## Elements with most … properties

>>Least density-Hydrogen (0.08988kg/m³ at standard condition)
Having the same volume for every mole of gases, hydrogen, which has the lowest molecular mass, is the least dense element, followed by Helium and Neon.
>>Most density-Osmium (22,610kg/m³)
The three noble metals, which exist together both in the periodic table and in nature, also hold the title of the densest material with osmium, iridium, platinum in order. This property is due to the lanthanide contraction.
>>Lowest Melting/Boiling Point-Helium(0.95K, 4.22K)
Helium is a noble gas, having very little attraction between its individual atoms, thus resulting in lowest mp and bp. It is followed by hydrogen and Neon.
>>Highest Melting/Boiling Point-Diamond/Carbon(3800K*/4300K), Rhenium(3459K/5869K*)
Carbon in its diamond form, due to its strong tetrahedral arrangement, has highest mp among all elements, placing over the metals like tungsten, rhenium, osmium, which place second to fourth in order. In terms of boiling point, the order is rhenium, tungsten, osmium, first to third. Because of the high mp and bp of tungsten and the availability in nature, it is used in light bulbs.
>>Highest thermal conductivity-Helium II (>100000W/m.K)
Helium II is liquid helium lower than 2.2K. Because of its high conductivity, it cannot be boiled, but instead directly evaporate into gas. Highest thermal conductivity at normal condition is found in diamond with 2320W/m.K, followed by silver.
>>Highest electrical conductivity-Silver(resistivity=15.87nΩm)
Ignoring superconductors, silver has highest conductivity, or lowest resistivity, followed by two other metals, copper and gold. However, it is notable that silver cannot achieve superconducting state, where resistivity=0. This fact is also employed in real life where almost all wires are made with copper.
>>Most common elements in the universe-Hydrogen
It is predictable since most hydrogen can be easily formed since the start of the universe, while heavier elements are formed by fusion of hydrogen in the stars. It is followed by helium and oxygen.
>>Most stable nucleus-Iron 56
Iron 56 is the most stable nucleus which cannot undergo any nuclear reaction. Any lower mass nucleus can be fused together, while any larger mass nucleus can undergo fission. This follows the fact that if the star has enough fuel, its final form will have iron core, which might lead to neutron star or black hole, without further fusion.

Note :
Data in this page are obtained from Wikipedia, and thus corresponding source.

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## The Color of the Sky

>>Color of the Sky
Everyone knows that the sky has blue color; however, not everyone knows why it is.
Sunlight has 7 colors of spectrum in white light. When light enters the earth atmosphere, it collides with the particles in the air, and arises to Rayleigh scattering. Rayleigh scattering occurs when light collides with the particles smaller than its wavelength, in this case air molecules. That scattering is inversely proportional to the fourth power of the wavelength. Thus, in the visible spectrum, blue light which has shorter wavelength suffers more scattering than red light which has longer wavelength. Thus, when we look up the sky, we all see the blue light scattered by the air. However, when we look at the sun, though the sun emits white light, it appears yellow because most of the blue are scattered away, remaining only yellow and red.

Rayleigh Scattering of Sunlight

>>Dawn and Dusk
During the sunset and sun rise, the sky appears to be red or orange rather different from the color of blue during daytime. This is because the sunlight has to pass through the atmosphere longer than during daytime. Thus, most of the blue light are scattered, and even some red and yellow light, and are mostly absorbed by the atmosphere because of the cooling and the long distance the light has passed. Thus, most of the light we can see are red and yellow, making it reddish orange.

The Orange Color of the Sky

>>Rainbow
Rainbow is resulted from the white light entering water droplet creating prism-like effect. When light enters a different medium, it refracts and the angle of refraction varies with varying wavelengths. Thus, when white light travels from air to water, and again to air, the color of different wavelengths split up resulting a rainbow.

>>Night sky and sky from space
The color of night sky and sky from the space is almost the same, both being dark. However, there are different causes. Night sky does not have any light resource; even the moon does not have enough light. Thus, the color of night sky is black. However, out in the space, even if there is a light source, the color of the sky is totally balck. This is because the light from those sources is not scattered by any particles but directly entering our eyes. Thus, there is no light from space other than the light sources which we will see as bright things.

Picture credits to en.wikipedia.org

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## Aurora

Aurora, one of the most beautiful scenes of nature, can be found in northern and southern polar (high latitude  regions. The colorful displays of light in the sky were once thought of a legend and created by Gods. Now, the scientific solutions have revealed the secrets of these mysterious lights.
>>Where Aurorae are found?
Aurorae are found in high latitudes of north and south poles, typically 10° to 20° from the magnetic poles of the earth. During a geomagnetic storm(a temporary disturbance of earth’s magnetosphere), the aurorae can spread to low latitudes as well. The aurora in the northern hemisphere is called aurora borealis, and in southern, aurora australis. Th aurorae are usually formed high above the sky, usually in thermosphere, a layer which resides within 100 km to 600 km from the earth surface. Aurorae can also be found in other planets, like Jupiter and Saturn, and have almost the same properties.
>>How Aurorae are created?
The sun is always emitting thousands of particles into the space every second. Those charged particles like electrons, protons, and alpha particles may be trapped within the earth’s magnetosphere, and travel towards the north and south poles. When the charged particles hit the atmosphere, they run into nitrogen and oxygen, which are dominant in the earth’s atmosphere, and cause ionization.
-When nitrogen atom is ionized(i.e. gain or lose the electron) and return to ground state(relose the electron it gained previously, or regain the electron it previously lost), it emits red or blue color.
-When oxygen atomis ionized and return to ground state, it emits green or brownish-red.
-If the earth has atmosphere dominant with hydrogen or other gas, we may see different colors than our aurorae.
The brightness of the aurora can be varied according to many factors—height or attitude where aurora is created, the geomagnetic storm, and the solar flare(the intensity of the emitting particles and waves from the sun). The aurorae can be as bright as the sunlight that we can read our books at night or can be so dim that they can’t be seeen with naked eyes.

Aurora

Aurora from outer space by NASA

Jupiter Aurora by NASA

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## Superconductivity

Superconductivity is a property of some substances, in which an electric current can flow eternally without any applying voltage under certain temperatures. It is considered as a quantum mechanical phenomenon.
>> Why Superconductivity will be important
Whenever we think about the usage of electricity, we will see light bulbs, stove, fans, etc., but we will never think that the wires in our house nor along the road are consuming electrical energy. Actually, those wires dissipate many energies that are sent through them. Thus, we want something that will not consume energy so that we can send electricity without any waste. But the question is how and why those wires consume energy. This is because an electric current is a flow of electrons. As every wire is made of metals, they have strong lattices of positive ions. As the electrons flow, they suffer deflection and collision with those ions resulting in loss of energies. This contribute to the resistance of every materials. Most of the household electrical appliances have resistance so that they can convert electrical energy(strictly speaking, the kinetic energy of electrons) to other energies. However, those resistances are also present in wires.
>> Properties of Superconductors
In that sense, superconductivity comes into play. Superconductor is something without any resistance. Thus, if an electric current is fed into it, it will continue flowing permanently unless there are external disturbances. Many substances become superconductors whenever their temperature is lowered under a certain point (critical temperature and critical magnetic field). The BCS theory said that, in a superconductor, a Cooper electron pair is formed in which they use phonons to attract each other, and that the Copper pair can flow through the metal lattice without any resistance. Besides, this energy of the paired electrons has to be larger than the thermal energy of the system so they can only be formed under a certain temperature. Thus, a critical temperature is present for all superconductors, ranging from less than 1K to 130K.

Another property of superconductor is to have magnetic field of zero inside it. This is known as Meissner effect, and exhibit perfect diamagnetism, which will expel all external magnetic fields. Thus, if an external magnetic field is applied, it can affect the flow of electrons, thus having a critical value for magnetic field, too. This also causes the levitation of magnets over superconductor and vice versa.

>>Difference between superconductor and ideal conductor
However, superconductors are different from theoretically ideal conductors. Perfect conductors have fixed magnetic field inside it, and thus will resist any changes in magnetic field due to Faraday and Lenz laws of induction. Superconductors, on the other hand, have the magnetic field decaying exponentially to zero, and have effective London penetration depth, which is usually 100nm for most superconductors. Thus, although they both have resistance of zero and repel magnetic fields, the microscopic effects are different.
>> History of Superconductors
Superconductivity was first discovered in mercury at 4.2K by Onnes in 1911. He is awarded Nobel Prize in 1913 for his work on low temperature of helium of which the discovery of superconductor is a part. Later many other substances are found to superconduct at a higher temperature. However, no theory is present to explain this, until the development of quantum theory. In 1933, Meissner and Ochsenfeld discovered Meissner effect, which was explained by F. and H. London. In 1952, Bardeen, Cooper and Schrieffer developed BCS theory which explained superconductivity using Cooper pair electrons. The three were also awarded Nobel Prize in 1972.
>> Substances known to have Superconductivity
Many metals such as mercury, lead, tin, aluminium etc can become superconductors at a certain critical temperatures. However, the metals known to have highest conductance (silver, gold, and copper) never show superconductivity. The superconductors from these metals are Type 1 which has critical temperature lower than 10K. Type 2 superconductors are formed from alloys and chemical compounds which has critical temperature as high as 50K. Later in 1986, scientists found a superconductor of temperature higher than 77K. They are possible candidates to be used for practical purposes because their temperature can be retained by liquid nitrogen.

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## Surface Tension

Have you ever heard of a pin floating on water? Have you ever noticed the water droplets are all rounded?
What is causing all these strange phenomena?
Before thinking about the problems, we should consider the molecules of water, H2O. Water molecules have strong dipole attractions and hydrogen bonds pulling the individual water molecules together strongly. These properties contribute to the relatively high melting points and boiling points compared to other covalent compounds like CO2, and its ability to dissolve ionic compounds.
Back to the topic, according to the dynamics of fluids, a pin is impossible to float on water since its density(~7gcm-3) is greater than that of water(1kgm-3). However, the molecule on the water surface is so strongly bonded by hydrogen bonds that the force exerted by the pin, i.e. the weight, is not enough to break the bonds. Thus, a pin is able to float on water though, if disturbed, it will sink. Water striders, an insect, which can stay atop water also use this mechanism.

The shape of the water droplet is also caused by surface tension. A perfect sphere is a geometrical shape that has the least surface area for the same volume. Thus, surface tension makes water round. A water drop falling to the earth is deformed from a perfect sphere because, as it moves, it suffers gravity and resistance of the air. Thus, without any external distortions, surface tension maintains the water droplets in a spherical shape.

water droplet

<Pictures credit to wikipedia>

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## Why can airplanes fly?

What, do you think, is making the airplanes fly? If your answer is wings, then you are correct. Here is an illustration of what a wing of an airplane looks like.

Every airplane has propellers or jet engines to move forward, which gives rise to “thrust”. This thrust has to be larger than ‘Drag’, frictional or resistance force. There is a downward force, weight by the Earth. If we want to fly, we need a larger force than this weight. The upward force acting on the wings, and thus, on the airplane is called “lift”. If “lift” is larger than the weight of the whole airplane, we can fly now.
There are two popular explanations on why airplanes can fly. The first is by Bernoulli’s principle, the other by Newton’s laws of motion.
>>The first explanation has been used for many years, and easily understood. According to the Bernoulli’s principle, the velocity of a fluid is inversely proportional to the pressure of the fluid.

As shown in the figure, as the airplane moves forward, when air have to travel above the wing, it takes a longer path than travelling from below. If they both reach the end of the wing at the same time, it can be concluded that the velocity of the above air is greater than the velocity of the lower air. Using the Bernoulli’s principle, the pressure below the wing is higher than the pressure above the wing, thus giving rise to “lift”. However, this explanation has some major flaws. The first is that many experiments have shown that the above and lower air doesn’t necessarily reach to the end of the wing together. The second is that even if they both at the same instant, the thrust provided is not large enough to overcome the weight of the whole airplane. Another is that inverted flying is theoretically impossible. Thus, another explanation is introduced.
>>The second explanation uses Newton’s laws of motions. The air from above and below doesn’t need to reach together in this theory, but they have to meet each other again at the end of the wing. This is logical because if they don’t meet up, there will be a vacuum created. Now this is the case!
When the air is on the wing, it changes its direction downward to meet the air from below. Newton stated that if there is no force acting on an object, it will continue to be at rest or to continue at the same constant velocity and direction. Since the air is directed downward, there must be a force which is acting downward on it. This should be provided by the wing since there is no other object to act that force. Again in Newton’s laws, whenever there is an action force, there is also a reaction force acting with the same magnitude and opposite directions. Thus, the air above the wing should act an upward force on the wing. That force now provides enough “lift” to overcome the weight.

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## What is a black hole?

“A black hole is a massive, highly dense object from which nothing can escape, even light.” Is this a clear description of black hole? No, a black hole is so complicated that it cannot be explained in just one sentence.
Mechanism – The first question arouse is why we call “black hole”. Is it really black-coloured? Think about it. When a thing doesn’t emit or reflect any kinds of visible spectrum, then we say the colour is black. The same goes to black hole. But it is more specific, i.e. it doesn’t emit or reflect any kinds of radiation including light. Why so? Here’s the concept of escape velocity.
$v_e\ = \sqrt{\frac{2GM}{r}}$
where ve=escape velocity, G=gravitational constant, M=mass of the substance, r=radius of that substance

If the escape velocity of an object is larger than the velocity of light, then every photon of light entering that object nor the photon originally inside that object will not be escaped. Then we call it black hole.
Mass and Density – We cannot measure the mass of the black hole directly since we cannot see them. However, scientists can calculate by measuring the force acting upon the stars nearby. We can also predict the mass and density of a black hole from escape velocity. If our earth is a black hole, what size it should be? When u calculate it, u will see an amazing fact—the earth has to be 8.855mm to be a black hole !!
Location – A super-massive black hole is supposed to exist in the center of every galaxy. There are many
other black holes in binary system with other stars or isolatedly.
Structure – There is no unique structure of a black hole since every object entering the black hole loses its identity. A black hole has an event horizon which is also the boundary of that black hole. Technically, every object including light entering event horizon can never escape. Outside of this event horizon, the gravitational force acts normally like a regular massive object. Thus, it will attract nearby substances to swallow them or to let them revolve around it. The event horizon is mostly spherical.
A black hole is identified by 3 properties : mass, charge, and angular momentum. Black hole can retain the charge of the substances it has swallowed and distributed around the event horizon. If the black hole rotates, it will have angular momentum. If all 3 properties of 2 black holes are equal, then the two black holes are indistinguishable in classical mechanics.
Evolution – The black hole is formed by gravitational collapse when massive stars run out of fuel, the pressure can’t retain the mass and collapse. Once the black hole is formed, it will absorb nearby matters and grow larger. Black holes can sometimes merge with other black holes or stars.
Black holes are supposed to emit Hawking radiation theoretically, evaporating gradually and losing its mass. But no event has still detected any radiation and they seem to last indefinitely.
Effect – The effect of black hole is obvious when it comes close to the event horizon. The gravitational force of a black hole acts normally outside the event horizon and will attract them. As a substance comes closer to the event horizon, the stationary observer will see it slowing down and eventually disappears but he will never see that substance goes pass the event horizon. Why so? The clock near the event horizon will tick slower than an ordinary clock, i.e. a second is longer near a black hole. When a person goes straight into the black hole, his clock will be slower and slower, but he may not notice it. He will be going through the event horizon in a period of time for his clock, but it is an infinite amount of time for a stationary observer outside. Besides, the light from that person will undergo red-shift, gradually becomes dimmer and eventually disappears. These are due to gravitational time dilation and gravitational red-shift.
There are many other effects caused by the black holes like gravitational lensing. Also, we are still curious about what is going on inside a black hole, but we still don’t have any ideas about it.
[I'll not go into more details of the black hole as I think it is enough. But there are still many other interesting facts about black hole I leave out. Go and check it out if you are interested.]

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## Euler’s identity

Euler’s identity is a famous mathematical identity. Many mathematicians term it as the most beautiful equation. Why?? It includes all five basic constants of mathematics

• e, base of the natural logarithms(e=2.718)
• π, ratio of the diameter and circumference of the circle, an important constant in Geometry(π=3.14)
• i, the imaginary unit(i=√-1)
• 1, the multiplicative identity
• 0, the additive identity(both 1 and 0 are the basic numbers)

Each constant has its uniqueness.
Euler’s identity comes from Euler’s equation, eix=cos(x)+i sin(x)
We can prove his equation in this way
$\frac{d}{dx} e^{ix}=ie^{ix} \\ \frac{d}{dx} cos{x}+i sin{x}=-sin{x}+i cos{x}=i(i sin{x}+cos{x}) (Note : i^{2}=-1)\\$
Since both functions solve these sets of equations,
$f^\prime (x)=i f(x), \\ e^{ix}=cos{x}+i sin{x}\\$
The last equation we obtain is Euler’s equation. Substituting x=π,
$e^{i\pi}=cos{\pi}+i sin{\pi} \\$
Since $cos{\pi}=-1$ and $sin{\pi}=0, \\$
$e^{i\pi}=-1 \\ e^{i\pi}+1=0 \\$

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We all know that lead can prevent radiations, but how many of them can lead stop and how?
Here comes the explanation.

• Gamma rays are the highest penetrating power of all rays; thus, a thick lead block is required to stop it. Gamma rays are mostly absorbed by electrons. Lead has 82 electrons and is the highest atomic number which is stable in nature, and also highly dense. Thus, lead is a good choice to shield gamma rays. Theoretically, nothing can never completely stop gamma rays. This is because gamma rays are dumped exponentially. If, for example, a 1 cm thick lead can reduce half the intensity of gamma rays, gamma rays intensity will be one-fourth after passing 2 cm lead, and so on. Therefore, however thick lead is, gamma rays will never completely stop. However, this sounds hair-splitting. Although it is not stopped completely, gamma ray will eventually come to a state which detectors cannot detect. Then we can say gamma ray is stopped.
• Alpha rays can be stopped by using even a thick sheet of paper. Thus, lead is not needed, but can stop alpha rays, too.
• In the case of e- beta rays, lead should not be used. Think about the X-ray production. X-rays are produced when electrons are shot at a dense target material, including lead. Therefore, if lead is used to stopped beta rays, bremsstrahlung radiation (X-rays) will be emitted. Either plastic or aluminium is good for shielding e- beta rays. They can also be used to stop e+ beta rays, but in this case, gamma rays will be emitted from electron-positron annihilation process.
• Lead is not a good shield against neutron rays. Elements with low atomic number are better in absorbing neutrons than high atomic number elements because of their difference in neutron numbers in their nucleus. However, if an low atomic number element absorbs neutrons they can re-emit gamma rays which has to be stopped by lead.

## Is there any substance travelling at or faster than the speed of light?

Well, light has an amazing speed high above any of the human speed. Whether human can reach this limit is an ambiguous question.

First, think of how a substance can gain high velocity. As its velocity gets higher, its kinetic energy becomes larger by KE=1/2 mv². So the energy needed to gain the velocity of light will be 1/2 mc² as seen from this equation. This means we can reach the velocity of light if we gain enough energy. However, it is not so simple after all.

The special relativity states that as the velocity of a substance gets near the speed of light, some of its physical characteristics will change including its mass. How can this be??? It is still true that KE=1/2 mv². The only problem here is its mass. According to relativity, the mass of any moving object can be described by

We call this γ as Lorentz factor. From this equation, you will see that as the velocity comes nearer to c, its mass will become infinitely large. As a result, the required KE will also be infinite. We will not experience this mass change in every day life although we are moving because the velocity we are moving is too small compared to c. Check the following graphs.

It is clear that we cannot give infinite energy to get the speed of light. Thus, it is impossible for human to travel at or greater than the speed of light.

So how can light, photons, travel at c? Because it is massless, however increase in its mass will not change the value of zero. Then, where is its energy? From KE=1/2 mv², if there is no mass, there will be no KE at all. Thus, Einstein formulated another formula to calculate the energy of massless particles, E²=p²c²+(mc²)². Now the problem is solved. Even if a particle has no mass, it can have its energy through its momentum. This is how photons manage to travel at the speed of light. As any object except light has masses, it seems impossible for human to reach the speed of light.

Then someone may hear of “tachyon”, a subatomic particle supposed to travel faster than light. Consider what is its mass from Eq(1). Since its v>c, γ becomes an imaginary number so its mass also becomes imaginary. Thus, although many has proposed “tachyon” hypothetically, no event approves of this particle’s existence practically.

To conclude, it is true till now that nothing can surpass this speed barrier of light.

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