Introduction
Ørsted had discovered that electricity and magnetism were linked, electric current gave rise to magnetic fields. However no one had succeded in generating electricity by using magnetic fields, until Michael Faraday found that moving a conductor in a magnetic field (or by moving the magnet field near a stationary conductor) created a voltage. The wire must be part of an electrical circuit. Otherwise the electrons have no place to go. In other words, there is no electrical current produced with a wire with open ends. But if the ends are attached to a light bulb, to an electrical meter or even to each other, the circuit is complete and electrical current is created.
Figure 1. Inducing a current in a wire by moving the wire in a magnetic field.
Direction of Current
The direction of the current is determined by Flemming's Right hand rule. The left-hand rule is used for motors and motion produced by a magnetic field. The right-hand rule is used for generators and current generated by a motion. Using the right-hand, the thumb is in the direction of the motion, the first finger points in the direction of the field and the second finger points in the direction of the current.
Flux and Flux Linkage
To create electricity all that was required was a coil of wire, ends of which may be connected to a voltmeter. The voltage created depends on the density of the magnetic field and the area of the loop cutting the magnetic field lines.
A quantity called the flux measures this and is give by &phi = BA where B is the magnetic flux density and A is the area of the coil in the magnetic field.
If there are more turns in the coil then the flux is termed the magnetic flux linkage. It is given by Nφ =BAN. This assumes that the loop cuts the magnetic field lines at an angle of 90°. If the loop cuts the magnetic field lines at a different angle say, θ then the flux linkage is defined as N&phi = BANcos θ where theta is the angle by the normal to the area and the magnetic field lines as shown in Figure 1.
Faraday's Law of Induction
We said that a voltage or Electro-Motive Force (EMF), is produced when the loop is moved in the magnetic field but more qualitatively, the voltage is produced is in responce to the change in the motion. The voltage produced depends on the rate of change of flux-linkage with time. In mathematical terms,
where E is the EMF. The other symbols have their usual meanings. The minus sign is a consequence of Lenz's Law which we shall discuss in the following section.
Lenz's Law
When we move a conductor in a magnetic field the current generated creates it's own magnetic field. If the magnetic field created had an additive effect to the original magnetic field then the magnetic field would become even stronger and this would create an even stronger current which would create an even strong magnetic field, and so on. If this were to happen we could get energy for free although the universe might explode. Unfortunately we cannot make free energy the reason is down to the Lenz's law. When a current is generated, the magnetic field produced by the current is in opposition to the original magnetic field. This produces a force opposes the motion of the conductor and brings it to a halt. This is why it becomes more difficult to turn a dynamo on a bicycle as you increase in speed. We express Lenz's law in as part of Faraday's Law by inserting a minus sign.
Tuesday, August 31, 2010
Equilibrium
A body which is in equilibrium is either moving at constant velocity in a straight line, or it is not moving. If it is not moving, it said to be in static equilibrium. The reason why the body does not move is because the forces acting on it cancel each other out. In this simple phrase we are expressing the two conditions necessary in order for a body to be in equilibrium:
The sum of rotational forces or moments must add to zero.
The vector sum of all external forces, is zero.
In mathematical terms, ΣFrot = 0 and ΣF = 0
Proving Equilibrium
To prove that a body is in equilibrium, we can follow a set proceedure.
Draw the free-body diagram, which shows the forces acting on the object.
Resolve the forces in any two conveinient directions, for example, ΣFx= 0 and ΣFy = 0, which will result in two equations from which the two unknowns can be found.
Theorems on Equilibrium
If three forces are in equilibrium, then the lines of force pass through a single point.
Equilibrium in three Dimensions
So far we have discussed equilibrium where the forces are coplanar. In three dimensions we need to ensure that the sum of the moments in the three independent directions are zero the sum of the vector forces also must be zero.
Σ Fx,y,z = 0
Στx,y,z = 0
The sum of rotational forces or moments must add to zero.
The vector sum of all external forces, is zero.
In mathematical terms, ΣFrot = 0 and ΣF = 0
Proving Equilibrium
To prove that a body is in equilibrium, we can follow a set proceedure.
Draw the free-body diagram, which shows the forces acting on the object.
Resolve the forces in any two conveinient directions, for example, ΣFx= 0 and ΣFy = 0, which will result in two equations from which the two unknowns can be found.
Theorems on Equilibrium
If three forces are in equilibrium, then the lines of force pass through a single point.
Equilibrium in three Dimensions
So far we have discussed equilibrium where the forces are coplanar. In three dimensions we need to ensure that the sum of the moments in the three independent directions are zero the sum of the vector forces also must be zero.
Σ Fx,y,z = 0
Στx,y,z = 0
Useful Mathematics
The Binomial Theorem
(1 + x)n = 1 + nx + [n(n-1) x2]/(2!) + [n(n-1)(n-2) x3]/(3!) + ...
If x << 1, then
(1 + x)n ≅ 1 + n x
(1 + x)-n ≅ 1 - n x
These approximations are useful when x2 is negliable.
Quadratic Equations
ax2 + bx + c = 0 has the solution,
x ={[-b ± (b2 - 4ac)]1/2} / (2a)
Trigonometry
π rad = 180 °
1 rad = 57.3 °
The quadrants in which trigonometrical functions are positive. Is shown below:
Figure 1. Signs of trigonometric functions.
A good way to remember this is the phrase clockwise ACTS. Clockwise gives the direction from the first quadrant is clockwise and each letter from the word ACTS stands for a trigonometric function: All, Cos, Tan and Sin. The direction of the angle increases in an anti-clockwise sense.
If A and B are angles then
tan A = sin A/cos A
sin2 A + cos2 A = 1
sec2A = 1 + tan2 A
cosec2 A = 1 + cot2 A
sin (A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B -/+; sin A sin B
tan (A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
If t= tan (1/2) A, sin A = (2t) / (1 + t2), cos A = (1 - t2) / (1 + t2)
2 sin A cos B = sin (A + B) + sin (A - B)
2 cos A cos B = cos (A + B) + cos (A - B)
2 sin A sin B = cos (A - B) - cos (A + B)
sin A + sin B = 2 sin [(A + B)/2] cos [(A - B)/2]
sin A - sin B = 2 cos [(A + B)/2] sin [(A - B)/2]
cos A + cos B = 2 cos [(A + B)/2] cos [(A - B)/2]
cos A - cos B = 2 sin [(A + B)/2] sin [(A - B)/2]
Power Series
for all x
ex = exp x = 1 + x + x2/(2!) + ... + xr/(r!) + ... for all x
(-1 < x <et; 1)
ln (1 + x) = x - x2/ 2 + x3/3 - ... + (-1)r+1xr/r + ... (-1 < x <et; 1)
for all x
cos x = (eix + e-ix)/2 = 1 - x2/(2!) + x4/(4!) - ... + (-1)rx2r/(2r)! + ... for all x
for all x
sin x = (eix - e-ix)/(2i) = x - x3/(3!) + x5/(5!) - ... + (-1)rx2r+1/(2r + 1)! + ... for all x
for all x
cosh x = (ex + e-x)/2 = 1 + x2/(2!) + x4/(4!) + ... + x2r/(2r)! + ... for all x
for all x
(1 + x)n = 1 + nx + [n(n-1) x2]/(2!) + [n(n-1)(n-2) x3]/(3!) + ...
If x << 1, then
(1 + x)n ≅ 1 + n x
(1 + x)-n ≅ 1 - n x
These approximations are useful when x2 is negliable.
Quadratic Equations
ax2 + bx + c = 0 has the solution,
x ={[-b ± (b2 - 4ac)]1/2} / (2a)
Trigonometry
π rad = 180 °
1 rad = 57.3 °
The quadrants in which trigonometrical functions are positive. Is shown below:
Figure 1. Signs of trigonometric functions.
A good way to remember this is the phrase clockwise ACTS. Clockwise gives the direction from the first quadrant is clockwise and each letter from the word ACTS stands for a trigonometric function: All, Cos, Tan and Sin. The direction of the angle increases in an anti-clockwise sense.
If A and B are angles then
tan A = sin A/cos A
sin2 A + cos2 A = 1
sec2A = 1 + tan2 A
cosec2 A = 1 + cot2 A
sin (A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B -/+; sin A sin B
tan (A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
If t= tan (1/2) A, sin A = (2t) / (1 + t2), cos A = (1 - t2) / (1 + t2)
2 sin A cos B = sin (A + B) + sin (A - B)
2 cos A cos B = cos (A + B) + cos (A - B)
2 sin A sin B = cos (A - B) - cos (A + B)
sin A + sin B = 2 sin [(A + B)/2] cos [(A - B)/2]
sin A - sin B = 2 cos [(A + B)/2] sin [(A - B)/2]
cos A + cos B = 2 cos [(A + B)/2] cos [(A - B)/2]
cos A - cos B = 2 sin [(A + B)/2] sin [(A - B)/2]
Power Series
for all x
ex = exp x = 1 + x + x2/(2!) + ... + xr/(r!) + ... for all x
(-1 < x <et; 1)
ln (1 + x) = x - x2/ 2 + x3/3 - ... + (-1)r+1xr/r + ... (-1 < x <et; 1)
for all x
cos x = (eix + e-ix)/2 = 1 - x2/(2!) + x4/(4!) - ... + (-1)rx2r/(2r)! + ... for all x
for all x
sin x = (eix - e-ix)/(2i) = x - x3/(3!) + x5/(5!) - ... + (-1)rx2r+1/(2r + 1)! + ... for all x
for all x
cosh x = (ex + e-x)/2 = 1 + x2/(2!) + x4/(4!) + ... + x2r/(2r)! + ... for all x
for all x
A Brief History of Cosmological Ideas
Aristotle
The Greek philospher Aristotle proposed that the heavens were literally composed of 55 concentric, crystalline spheres to which the celestial objects were attached and which rotated at different velocities (but the angular velocity was constant for a given sphere), with the Earth at the center.
The Ptolemaic System
The prevailing theory in Europe as Copernicus was writing was that created by Ptolemy in his Almagest, dating from about 150 A.D. The Ptolemaic system drew on many previous theories that viewed Earth as a stationary center of the universe. Stars were embedded in a large outer sphere which rotated relatively rapidly, while the planets inhabited smaller spheres as . The idea that the Earth was at the centre of the universe with everything revolving around it was one that fitted with religious beleifs. Afterall, man is the most important of God's creations and so it was proper that the Earth should be at the center of a perfect and uniform universe.
Retrograde motion of Mars
The Ptolemaic model of the universe, had the Earth at the center of the universe with the Sun and the planets travelling around in circular orbits. To accurately describe the observed data, the planets travelled in smaller orbits known as epicycles as they orbited around the Earth. This reproduced the motion of the planets as the travelled across the sky. In particular, the phenomena of retrograde motion, in which the planet sometimes appears to travel backwards or loop the loop. The illustration shows the orbit of Mars over a period of several month.
Nicolaus Copernicus (1473-1543)
In 1543, Copernicus formulated another model of the universe in which the Earth went around the Sun, the Heliocentric Model of the Universe. The publication of his book, De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres) is often taken to be the beginning of the Scientific Revolution.
The orbit of the planets was still circular but in his model the problem of retrograde motion was incoporated naturally, as shown in Figure 1. The orbit of the Earth and Mars is shown. Since Mars is further from the Sun it takes longer to travel in its orbit than the Earth. Therefore, there will be ocassions when the Earth overtakes Mars in its orbit. The line of sight is shown from the Earth to Mar is shown at different intervals. As viewed from the Earth, does Mars appears to go backward in its orbit as the Earth overtakes Mars but it is no more unusual than seeing a cyclist go backwards past your side window when you are in a car travelling in the same direction at a greater speed.
Figure 1. Retrograde motion of Mars in the Copernican model.
Tycho Brahe (1546-1601)
Tycho Brahe observed the planets and made accurate observation of the stars.
Johannes Kepler (1571-1630)
Originally, Kepler had planned to be ordained as a Lutheran minister. He saw it as his Christian duty to understand the universe in terms of mathematical rules and thus understand the works of God. He first attempt at explaining the cosmos was in the work, Mysterium Cosmographicum (Mystery of the Sacred Cosmos). He devised a model of the solar system in which the orbits of the planets fitted inside spheres with radii that could accomodate each of the Platonic solids. This idea is wrong.
Kepler's model of the Solar System had the planets orbit in spheres of the same radius that could accomodate each of the Platonic solids.
In 1601 he was hired as Tycho Brahe's assistant. As a mathematician it was his job to make sense of Brahe's extremely accurate observational data for the orbit for Mars. Kepler described the effort as his "War with Mars" but it resulted in the three laws that bare his name.
Keplers First Law - the planets do not move in circles but rather in elliptic orbit, with the Sun at one focus.
Keplers Second Law - the radius vector sweeps equal areas in equal times
Keplers Third Law - the time for the period square is proportional to the cube of their average distance from the Sun cubed. T2 ∝ R3
Galileo Galilei (1564-1642)
Galileo's drawing of the Moon.
The phases of Venus.
Galileo's notebook on the Jupiter
Galileo supported the Copernican model of the universe and had used the newly invented telescope to discover evidence to support his belief. The telescope allowed Galileo to see that there were mountains on the moon, which went against the accepted religous view that the universe was perfect. Galileo also discovered spots on the Sun. People really tried hard to account for these observations without making the heavens imperfect; one suggestion was that over the mountains of the Moon there was a layer of clear crystal so the final surface would be smooth and perfect!
One observation definitely disproved the Ptolemaic model, although it didn't prove that Copernicus was right (as Tycho Brahe pointed out). This was the observation that Venus has phases, much like our Moon does. To the naked eye, Venus always appears as a bright dot in the sky. With a telescope, however, it is fairly easy to see the phases of Venus. Just as the Moon has phases, Venus too has phases based on the planet’s position relative to us and the Sun. There was no way for the Ptolemaic model (Earth centered solar system) to account for these phases. They can only occur as Galileo saw them if Venus is circling the Sun, not the Earth.
Galileo saw near Jupiter what he first thought to be stars. When he realized that the stars were actually going around Jupiter, it negated a major argument of the Ptolemaic model. Not only did this mean that the Earth could not be the only center of motion, but also it knocked a hole in another argument. The supporters of the Ptolemaic model argued that if the Earth were moving through space, the Moon would be left behind. Galileo’s observations showed that the moons of Jupiter were not being left behind as Jupiter moved.
Galileo's advocasy of the heliocentric model of the universe brought him into conflict with the Church. In 1616, the theologians of the Holy Office decleared Copernicanism, 'false and erronious' and the Pope admonished Galileo for not defending its doctrines.
Galileo was asked to published a book which was supposed to support the Geocentric view, however when Dialogo Sopra I Due Massimi Sistemi Del Mondo (Dialogues on the Two Chief Systems of the World) was published, it was an outright argument for Copernican view. The book was an imaginary conversation between three people. The Geocentric position was argued for by a doggmatic, arogant character named Simplicio. The Copernican view was supported by intelligent and wise character named, Salvanti, representing Galileo. A neutral character who was receptive to either position was also written.
The Church banned the book and ordered for Galileo to appear before the Inquisition for herecy. Threatened with torture, Galileo confessed that he was wrong. By this time, Galileo was an old man of 68 years. A death sentence would certainly not have been an unusual punishment for herecy, however Galileo was lucky and was sentenced to life imprisonment, which was latter commuted to being held under house arrest at his home outside Florence. He died in 1642.
Some 359 years after Galileo death, the Vatican cleared Galileo of any wrongdoing. Pope John Paul II said,
Thanks to his intuition as a brilliant physicist and by relying on different arguments, Galileo, who practically invented the experimental method, understood why only the sun could function as the centre of the world, as it was then known, that is to say, as a planetary system. The error of the theologians of the time, when they maintained the centrality of the Earth, was to think that our understanding of the physical world's structure was, in some way, imposed by the literal sense of Sacred Scripture
Which of course it isn't.
Isaac Newton (1642-1727)
It was Sir Isaac Newton who was able to show that Kepler's laws of planetary motion are a natural consequence of simpler and more general descriptions of motion in nature. This brought into one theory both our observations of how things move on Earth and how the planets move in the heavens. These motions are described formally as Newton's laws of motion and gravity.
Newton applied this idea to the Sun and planets and took Keplers laws and calculated that the force falls off with the square of the distance from the Sun.
Newton's law of universal gravitation states that there is a force acting between objects that pulls them together. This force is proportional to the mass of the objects and inversely proportional to the square of their distance apart.
F = (GMmr^)/r2
Newton looked at the motion of the moon around the Sun and reasoned that the force responsible for gravity on Earth might be responsible for keeping the moon in orbit around the Earth. It turned out to be so
The Greek philospher Aristotle proposed that the heavens were literally composed of 55 concentric, crystalline spheres to which the celestial objects were attached and which rotated at different velocities (but the angular velocity was constant for a given sphere), with the Earth at the center.
The Ptolemaic System
The prevailing theory in Europe as Copernicus was writing was that created by Ptolemy in his Almagest, dating from about 150 A.D. The Ptolemaic system drew on many previous theories that viewed Earth as a stationary center of the universe. Stars were embedded in a large outer sphere which rotated relatively rapidly, while the planets inhabited smaller spheres as . The idea that the Earth was at the centre of the universe with everything revolving around it was one that fitted with religious beleifs. Afterall, man is the most important of God's creations and so it was proper that the Earth should be at the center of a perfect and uniform universe.
Retrograde motion of Mars
The Ptolemaic model of the universe, had the Earth at the center of the universe with the Sun and the planets travelling around in circular orbits. To accurately describe the observed data, the planets travelled in smaller orbits known as epicycles as they orbited around the Earth. This reproduced the motion of the planets as the travelled across the sky. In particular, the phenomena of retrograde motion, in which the planet sometimes appears to travel backwards or loop the loop. The illustration shows the orbit of Mars over a period of several month.
Nicolaus Copernicus (1473-1543)
In 1543, Copernicus formulated another model of the universe in which the Earth went around the Sun, the Heliocentric Model of the Universe. The publication of his book, De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres) is often taken to be the beginning of the Scientific Revolution.
The orbit of the planets was still circular but in his model the problem of retrograde motion was incoporated naturally, as shown in Figure 1. The orbit of the Earth and Mars is shown. Since Mars is further from the Sun it takes longer to travel in its orbit than the Earth. Therefore, there will be ocassions when the Earth overtakes Mars in its orbit. The line of sight is shown from the Earth to Mar is shown at different intervals. As viewed from the Earth, does Mars appears to go backward in its orbit as the Earth overtakes Mars but it is no more unusual than seeing a cyclist go backwards past your side window when you are in a car travelling in the same direction at a greater speed.
Figure 1. Retrograde motion of Mars in the Copernican model.
Tycho Brahe (1546-1601)
Tycho Brahe observed the planets and made accurate observation of the stars.
Johannes Kepler (1571-1630)
Originally, Kepler had planned to be ordained as a Lutheran minister. He saw it as his Christian duty to understand the universe in terms of mathematical rules and thus understand the works of God. He first attempt at explaining the cosmos was in the work, Mysterium Cosmographicum (Mystery of the Sacred Cosmos). He devised a model of the solar system in which the orbits of the planets fitted inside spheres with radii that could accomodate each of the Platonic solids. This idea is wrong.
Kepler's model of the Solar System had the planets orbit in spheres of the same radius that could accomodate each of the Platonic solids.
In 1601 he was hired as Tycho Brahe's assistant. As a mathematician it was his job to make sense of Brahe's extremely accurate observational data for the orbit for Mars. Kepler described the effort as his "War with Mars" but it resulted in the three laws that bare his name.
Keplers First Law - the planets do not move in circles but rather in elliptic orbit, with the Sun at one focus.
Keplers Second Law - the radius vector sweeps equal areas in equal times
Keplers Third Law - the time for the period square is proportional to the cube of their average distance from the Sun cubed. T2 ∝ R3
Galileo Galilei (1564-1642)
Galileo's drawing of the Moon.
The phases of Venus.
Galileo's notebook on the Jupiter
Galileo supported the Copernican model of the universe and had used the newly invented telescope to discover evidence to support his belief. The telescope allowed Galileo to see that there were mountains on the moon, which went against the accepted religous view that the universe was perfect. Galileo also discovered spots on the Sun. People really tried hard to account for these observations without making the heavens imperfect; one suggestion was that over the mountains of the Moon there was a layer of clear crystal so the final surface would be smooth and perfect!
One observation definitely disproved the Ptolemaic model, although it didn't prove that Copernicus was right (as Tycho Brahe pointed out). This was the observation that Venus has phases, much like our Moon does. To the naked eye, Venus always appears as a bright dot in the sky. With a telescope, however, it is fairly easy to see the phases of Venus. Just as the Moon has phases, Venus too has phases based on the planet’s position relative to us and the Sun. There was no way for the Ptolemaic model (Earth centered solar system) to account for these phases. They can only occur as Galileo saw them if Venus is circling the Sun, not the Earth.
Galileo saw near Jupiter what he first thought to be stars. When he realized that the stars were actually going around Jupiter, it negated a major argument of the Ptolemaic model. Not only did this mean that the Earth could not be the only center of motion, but also it knocked a hole in another argument. The supporters of the Ptolemaic model argued that if the Earth were moving through space, the Moon would be left behind. Galileo’s observations showed that the moons of Jupiter were not being left behind as Jupiter moved.
Galileo's advocasy of the heliocentric model of the universe brought him into conflict with the Church. In 1616, the theologians of the Holy Office decleared Copernicanism, 'false and erronious' and the Pope admonished Galileo for not defending its doctrines.
Galileo was asked to published a book which was supposed to support the Geocentric view, however when Dialogo Sopra I Due Massimi Sistemi Del Mondo (Dialogues on the Two Chief Systems of the World) was published, it was an outright argument for Copernican view. The book was an imaginary conversation between three people. The Geocentric position was argued for by a doggmatic, arogant character named Simplicio. The Copernican view was supported by intelligent and wise character named, Salvanti, representing Galileo. A neutral character who was receptive to either position was also written.
The Church banned the book and ordered for Galileo to appear before the Inquisition for herecy. Threatened with torture, Galileo confessed that he was wrong. By this time, Galileo was an old man of 68 years. A death sentence would certainly not have been an unusual punishment for herecy, however Galileo was lucky and was sentenced to life imprisonment, which was latter commuted to being held under house arrest at his home outside Florence. He died in 1642.
Some 359 years after Galileo death, the Vatican cleared Galileo of any wrongdoing. Pope John Paul II said,
Thanks to his intuition as a brilliant physicist and by relying on different arguments, Galileo, who practically invented the experimental method, understood why only the sun could function as the centre of the world, as it was then known, that is to say, as a planetary system. The error of the theologians of the time, when they maintained the centrality of the Earth, was to think that our understanding of the physical world's structure was, in some way, imposed by the literal sense of Sacred Scripture
Which of course it isn't.
Isaac Newton (1642-1727)
It was Sir Isaac Newton who was able to show that Kepler's laws of planetary motion are a natural consequence of simpler and more general descriptions of motion in nature. This brought into one theory both our observations of how things move on Earth and how the planets move in the heavens. These motions are described formally as Newton's laws of motion and gravity.
Newton applied this idea to the Sun and planets and took Keplers laws and calculated that the force falls off with the square of the distance from the Sun.
Newton's law of universal gravitation states that there is a force acting between objects that pulls them together. This force is proportional to the mass of the objects and inversely proportional to the square of their distance apart.
F = (GMmr^)/r2
Newton looked at the motion of the moon around the Sun and reasoned that the force responsible for gravity on Earth might be responsible for keeping the moon in orbit around the Earth. It turned out to be so
Binding Energy Curve
The mass of a nucleus is less than the sum of it constituent protons and neutrons. If we took the same number of protons and neutrons as in the nucleus we were trying to recreate, we would find the total mass of the individual protons and neutrons is greater than when they are arranged as a nucleus. The difference in mass between the products and sum of the individual nucleons is known as the mass defect. The binding energy is the amount of energy required to break the nucleus into protons and neutrons again; the larger the binding energy, the more difficult that would be. Figure. 1. Shows the binding energy for each element, against their atomic number.
Figure 1. Binding energy of the elements.
Starting from Hydrogen, as we increase the atomic number, the binding energy increases. So Helium has a greater binding energy per nucleon than Hydrogen while Lithium has a greater binding energy than Helium, and Berilium has a greater binding energy than Lithium, and so on. This trend continues, until we reach iron. It begins to decrease slowly.
The binding energy curve is obtained by dividing the total nuclear binding energy by the number of nucleons. The fact that there is a peak in the binding energy curve in the region of stability near iron means that either the breakup of heavier nuclei (fission) or the combining of lighter nuclei (fusion) will yield nuclei which are more tightly bound (less mass per nucleon).
The binding energy is intimately linked with fusion and fission. The lighter elements up to Fe are available will release energy via the fusion process, while in the opposite direction the heaviest elements down Fe are more susceptable to liberate energy via fission.
Figure 1. Binding energy of the elements.
Starting from Hydrogen, as we increase the atomic number, the binding energy increases. So Helium has a greater binding energy per nucleon than Hydrogen while Lithium has a greater binding energy than Helium, and Berilium has a greater binding energy than Lithium, and so on. This trend continues, until we reach iron. It begins to decrease slowly.
The binding energy curve is obtained by dividing the total nuclear binding energy by the number of nucleons. The fact that there is a peak in the binding energy curve in the region of stability near iron means that either the breakup of heavier nuclei (fission) or the combining of lighter nuclei (fusion) will yield nuclei which are more tightly bound (less mass per nucleon).
The binding energy is intimately linked with fusion and fission. The lighter elements up to Fe are available will release energy via the fusion process, while in the opposite direction the heaviest elements down Fe are more susceptable to liberate energy via fission.
A self-optimising microreactor system
Chemists in the US have developed a microreactor system which automatically calculates the optimal conditions for the chemical reaction it is undertaking. Once computed, the conditions can then be applied to a larger-scale reaction system. The researchers say their approach can save hours or days of tedium in the laboratory, by eliminating many manual experiments that would otherwise be required, as well as reducing the amounts of reagent needed.
To demonstrate the system, the research team from the Massachusetts Institute of Technology used the reaction of 4-chlorobenzotrifluoride with 2,3-dihydrofuran - an example of a Heck reaction, widely used in organic synthesis. Three syringe pumps containing the various components of the reaction were fed into a mixer, which in turn was connected to a 140ul microreactor. The yield of product was measured by high performance liquid chromatography (HPLC), whose results were passed to a computer programmed with an 'optimisation algorithm'. This enables the computer to take information about parameters such as flow rate, temperature and concentration of reactants, relate them to the yield, and then adjust them intelligently - based on the readings from the previous cycle - to produce gradually higher yields of product. The computer is also connected to the apparatus that controls flow rate, temperature, reactant concentration and so on, enabling these adjustments to the experimental conditions to be made automatically.
Within 2 days and after multiple cycles the system had arrived at the optimal conditions for a product yield, in this case, of 83 per cent.'We then wanted to see if we could use this information to scale the experiment up,' says team member Klavs Jensen. Using the conditions calculated by the microreactor system, the experiment was scaled up to a reactor representing a 50-fold increase in volume. 'The same optimal conditions applied at this larger scale,' says Jensen.
The researchers say that their system should be applicable to many reactions that can be conducted in a microreactor and could result in far less time and material being expended on finding the optimal conditions for a reaction - something that is key in organic chemistry. Furthermore a range of optimisation algorithms exist which can be applied to a variety of complex reaction scenarios. An added bonus, says Jensen, is that the system automatically calibrates the HPLC - 'one of the more tedious parts of doing this kind of work by hand.'
Commenting on the work, Kaspar Koch, managing director of FutureChemistry, a company based in the Netherlands specialising in microreactor technology, says, 'Conventional industrial optimisation methods are still laborious and environmentally unfriendly due to the large consumption of chemicals required. This new research exemplifies the advantages of microreactor technology in a low-waste reaction self-optimisation system consuming only minute amounts of starting materials - another significant step forward to smarter and greener chemistry.'
To demonstrate the system, the research team from the Massachusetts Institute of Technology used the reaction of 4-chlorobenzotrifluoride with 2,3-dihydrofuran - an example of a Heck reaction, widely used in organic synthesis. Three syringe pumps containing the various components of the reaction were fed into a mixer, which in turn was connected to a 140ul microreactor. The yield of product was measured by high performance liquid chromatography (HPLC), whose results were passed to a computer programmed with an 'optimisation algorithm'. This enables the computer to take information about parameters such as flow rate, temperature and concentration of reactants, relate them to the yield, and then adjust them intelligently - based on the readings from the previous cycle - to produce gradually higher yields of product. The computer is also connected to the apparatus that controls flow rate, temperature, reactant concentration and so on, enabling these adjustments to the experimental conditions to be made automatically.
Within 2 days and after multiple cycles the system had arrived at the optimal conditions for a product yield, in this case, of 83 per cent.'We then wanted to see if we could use this information to scale the experiment up,' says team member Klavs Jensen. Using the conditions calculated by the microreactor system, the experiment was scaled up to a reactor representing a 50-fold increase in volume. 'The same optimal conditions applied at this larger scale,' says Jensen.
The researchers say that their system should be applicable to many reactions that can be conducted in a microreactor and could result in far less time and material being expended on finding the optimal conditions for a reaction - something that is key in organic chemistry. Furthermore a range of optimisation algorithms exist which can be applied to a variety of complex reaction scenarios. An added bonus, says Jensen, is that the system automatically calibrates the HPLC - 'one of the more tedious parts of doing this kind of work by hand.'
Commenting on the work, Kaspar Koch, managing director of FutureChemistry, a company based in the Netherlands specialising in microreactor technology, says, 'Conventional industrial optimisation methods are still laborious and environmentally unfriendly due to the large consumption of chemicals required. This new research exemplifies the advantages of microreactor technology in a low-waste reaction self-optimisation system consuming only minute amounts of starting materials - another significant step forward to smarter and greener chemistry.'
Subscribe to:
Posts (Atom)