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From Wikipedia

Magnetic moment

The magnetic moment of a magnet is a quantity that determines the force the magnet can exert on electric currents and the torque that a magnetic field will exert on it. A loop of electric current, a bar magnet, an electron, a molecule, and a planet all have magnetic moments.

Both the magnetic moment and magnetic field may be considered to be vectors having a magnitude and direction. The direction of the magnetic moment points from the south to north pole of a magnet. The magnetic field produced by a magnet is proportional to its magnetic moment as well. More precisely, the term magnetic moment normally refers to a system's magnetic dipole moment, which produces the first term in the multipole expansion of a general magnetic field. The dipole component of an object's magnetic field is symmetric about the direction of its magnetic dipole moment, and decreases as the inverse cube of the distance from the object.

## Two definitions of moment

In textbooks, two complementary approaches have been used to define magnetic moments. In pre-1930's textbooks, they were defined using magnetic poles. Most recent textbooks define it in terms of AmpÃ¨rian currents.

### Magnetic pole definition

The sources of magnetic moments in materials can be represented by poles in analogy to electrostatics. Consider a bar magnet which has magnetic poles of equal magnitude but opposite polarity. Each pole is the source of magnetic force which weakens with distance. Since magnetic poles always come in pairs, their forces partially cancel each other because while one pole pulls, the other repels. This cancellation is greatest when the poles are close to each other i.e. when the bar magnet is short. The magnetic force produced by a bar magnet, at a given point in space, therefore depends on two factors: on both the strength p of its poles, and on the vector \mathbf{l} separating them. The moment is defined as

\boldsymbol{\mu}=p\mathbf{l}.

The analogy with electric dipoles cannot be taken too far. Magnetic dipoles are associated with angular momentum, as demonstrated by the Einstein-de Haas effect and the Barnett effect, for example. Therefore, they do not behave like ideal magnetic dipoles. In particular, although a magnetic dipole is subject to a torque in a magnetic field that tends to align its magnetic moment with the applied magnetic field, as a consequence of the associated angular momentum, the magnetic dipole precesses, that is, its direction rotates about the axis of the applied field. Nevertheless, magnetic poles are very useful for magnetostatic calculations in ferromagnets.

### Current loop definition

Suppose a planar closed loop carries an electric current I and has vector area \mathbf{S} (x, y, and z coordinates of this vector are the areas of projections of the loop onto the yz, zx, and xy planes). Its magnetic moment \boldsymbol{\mu}, vector, is defined as:

\boldsymbol{\mu}=I \mathbf{S}.

By convention, the direction of the vector area is given by the right hand grip rule (curling the fingers of one's right hand in the direction of the current around the loop, when the palm of the hand is "touching" the loop's outer edge, and the straight thumb indicates the direction of the vector area and thus of the magnetic moment).

## Units

The unit for magnetic moment is not a base unit in the International System of Units (SI) and it can be represented in more than one way. For example, in the current loop definition, the area is measured in square meters and I is measured in amperes, so the magnetic moment is measured in ampereâ€“square meters (\text{A m}^2). In the equation for torque on a moment, the torque is measured in joules and the magnetic field in tesla, so the moment is measured in Joules per Tesla (\text{J T}^{-1}). These two representations are equivalent:

1\,\text{A m}^2 = 1\,\text{J T}^{-1}.

In the CGS system, there are several different sets of electromagnetism units, of which the main ones are ESU, Gaussian, and EMU. Among these, there are two alternative (non-equivalent) units of magnetic dipole moment in CGS:

(ESU CGS) 1 statAÂ·cmÂ² = 3.33564095Ã—10-14 (m2Â·A or J/T)

and (more frequently used)

(EMU CGS and Gaussian-CGS) 1 erg/G = 1 abAÂ·cmÂ² = 10-3 (

Atoms in molecules

The atoms in molecules or atoms-in-molecules or quantum theory of atoms in molecules (Qtaim) approach is a quantumchemical model that characterizes the chemical bonding of a system based on the topology of the quantum charge density. In addition to bonding, AIM allows the calculation of certain physical properties on a per-atom basis, by dividing space up into atomic volumes containing exactly one nucleus. Developed by Professor Richard Bader since the early 1960s, during the past decades QTAIM has gradually become a theory for addressing possible questions regarding chemical systems, in a variety of situations hardly handled before by any other model or theory in Chemistry . In QTAIM an atom is defined as a proper open system, i.e. a system that can share energy and electron density, which is localized in the 3D space. Each atom acts as a local attractor of the electron density, and therefore it can be defined in terms of the local curvatures of the electron density. The mathematical study of these features is usually referred in the literature as charge density topology. Nevertheless, the term topology is used in a different sense in Mathematics.

According to the theorems of QTAIM, the molecular structure is given by the stationary points of the electron density.

## Main results

The major conclusions of the AIM approach are:

• A molecule can be uniquely divided into a set of atomic volumes. These volumes are divided by a series of surfaces through which the gradient vector field of the electron density has no flux. Atomic properties such as atomic charge, dipole moment, and energies can be calculated by integrating their corresponding operators over the atomic volume.
• Two atoms are bonded if their atomic volumes share a common interatomic surface, and there is a (3,&nbsp;&minus;1) critical point on this surface. A critical point is defined as a point in space where the gradient is zero. A (3,&nbsp;&minus;1) critical point is defined as a critical point at which two of the eigenvalues of the Hessian matrix at the critical point are negative, while the other eigenvalue is positive. In other words, a bonding critical point is a first-order saddle point in the electron density scalar field. A bond path is the line along which the electron density is a maximum with respect to a neighboring line. Along the associated virial path the potential energy is maximally stabilizing.
• The interatomic bonds are classified as either closed shell or shared, if the Laplacian of the electron density at the critical point is positive or negative, respectively.
• Geometric bond strain can be gauged by examining the deviation of the bonding critical point from the interatomic axis between the two atoms. A large deviation implies larger bond strain.

## Applications

QTAIM is applied to the description of certain organic crystals with unusually short distances between neighboring molecules as observed by X-ray diffraction. For example in the crystal structure of molecular chlorine the experimental Cl...Cl distance between two molecules is 327 picometres which is less than the sum of the van der Waals radii of 350 picometres. In one Qtaim result 12 bond paths start from each chlorine atom to other chlorine atoms including the other chlorine atom in the molecule. The theory also aims to explain the metallic properties of metallic hydrogen in much the same way.

The theory is also applied to so-called hydrogen-hydrogen bonds as they occur in molecules such as phenanthrene and chrysene. In these compounds the distance between two ortho hydrogen atoms again is shorter than their van der Waals radii and according to in silico experiments based on this theory, a bond path is identified between them. Both hydrogen atoms have identical electron density and are closed shell and therefore they are very different from the so-called dihydrogen bonds which are postulated for compounds such as (CH3)2NHBH3 and also different from so-called agostic interactions.

In mainstream chemistry close proximity of two nonbonding atoms leads to destabilizing steric repulsion but in QTAIM the observed hydrogen hydrogen interactions are in fact stabilizing. It is well known that both kinked phenanthrene and chrysene are around 6 kcal/mol (25 kJ/mol) more stable than their linear isomers anthracene and tetracene. One traditional explanation is given by Clar's rule. QTAIM shows that a calculated stabilization for phenanthrene by 8 kcal/mol (33 kJ/mol) is the result of destabilization of the compound by 8 kcal/mol (33 kJ/mol) originating from electron transfer from carbon to hydrogen, offset by 12.1 kcal (51 kJ/mol) of stabilization due to a H..H bond path. The electron density at the critical point between the two hydrogen atoms is low, 0.012 e for phenanthrene. Another property of the bond path is its curvature.

An

Diatomic molecule

Diatomic molecules are molecules composed only of two atoms, of either the same or different chemical elements. The prefix di- means two in Greek. Common diatomic molecules are hydrogen (H2), nitrogen (N2), oxygen (O2), and carbon monoxide (CO). Seven elements exist in the diatomic state in the liquid and solid forms: H2, N2, O2, F2, Cl2, Br2, and I2. Most elements (and many chemical compounds) aside from the form diatomic molecules when evaporated, although at very high temperatures, all materials disintegrate into atoms. The noble gases do not form diatomic molecules.

## Occurrence

Hundreds of diatomic molecules have been characterized in the terrestrial environment, laboratory, and interstellar medium. About 99% of the Earth's atmosphere is composed of diatomic molecules, specifically oxygen and nitrogen at 21 and 78%, respectively. The natural abundance of hydrogen (H2) in the Earth's atmosphere is only on the order of parts per million, but H2 is, in fact, the most abundant diatomic molecule seen in nature. The interstellar medium is, indeed, dominated by hydrogen atoms.

Elements that consist of diatomic molecules, under typical laboratory conditions of 1 bar and 25 Â°C, include hydrogen (H2), nitrogen (N2), oxygen (O2), and the halogens. Again, many other diatomics are possible and form when elements are evaporated, but these diatomic species repolymerize at lower temperatures. For example, heating ("cracking") elemental phosphorus gives diphosphorus.

If a diatomic molecule consists of two atoms of the same element, such as H2 and O2, then it is said to be homonuclear, but otherwise it is heteronuclear, such as with CO or NO. The bond in a homonuclear diatomic molecule is non-polar and covalent. In most diatomic molecules, the elements are nonidentical. Prominent examples include carbon monoxide, nitric oxide, and hydrogen chloride, but other important examples include MgO and NaCl.

## Molecular geometry

Diatomic molecules cannot have any geometry but linear, as any two points always lie in a line. This is the simplest spatial arrangement of atoms after the sphericity of single atoms.

## Historical significance

Diatomic elements played an important role in the elucidation of the concepts of element, atom, and molecule in the 19th century, because some of the most common elements, such as hydrogen, oxygen, and nitrogen, occur as diatomic molecules. John Dalton's original atomic hypothesis assumed that all elements were monatomic and that the atoms in compounds would normally have the simplest atomic ratios with respect to one another. For example, Dalton assumed that water's formula was HO, giving the atomic weight of oxygen as 8 times that of hydrogen, instead of the modern value of about 16. As a consequence, confusion existed regarding atomic weights and molecular formulas for about half a century.

As early as 1805, Gay-Lussac and von Humboldt showed that water is formed of two volumes of hydrogen and one volume of oxygen, and by 1811 Amedeo Avogadro had arrived at the correct interpretation of water's composition, based on what is now called Avogadro's law and the assumption of diatomic elemental molecules. However, these results were mostly ignored until 1860. Part of this rejection was due to the belief that atoms of one element would have no chemical affinity towards atoms of the same element, and part was due to apparent exceptions to Avogadro's law that were not explained until later in terms of dissociating molecules.

At the 1860 Karlsruhe Congress on atomic weights, Cannizzaro resurrected Avogadro's ideas and used them to produce a consistent table of atomic weights, which mostly agree with modern values. These weights were an important pre-requisite for the discovery of the periodic law by Dmitri Mendeleev and Lothar Meyer.

## Energy levels

It is convenient, and common, to represent a diatomic molecule as two point masses (the two atoms) connected by a massless spring. The energies involved in the various motions of the molecule can then be broken down into three categories.

• The translational energies
• The rotational energies
• The vibrational energies

### Translational energies

The translational energy of the molecule is simply given by the kinetic energy expression:

E_{trans}=\frac{1}{2}mv^2

where m is the mass of the molecule and v is its velocity.

### Rotational energies

Classically, the kinetic energy of rotation is

E_{rot} = \frac{L^2}{2 I} \,
where
L \, is the angular momentum
I \, is the moment of inertia of the molecule

For microscopic, atomic-level systems like a molecule, angular momentum can only have specific discrete values given by

L^2 = l(l+1) \hbar^2 \,
where l is a non-negative integer and \hbar is Planck's reduced constant.

Also, for a diatomic molecule the moment of inertia is

I = \mu r_{0}^2 \,
where
\mu \, is the reduced mass of the molecule and

Question:What does it mean to say a molecule is polar? What factors determine whether or not a molecule has a dipole moment? I understand what a polar molecule is but I don't know what the dipole movement factors are. Please help. Thank you.

Answers:Any molecule that, as a whole, has an asymmetrical charge distribution, will be polar (have a dipole moment) Any bond between two elements with different electronegativities will have some ionic character and thus produce a dipole moment. The spatial direction and strengths of these dipole moments will average out to form the molecule's dipole moment. A few examples: Water has a dipole moment because it is a bent molecule. CO2 does not have a dipole moment, because it is linear and the dipole moments of the two C=O bonds cancel. BCl3 is flat and triangular, so the dipole moments of the 3 B-Cl bonds cancel. NH3 and NCl3 have a trigonal pyramid shape, so the dipole moments of their 3 bonds do not cancel.

Question:The permanent electric dipole moment of the water molecule (H20) is 6.2X10^-30Cm. WHat is the maximum possible torque on a water molecule 5.0 X 10^8 N/C electric field.

Answers:The maximum possible torque is the product of the dipole moment with the E-field. Why? Think of the dipole moment as being two charges, +q and -q, separated by distance d. When an E-field is turned on, the forces on them are +qE and -qE respectively. If the separation is aligned with the E-field, there is no torque, but if it is perpendicular, each force has lever-arm of d/2 from the center.So the total torque is: 2*(d/2)*qE = d*q*E = p*E. So the maximum possible torque is: 6.2e-30 * 5e8 = 31e-22 = 3.1e-21 (Nm)

Question:Are there specific characteristics? Bigger molecule, more electrons, ect? For example, Which of the following molecules has the largest dipole moment? a) H2O2 b) HCl c) SO2 d) NO e) N2 Thanks!

Answers:the dipole moment is a measure of the disproportionate distribution of electrons in the molecule. it is also affected by the shape of the molecule. net molecular dipoles can be zero even if the individual bonds are polar. CO2 is a good example of this, as it it is a linear molecule. of the list you gave, the HCl will have the largest dipole moment because of the electron affinity of the chlorine over the hydrogen. I would place the peroxide next.

Question:The dipole moment of an HCl molecule is 3.4 X 10^-30 C*m. How do you calculate the magnitude of the Torque if an electric field exerts 2.0 X 10^6 N/C on the molecule and the angle between the electric field and the logitudinal axis of the molecule is 45 ?

Answers:The magnitude of the torque exerted by an electric field on an electric dipole is: |T| = |p|*|E|*sin(theta) where |p| is the magnitude of the dipole moment, |E| is the magnitude of the electric field, and theta (0 <= theta <= pi/2) is the angle between the axis of the dipole and the electric field vector. In this case, |T| = (3.4*10^-30 C*m) * (2.0*10^6 N/C) * sin(pi/4) |T| = [(6.8*10^-24)/sqrt(2)] N*m = 4.8*10^-24 N*m