - (-e)(1 V) = 1.6 x 10^-19 Joule = 1 eV

Potential due to a point charge

For test charge q near a point charge Q, potential

Several point charges

Similarly, for several point charges:

Potentials are scalars and simply add.

Distribution of Charge

For a continuous distribution of charge, the sum becomes an integral:

Treat each infinitesimal element of charge as a point charge:

Potential at point P:

Equipotential surfaces

A set of points with same potential forms equipotential surface.

For a point charge, equipotentials are spheres at fixed radius r

Along AB, W = -qDV = zero!

Diels-Alder Reaction

The [4+2]-cycloaddition of a conjugated diene and a dienophile (an alkene or alkyne), an electrocyclic reaction that involves the 4 ?-electrons of the diene and 2 ?-electrons of the dienophile. The driving force of the reaction is the formation of new ?-bonds, which are energetically more stable than the ?-bonds.

In the case of an alkynyl dienophile, the initial adduct can still react as a dienophile if not too sterically hindered. In addition, either the diene or the dienophile can be substituted with cumulated double bonds, such as substituted allenes.

With its broad scope and simplicity of operation, the Diels-Alder is the most powerful synthetic method for unsaturated six-membered rings.

A variant is the hetero-Diels-Alder, in which either the diene or the dienophile contains a heteroatom, most often nitrogen or oxygen. This alternative constitutes a powerful synthesis of six-membered ring heterocycles.

Mechanism

Overlap of the molecular orbitals (MOs) is required:

Overlap between the highest occupied MO of the diene (HOMO) and the lowest unoccupied MO of the dienophile (LUMO) is thermally allowed in the Diels Alder Reaction, provided the orbitals are of similar energy. The reaction is facilitated by electron-withdrawing groups on the dienophile, since this will lower the energy of the LUMO. Good dienophiles often bear one or two of the following substituents: CHO, COR, COOR, CN, C=C, Ph, or halogen. The diene component should be as electron-rich as possible.

There are ?inverse demand? Diels Alder Reactions that involve the overlap of the HOMO of the dienophile with the unoccupied MO of the diene. This alternative scenario for the reaction is favored by electron-donating groups on the dienophile and an electron-poor diene.

The reaction is diastereoselective.

Cyclic dienes give stereoisomeric products. The endo product is usually favored by kinetic control due to secondary orbital interactions.

Recent Applications

Knoevenagel Condensation
Doebner Modification

The condensation of carbon acid compounds with aldehydes to afford unsaturated compounds.

The Doebner Modification, which is possible in the presence of carboxylic acid groups, includes a pyridine-induced decarboxylation.

Mechanism

An enol intermediate is formed initially:

This enol reacts with the aldehyde, and the resulting aldol undergoes subsequent base-induced elimination:

A reasonable variation of the mechanism, in which piperidine acts as organocatalyst, involves the corresponding iminium intermediate as the acceptor:

The Doebner-Modification in refluxing pyridine effects concerted decarboxylation and elimination:

Scope

A Knoevenagel condensation is demonstrated in the reaction of 2-methoxybenzaldehyde 1 with the barbituric acid 2 in ethanol using

Introduction and Examples

Matrix Addition and Subtraction

Solving Systems of Equations using Matrices

Inverse Matrix Method

Cramer's Rule

Evidence for the Structure of Atoms

"Everything that has a beginning, has an end"

Diels-Alder Reaction

Knoevenagel CondensationDoebner Modification

Knoevenagel Condensation
Doebner Modification