Orbital-shape quantum

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). 

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. 

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 first shell or energy level out from the nucleus is called the K shell or energy level and contains a maximum of two electrons in the s orbital— that is, K = s2, where the K represents the shell number (or principle quantum number), the s describes the orbital shape of the angular momentum quantum number, and the 2 is the maximum number of electrons that the s orbital can contain. This particular sequence is K = s2, which means K shell contains 2 electrons in the s orbital. This is the sequence for the element helium. Look up helium in the text for more information. 

The second quantum number describes the shape of the orbital as s, p, d, f or g. These shapes do not describe the electron s path but rather are mathematical models showing the probability of the electron s location. The s and p orbital shapes are shown in Figure 8.9, but descriptions of the d and f orbitals are reserved for more advanced texts. 

Azimuthal quantum number (1)—This number describes the shape of the orbital. The azimuthal quantum number can have values, from 0 to n-1, and these values correspond to certain orbital shapes. While the value can theoretically have a value as high as 6, we will see later that no values higher than 3 are found. The values that do exist, 0, 1, 2, and 3, correspond to particular shapes and are commonly designated as s, p, d, and /orbitals, respectively. In our house analogy, this quantum number would correspond roughly to the City. That is, it is a bit more specific than the State, but it still doesn t tell us exactly where the house is. 

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. 

PRINCIPAL QUANTUM NUMBER (/j) (Denotes shell) ANGULAR MOMENTUM QUANTUM NUMBER ( ) (Denotes subshell) ORBITAL SHAPE DESIGNATION MAGNETIC QUANTUM NUMBER (m ) NUMBER OF ORBITALS... 

The magnetic quantum number determines how the s, p, d, and / orbitals are oriented in space. The shapes of the first three s orbitals are shown in Figure 2.2. The orbitals are spherical, with the lower-energy orbitals nested inside the higher-energy orbitals. Figure 2.3 shows the p and d orbitals. The p orbitals are dumbbell shaped, and all but one of the d orbitals have four lobes. The orbital shapes represent electron probabilities. The shaded areas are regions where an electron is most likely to be found. 

All of the information that was used in the argument to derive the >2/1 arrangement of nuclei in ethylene is contained in the molecular wave function and could have been identified directly had it been possible to solve the molecular wave equation. It may therefore be correct to argue [161, 163] that the ab initio methods of quantum chemistry can never produce molecular conformation, but not that the concept of molecular shape lies outside the realm of quantum theory. The crucial structure-generating information carried by orbital angular momentum must however, be taken into account. Any quantitative scheme that incorporates, not only the molecular Hamiltonian, but also the complex phase of the wave function, must produce a framework for the definition of three-dimensional molecular shape. The basis sets of ab initio theory, invariably constructed as products of radial wave functions and real spherical harmonics [194], take account of orbital shape, but not of angular momentum. 

The second quantum number is the azimuthal quantum number, t The azimuthal quantum number designates the subshell. These are the orbital shapes with which we are familiar such as s, p, d, and f. If C = 0, we are in the s subshell if = 1, we are in the p subshell and so on. For each new shell, there exists an additional subshell with the azimuthal quantum number f = it -1. Each subshell has a peculiar shape to its orbitals. The shapes are based on probability functions of the position of the electron. There is a 90% chance of finding the electron somewhere inside a given shape. You should recognize the shapes of the orbitals in the s and p subshells. 

The angular momentum quantum number (/) is an integer from 0 to n — 1. It is related to the shape of the orbital and is sometimes called the orbital-shape (or azimuthal) quantum number. Note that the principal quantum number sets a limit on the values for the angular momentum quantum number that is, n limits 1. For an orbital with n =, I can have a value of only 0. For orbitals with n = 2, I can have a value of 0 or 1 for those with n = 3, I can be 0, 1, or 2 and so forth. Note that the number of possible I values equals the value of n. 

Ans. Three new quantum numbers appear, which characterize atomic structure in finer detail. These are /, the subshell number, a quantum number which specifies orbital shape, m, which specifies the orbitals orientation in space, and s, the spin quantum number, which describes the fact that electrons appear to rotate or spin on their axes. 

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