Trigonometry/Solving triangles by half-angle formulae;Sin(A) and Heron's formula

Trigonometry/Solving triangles by half-angle formulae

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In this section, we present alternative ways of solving triangles by using half-angle formulae.

Given a triangle with sides a, b and c, define

s = ¹⁄₂(a+b+c).

Note that

a+b-c = 2s-2c = 2(s-c)

and similarly for a and b.

We have from the cosine theorem

${\displaystyle \displaystyle \cos(A)={{b^{2}+c^{2}-a^{2}} \over {2bc}}}$

Sin(A/2)

${\displaystyle \displaystyle 2\sin ^{2}\left({\frac {A}{2}}\right)=1-\cos(A)=1-{{b^{2}+c^{2}-a^{2}} \over {2bc}}={{(a+b-c)(a-b+c)} \over {2bc}}={{2(s-b)(s-c)} \over {bc}}}$

So

${\displaystyle \displaystyle \sin \left({\frac {A}{2}}\right)={\sqrt {{(s-b)(s-c)} \over {bc}}}}$.

By symmetry, there are similar expressions involving the angles B and C.

Note that in this expression and all the others for half angles, the positive square root is always taken. This is because a half-angle of a triangle must always be less than a right angle.

Cos(A/2) and tan(A/2)

${\displaystyle \displaystyle 2\cos ^{2}\left({\frac {A}{2}}\right)=1+\cos(A)=1+{{b^{2}+c^{2}-a^{2}} \over {2bc}}={{(a+b+c)(b+c-a)} \over {2bc}}={{2s(s-a)} \over {bc}}}$

So

${\displaystyle \displaystyle \cos \left({\frac {A}{2}}\right)={\sqrt {{s(s-a)} \over {bc}}}}$.

${\displaystyle \displaystyle \tan \left({\frac {A}{2}}\right)={\sin({\frac {A}{2}}) \over \cos({\frac {A}{2}})}={\sqrt {{(s-b)(s-c)} \over {s(s-a)}}}}$.

Again, by symmetry there are similar expressions involving the angles B and C.

A formula for sin(A) can be found using either of the following identities:

${\displaystyle \displaystyle \sin(A)=2\sin \left({\frac {A}{2}}\right)\cos \left({\frac {A}{2}}\right)}$

${\displaystyle \displaystyle \sin(A)={\sqrt {(1+\cos(A))(1-\cos(A))}}}$

These both lead to

${\displaystyle \displaystyle \sin(A)={\frac {2}{bc}}{\sqrt {s(s-a)(s-b)(s-c)}}}$

The positive square root is always used, since A cannot exceed 180º. Again, by symmetry there are similar expressions involving the angles B and C. These expressions provide an alternative proof of the sine theorem.

Since the area of a triangle

${\displaystyle \displaystyle \Delta ={\frac {1}{2}}bc\sin(A)}$,

${\displaystyle \displaystyle \Delta ={\sqrt {s(s-a)(s-b)(s-c)}}}$

which is Heron's formula.

What is Gravitational Force? Newton’s Law of Universal Gravitation;

What is Gravitational Force?

Newton’s Law of Universal Gravitation is used to explain gravitational force. This law states that every massive particle in the universe attracts every other massive particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. This general, physical law was derived from observations made by induction. Another way, more modern, way to state the law is: ‘every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between the point masses’.

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Gravitational force surrounds us. It is what decides how much we weigh and how far a basketball will travel when thrown before it returns to the surface. The gravitational force on Earth is equal to the force the Earth exerts on you. At rest, on or near the surface of the Earth, the gravitational force equals your weight. On a different astronomical body like Venus or the Moon, the acceleration of gravity is different than on Earth, so if you were to stand on a scale, it would show you that you weigh a different amount than on Earth.

When two objects are gravitationally locked, their gravitational force is centered in an area that is not at the center of either object, but at the barycenter of the system. The principle is similar to that of a see-saw. If two people of very different weights sit on opposite sides of the balance point, the heavier one must sit closer to the balance point so that they can equalize each other's mass. For instance, if the heavier person weighs twice as much as the lighter one, they must sit at only half the distance from the fulcrum. The balance point is the center of mass of the see-saw, just as the barycenter is the balance point of the Earth-Moon system. This point that moves around the Sun in the orbit of the Earth, while the Earth and Moon each move around the barycenter, in their orbits.

Hydrogen bonding in alcohols

Hydrogen bonding in alcohols

An alcohol is an organic molecule containing an -O-H group.

Any molecule which has a hydrogen atom attached directly to an oxygen or a nitrogen is capable of hydrogen bonding. Such molecules will always have higher boiling points than similarly sized molecules which don't have an -O-H or an -N-H group. The hydrogen bonding makes the molecules "stickier", and more heat is necessary to separate them.

Ethanol, CH3CH2-OH, and methoxymethane, CH3-O-CH3, both have the same molecular formula, C2H6O.

They have the same number of electrons and a similar length to the molecule. The van der Waals attractions (both dispersion forces and dipole-dipole attractions) in each will be much the same.

However, ethanol has a hydrogen atom attached directly to an oxygen - and that oxygen still has the same two lone pairs as in a water molecule. Hydrogen bonding can occur between ethanol molecules, although not as effectively as in water. The hydrogen bonding is limited by the fact that there is only one hydrogen in each ethanol molecule with sufficient δ+ charge.

In methoxymethane, the lone pairs on the oxygen are still there, but the hydrogens aren't sufficiently δ+ for hydrogen bonds to form. Except in some rather unusual cases, the hydrogen atom has to be attached directly to the very electronegative element for hydrogen bonding to occur.

The boiling points of ethanol and methoxymethane show the dramatic effect that the hydrogen bonding has on the stickiness of the ethanol molecules:

ethanol (with hydrogen bonding)        78.5°C

methoxymethane (without hydrogen bonding)        -24.8°C

The hydrogen bonding in the ethanol has lifted its boiling point about 100°C.

It is important to realise that hydrogen bonding exists in addition to van der Waals attractions. For example, all the following molecules contain the same number of electrons, and the first two are much the same length. The higher boiling point of the button-1-ol is due to the additional hydrogen bonding.

Comparing the two alcohols (containing -OH groups), both boiling points are high because of the additional hydrogen bonding due to the hydrogen attached directly to the oxygen - but they aren't the same.

The boiling point of the 2-methylpropan-1-ol isn't as high as the butan-1-ol because the branching in the molecule makes the van der Waals attractions less effective than in the longer butan-1-ol.

More complex examples of hydrogen bonding

More complex examples of hydrogen bonding

The hydration of negative ions

When an ionic substance dissolves in water, water molecules cluster around the separated ions. This process is called hydration.

Water frequently attaches to positive ions by co-ordinate (dative covalent) bonds. It bonds to negative ions using hydrogen bonds.

The diagram shows the potential hydrogen bonds formed to a chloride ion, Cl-. Although the lone pairs in the chloride ion are at the 3-level and wouldn't usually be active enough to form hydrogen bonds, in this case, they are made more attractive by the full negative charge on the chlorine.

INTERMOLECULAR BONDING - HYDROGEN BONDS

INTERMOLECULAR BONDING - HYDROGEN BONDS

The evidence for hydrogen bonding

Many elements form compounds with hydrogen. If you plot the boiling points of the compounds of the Group 4 elements with hydrogen, you find that the boiling points increase as you go down the group.

The increase in boiling point happens because the molecules are getting larger with more electrons, and so van der Waals dispersion forces become greater.

Note:  If you aren't sure about van der Waals dispersion forces, it would pay you to follow this link before you go on.

If you repeat this exercise with the compounds of the elements in Groups 5, 6 and 7 with hydrogen, something odd happens.

Although for the most part, the trend is the same as in group 4 (for the same reasons), the boiling point of the compound of hydrogen with the first element in each group is abnormally high.

In the cases of NH3, H2O, and HF there must be some additional intermolecular forces of attraction, requiring significantly more heat energy to break. These relatively powerful intermolecular forces are described as hydrogen bonds.

 

The origin of hydrogen bonding

The molecules which have this extra bonding are:

Note:  The solid line represents a bond in the plane of the screen or paper. Dotted bonds are going back into the screen or paper away from you, and wedge-shaped ones are coming out towards you.

Notice that in each of these molecules:

The hydrogen is attached directly to one of the most electronegative elements, causing the hydrogen to acquire a significant amount of positive charge.

Each of the elements to which the hydrogen is attached is not only significantly negative but also has at least one "active" lone pair.

Lone pairs at the 2-level have the electrons contained in a relatively small volume of space which therefore has a high density of negative charge. Lone pairs at higher levels are more diffuse and not so attractive to positive things.

Note:  If you aren't happy about electronegativity, you should follow this link before you go on.

Consider two water molecules coming close together.

The δ+ hydrogen is so strongly attracted to the lone pair that it is almost as if you were beginning to form a co-ordinate (dative covalent) bond. It doesn't go that far, but the attraction is significantly stronger than an ordinary dipole-dipole interaction.

Hydrogen bonds have about a tenth of the strength of an average covalent bond and are being broken continuously and reformed in liquid water. If you liken the covalent bond between the oxygen and hydrogen to a stable marriage, the hydrogen bond has "just good friends" status.

Water as a "perfect" example of hydrogen bonding

Notice that each water molecule can potentially form four hydrogen bonds with surrounding water molecules. There are exactly the right numbers of δ+ hydrogens and lone pairs so that every one of them can be involved in hydrogen bonding.

This is why the boiling point of water is higher than that of ammonia or hydrogen fluoride.

Note:  You will find more discussion on the effect of hydrogen bonding on the properties of water in the page on molecular structures.

In the case of ammonia, the amount of hydrogen bonding is limited by the fact that each nitrogen only has one lone pair. In a group of ammonia molecules, there aren't enough lone pairs to go around to satisfy all the hydrogens.

That means that on average each ammonia molecule can form one hydrogen bond using its lone pair and one involving one of its δ+ hydrogens. The other hydrogens are wasted.

In hydrogen fluoride, the problem is a shortage of hydrogens. On average, then, each molecule can only form one hydrogen bond using its δ+ hydrogen and one involving one of its lone pairs. The other lone pairs are mostly wasted.

In water, there is precisely the right number of each. Water could be considered as the "perfect" hydrogen bonded system.

Warning:  It has been pointed out to me that some sources (including one of the UK A level Exam Boards) count the number of hydrogen bonds formed by water, say, differently. They say that water forms 2 hydrogen bonds, not 4. That is often accompanied by a diagram of ice next to this statement clearly showing 4 hydrogen bonds!

Reading what they say, it appears that they only count a hydrogen bond as belonging to a particular molecule if it comes from a hydrogen atom on that molecule. That seems to me to be illogical. A hydrogen bond is made from two parts - a δ+ hydrogen attached to a sufficiently electronegative element, and an active lone pair. These interact to make a hydrogen bond, and it is still a hydrogen bond irrespective of which end you look at it from.

The IUPAC definitions of a hydrogen bond do not refer at all to any of this, so there doesn't seem to be any "official" backing for this one way or the other.

However, it is essential that you find out what your examiners are expecting. They make the rules for the exam you will be sitting, and you have no choice other than to play by those rules.

Hydrogen Bond Definition and Examples of Hydrogen Bonds

Hydrogen Bond Definition

A hydrogen bond is a type of attractive (dipole-dipole) interaction between an electronegative atom and a hydrogen atom bonded to another electronegative atom. This bond always involves a hydrogen atom. Hydrogen bonds can occur between molecules or within parts of a single molecule.

A hydrogen bond tends to be stronger than van der Waals forces, but weaker than covalent bonds or ionic bonds. It is about 1/20th (5%) the strength of the covalent bond formed between O-H. However, even this weak bond is strong enough to withstand slight temperature fluctuation.

But the Atoms Are Already Bonded

How can hydrogen be attracted to another atom when it is already bonded? In a polar bond, one side of the bond still exerts a slight positive charge, while the other side has a slightly negative electrical charge. Forming a bond doesn't neutralize the electrical nature of the participant atoms.

Examples of Hydrogen Bonds

Hydrogen bonds are found in nucleic acids between base pairs and between water molecules. This type of bond also forms between hydrogen and carbon atoms of different chloroform molecules, between hydrogen and nitrogen atoms of neighboring ammonia molecules, between repeating subunits in the polymer nylon, and between hydrogen and oxygen in acetylacetone. Many organic molecules are subject to hydrogen bonds. Hydrogen bond:

Help bind transcription factors to DNA

Aid antigen-antibody binding

Organize polypeptides into secondary structures, such as alpha helix and beta sheet

Hold together the two strands of DNA

Bind transcription factors to each other

Hydrogen Bonding in Water

Although hydrogen bonds form between hydrogen and any other electronegative atom, the bonds within the water are the most ubiquitous (and some would argue, the most important). Hydrogen bonds form between neighboring water molecules when the hydrogen of one atom comes between the oxygen atoms of its own molecule and that of its neighbor. This happens because the hydrogen atom is attracted to both its own oxygen and other oxygen atoms that come close enough. The oxygen nucleus has 8 "plus" charges, so it attracts electrons better than the hydrogen nucleus, with its single positive charge. So, neighbor oxygen molecules are capable of attracting hydrogen atoms from other molecules, forming the basis of hydrogen bond formation.

The total number of hydrogen bonds formed between water molecules is 4. Each water molecule can form 2 hydrogen bonds between oxygen and the two hydrogen atoms in the molecule. An additional two bonds can be formed between each hydrogen atom and nearby oxygen atoms.

A consequence of hydrogen bonding is that hydrogen bonds tend to arrange in a tetrahedron around each water molecule, leading to the well-known crystal structure of snowflakes. In liquid water, the distance between adjacent molecules is more significant and the energy of the molecules is high enough that hydrogen bonds are often stretched and broken. However, even liquid water molecules average out to a tetrahedral arrangement. Because of hydrogen bonding, the structure of liquid water becomes ordered at a lower temperature, far beyond that of other liquids. Hydrogen bonding holds water molecules about 15% closer than if the bonds weren't present. The bonds are the primary reason water displays exciting and unusual chemical properties.

Hydrogen bonding reduces extreme temperature shifts near large bodies of water.

Hydrogen bonding allows animals to cool themselves using perspiration because such a large amount of heat is needed to break hydrogen bonds between water molecules.

Hydrogen bonding keeps water in its liquid state over a wider temperature range than for any other comparable-sized molecule.

The bonding gives water an exceptionally high heat of vaporization, which means considerable thermal energy is needed to change liquid water into water vapor.

Hydrogen bonds within the heavy water are even stronger than those within ordinary water made using normal hydrogen (protium). Hydrogen bonding in tritiated water is stronger still.

Universal Law of Gravitation

The universe has a lot of forces, a lot of pushes and pulls. We're always pushing or pulling something, even if only the ground. But it turns out that in physics, there are really only four fundamental forces from which everything else is derived: the strong force, the weak force, the electromagnetic force, and the gravitational force.

The gravitational force is a force that attracts any two objects with mass. We call the gravitational force attractive because it always tries to pull masses together, it never pushes them apart. In fact, every object, including you, is pulling on every other object in the entire universe! This is called Newton's Universal Law of Gravitation. Admittedly, you don't have a very large mass and so, you're not pulling on those other objects much. And objects that are really far apart from each other don't pull on each other noticeably either. But the force is there and we can calculate it.

Hydrogen bonding,S different picture