quantum numbers

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The electron capacity of each shell is indicated by 2n2 f or a given n For each solution to the quantum mechanical equation, we can calculate the electron probability density (sometimes just called the electron density) at each point in the atom. This is the probability of finding an electron at that point. It can be shown that this electron density is proportional to r2ψ2, where r is the distance from the nucleus. The electron probability density at a given distance from the nucleus is plotted against distance from the nucleus, for s orbitals. It is found that the electron probability density curve is the same regardless of the direction in the atom. An electron cloud surrounding an atomic nucleus. The electron density drops off rapidly but smoothly as distance from the nucleus increases.

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Page 1: Quantum Numbers

The electron capacity of each shell is indicated by 2n2 f or a given n

For each solution to the quantum mechanical equation, we can calculate the electronprobability density (sometimes just called the electron density) at each point in the atom.This is the probability of finding an electron at that point. It can be shown that this electrondensity is proportional to r2ψ2, where r is the distance from the nucleus.

The electron probability density at a given distance from the nucleus is plotted against distance from the nucleus, for s orbitals. It is found that the electron probability density curve is the same regardless of the direction in the atom.

An electron cloud surrounding anatomic nucleus. The electron density drops off rapidlybut smoothly as distance from the nucleus increases.

Page 2: Quantum Numbers

Plots of the electron density distributions associated with s orbitals. For any s orbital, this plot is the same in any direction (orbital is spherically symmetrical). The sketch below each plot shows a cross section, in the plane of the atomic nucleus, of the electron cloud associated with that orbital. Electron density is proportional to r2c2. s orbitals with n . 1 have n 2 1 regions where the electron density drops to zero. These are indicated by blue vertical lines on the plots, indicating where the electron density drops to zero (nodes), and by the white regions on the cross section.

Page 3: Quantum Numbers

Two representations of the hydrogen1s, 2s, and 3s orbitals. (a) The electronprobability distribution. (b) Thesurface that contains 90% of the totalelectron probability (the size of theorbital, by definition).

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s orbital as spherically symmetrical; that is, it is round like a basketball .The electron clouds (electron densities) associated with the 1s, 2s, and 3s atomic orbitals are shown just below the plots

The shape of an s orbital

P ORBITAL:

Beginning with the second shell, each shell also contains a p subshell, defined by , l=1. Each of these subshells consists of a set of three p atomic orbitals, corresponding to thethree allowed values of m, (-1, 0, and +1) when , l= 1. The sets are referred to as 2p,3p, 4p, 5p, . . . orbitals to indicate the main shells in which they are found. Each set ofatomic p orbitals resembles three mutually perpendicular equal-arm flattened dumbbells.

Each p orbital has a nodal plane (blue lines/planes in Figure 4-24) in which there is zero probability of finding the electron. The two regions in which the electron probability is nonzero are on opposite sides of this nodal plane and are referred to as the lobes of the p orbital. Each electron in a p orbital has equal probability of being in either lobe. In the two lobes, the wave c represents that the electron has opposite phases, corresponding to the crests and troughs of water waves in Figure 4-12. These phases correspond to mathematical wave functions with positive and negative signs, but these signs do not represent charges. The nucleus defines the origin of a set of Cartesian coordinates with the usual x, y, and z axes (see Figure 4-25a). The subscript x, y, or z indicates the axis along which each of the orbitals is directed. A set of three p atomic orbitals may be represented as in Figure 4-25b.

Page 5: Quantum Numbers

Figure 4-24 Representations of the shapeof a p orbital. The nodal plane for thisorbital is shown (edge on) as a blue line inthese drawings. (a) A probability diagram,where the density of dots signifies theprobability of finding the electron in thatregion. (b) The shape of this p orbital. Thetwo lobes are shown in different colors torepresent the different mathematical phases(+ and –) of the wave function. (c) A plotof electron probability along the axis ofmaximum electron density for this orbital.A plot along any other direction would bedifferent because a p orbital is not sphericallysymmetrical like an s orbital.

Page 6: Quantum Numbers

Representation of the 2p orbitals. (a) The electron probability distribution for a 2p orbital. (b) The boundary surface representations of all three 2p orbitals. Note that the signs inside the surface indicate the phases (signs) of the orbital in that region of space.

Page 7: Quantum Numbers

A cross-section of the electron probabilitydistribution for a 3p orbital.

all orbitals with the same value of n have the same energy—they are said to be degenerate.

Page 8: Quantum Numbers

Beginning at the third shell, each shell also contains a third subshell (, 5 2) composedof a set of five d atomic orbitals (m, 5 22,21, 0,11,12) . They are designated 3d, 4d,5d, . . . to indicate the shell in which they are found. The shapes of the members of a set areindicated in Figure 4-26.

Page 9: Quantum Numbers

Representation of the 3d orbitals. (a) Electron density plots of selected 3d orbitals. on (b) The boundary surfaces of all five 3d orbitals, with the signs (phases) indicated.

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spin quantum number, ms. Because ms has two possible values, 112and 212 , each atomic orbital, defined by the values of n, ,, and m,, has a capacity of two electrons. Electrons are negatively charged, and they behave as though they were spinning about axes through their centers, so they act like tiny magnets. The motions of electrons produce magnetic fields, and these can interact with one another. Two electrons in the same orbital having opposite ms values are said to be spin-paired, or simply paired (Figure 4-29).

Let’s summarize, in tabular form, some of the information we have developed to this point. The principal quantum number n indicates the main shell.

The number of subshells per shell is equal to n, the number of atomic orbitals per shell is n2, and the maximum number of electrons per shell is 2n2, because each atomic orbital can hold two electrons.

Page 14: Quantum Numbers

Electron spin. Electrons act as though they spin about an axis through their centers. Becauseelectrons may spin in two directions, the spin quantum number has two possible values, 112 and 2 12 , sometimes referred to as “spin up” or “spin down.”

Each electron spin produces a magnetic field. When two electrons have opposite spins, the attraction due to their opposite magnetic fields (gray arrows) helps to overcome the repulsion of their like charges. This permits two electrons to occupy the same region (orbital).

Page 15: Quantum Numbers

Orbital energy levels for the hydrogen atom.

This is not the case for polyelectronic atoms, where we fi nd that for a given principal quantum level the orbitals vary in energy as follows: Ens < Enp < End< Enf

In other words, when electrons are placed in a particular quantum level, they “prefer”the orbitals in the order s, p, d, and then f.

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