Orbital-shape quantum number

The second quantum number describes an orbital s shape, and is a positive integer that ranges in value from 0 to (n - 1). Chemists use a variety of names for the second quantum number. For example, you may see it referred to as the angular momentum quantum number, the azimuthal quantum number, the secondary quantum number, or the orbital-shape quantum number. 

Regardless of its name, the second quantum number refers to the energy sublevels within each principal energy level. The name that this hook uses for the second quantum number is orbital-shape quantum number (i), to help you remember that the value of 1 determines orbital shape. (You will see examples of orbital shapes near the end of this section.)... 

The principal quantum number n is the most important determinant of the radius and energy of the electron atomic orbital. The orbital shape quantum number I determines the shape of the atomic orbital. When / = 1, the atomic orbital is called an s orbital there are two s orbitals for each value of n, and they are spherically symmetric in space around the nucleus. When I = 2, the orbitals are called the p orbitals there are six p orbitals, and they have a dumbbell shape of two lobes that are diametrically opposed. When I = 3 and 4, we have 10 d orbitals and 14 f orbitals. The orbital orientation quantum number m controls the orientation of the orbitals. For the simplest system of a single electron in a hydrogen atom, the most stable wave function Is has the following form ... 

Sometimes / is referred to as the orbital-shape quantum number. 

I is the orbital shape quantum number (see Chapter 3). The spin magnetic moment of an electron is... 

In addition to size, an atomic orbital also has a specific shape. The solutions for the Schrodinger equation and experimental evidence show that orbitals have a variety of shapes. A second quantum number indexes the shapes of atomic orbitals. This quantum number is the azimuthal quantum number (1). 

Orbitals have a variety of different possible shapes. Therefore, scientists use three quantum numbers to describe an atomic orbital. One quantum number, n, describes an orbital s energy level and size. A second quantum number, I, describes an orbital s shape. A third quantum number, mi, describes an orbital s orientation in space. These three quantum numbers are described further below. The Concept Organizer that follows afterward summarizes this information. (In section 3.3, you will learn about a fourth quantum number, mg, which is used to describe the electron inside an orbital.)... 

The orbital angular momentum quantum number, , determines, as you might guess, the angular momentum of the electron as it moves in its orbital. This quantum number tells us the shape of the orbital, spherical or whatever. The values that can take depend on the value of n can have any value from 0 up to - 1 — 0,... 

All this explains why the shape of an orbital depends on the orbital angular quantum number, t. All s orbitals ( = 0) are spherical, all p orbitals ( - 1) are shaped like a figure eight, and d orbitals ( = 2) are yet another different shape. The problem is that these probability density plots take a long time to draw—organic chemists need a simple easy way to represent orbitals. The contour diagrams were easier to draw but even they were a little tedious. Even simpler still is to draw just one contour within which there is, say, a 90% chance of finding the electron. This means that all s orbitals can be represented by a circle, and all p orbitals by a pair of lobes. 

Thinking it Through For the second period, the principal quantum number, n, is equal to 2. For a principal quantum number of 2, the angular momentum (shape) quantum number, I, has allowed values of 0 or 1. The letter s is associated with / = 0, and the letter p is associated with / = 1. Choice (B) cannot be correct, since only s and p orbitals arc possible when n = 2. Choice (D) is eliminated because the 2s orbital mu.st fill before the 2p orbital. The 2s orbital has room for two electrons, which must be occupied before the 2p orbital begins to fill. Choice (A) is eliminated because the order of filling is incorrect. Choice (C) has the correct orbitals present for a second period... 

The kinetic energy operator,however,is almost separable in spherical polar coordinates, and the actual method of solving the differential equation can be found in a number of textbooks. The bound solutions (negative total energy) are called orbitals and can be classified in terms of three quantum numbers, n, I and m, corresponding to the three spatial variables r, d and q>. The quantum numbers arise from the boundary conditions on the wave function, i.e. it must be periodic in the 0 and q> variables, and must decay to zero as r oo. Since the Schrodinger equation is not completely separable in spherical polar coordinates, there exist the restrictions n > /> m. The n quantum number describes the size of the orbital, the / quantum number describes the shape of the orbital, while the m quantum number describes the orientation of the orbital relative to a fixed coordinate system. The / quantum number translates into names for the orbitals ... 

An s orbital is spherically symmetrical and can contain a maximum of two electrons with opposed spins. A p orbital has a solid figure-of-eight shape there are three equivalent p orbitals for each principal quantum number they correspond to the three axes of rectangular coordinates. 

The d and f orbitals have more complex shapes there are five equivalent d orbitals and seven equivalent f orbitals for each principal quantum number, each orbital containing a maximum of 2 electrons with opposed spins. 

Cartesian Gaussian-type orbitals (GTOs) Jfa.i.f( ( characterized by the quantum numbers a, b and c, which detail the angular shape and direction of the orbital, and the exponent a which governs the radial size . 

Orbitals are described by specifying their size shape and directional properties Spherically symmetrical ones such as shown m Figure 1 1 are called y orbitals The let ter s IS preceded by the principal quantum number n n = 2 3 etc ) which speci ties the shell and is related to the energy of the orbital An electron m a Is orbital is likely to be found closer to the nucleus is lower m energy and is more strongly held than an electron m a 2s orbital... 

The arrangement of electrons in an atom is described by means of four quantum numbers which determine the spatial distribution, energy, and other properties, see Appendix 1 (p. 1285). The principal quantum number n defines the general energy level or shell to which the electron belongs. Electrons with n = 1.2, 3, 4., are sometimes referred to as K, L, M, N,. .., electrons. The orbital quantum number / defines both the shape of the electron charge distribution and its orbital angular... 

The Schrodinger equation can be solved approximately for atoms with two or more electrons. There are many solutions for the wave function, ij/, each associated with a set of numbers called quantum numbers. Three such numbers are given the symbols n, , and mi. A wave function corresponding to a particular set of three quantum numbers (e.g., n = 2, = 1, mi = 0) is associated with an electron occupying an atomic orbital. From the expression for ij/y we can deduce the relative energy of that orbital, its shape, and its orientation in space. 

Although it is not shown in Figure 6.7, p orbitals, like s orbitals, increase in size as the principal quantum number n increases. Also not shown are the shapes and sizes of d and f orbitals. We will say more about the nature of d orbitals in Chapter 15. 

The expressions for a number of other atomic orbitals are shown in Table 1.2a (for R) and Table 1.2b (for Y). To understand these tables, we need to know that each wavefunction is labeled by three quantum numbers n is related to the size and energy of the orbital, / is related to its shape, and mt is related to its orientation in space. 

The value of / correlates with the number of preferred axes in a particular orbital and thereby identifies the orbital shape. According to quantum theoiy, orbital shapes are highly restricted. These restrictions are linked to energy, so the value of the principal quantum number ( ) limits the possible values of /. The smaller U is, the more compact the orbital and the more restricted its possible shapes ... 

Among atomic orbitals, s orbitals are spherical and have no directionality. Other orbitals are nonspherical, so, in addition to having shape, every orbital points in some direction. Like energy and orbital shape, orbital direction is quantized. Unlike footballs, p, d, and f orbitals have restricted numbers of possible orientations. The magnetic quantum number (fflj) indexes these restrictions. 

The quantum numbers and / determine the size and shape of an orbital. As a increases, the size of the orbital Increases, and as / Increases, the shape of the orbital becomes more elaborate. 

The quantum number / — 1 corresponds to a p orbital. A p electron can have any of three values for Jitt/, so for each value of tt there are three different p orbitals. The p orbitals, which are not spherical, can be shown in various ways. The most convenient representation shows the three orbitals with identical shapes but pointing in three different directions. Figure 7-22 shows electron contour drawings of the 2p orbitals. Each p orbital has high electron density in one particular direction, perpendicular to the other two orbitals, with the nucleus at the center of the system. The three different orbitals can be represented so that each has its electron density concentrated on both sides of the nucleus along a preferred axis. We can write subscripts on the orbitals to distinguish the three distinct orientations Px, Py, and Pz Each p orbital also has a nodal plane that passes through the nucleus. The nodal plane for the p orbital is the J z plane, for the Py orbital the nodal plane is the X Z plane, and for the Pz orbital it is the Jt plane. 

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