Quantum Numbers of an Atomic Orbital

So far we have discussed the electron density for the ground state of the H atom. When the atom absorbs energy, it exists in an excited state and the region of space occupied by the electron is described by a different atomic orbital (wave function). As you ll see, each atomic orbital has a distinctive radial probability distribution and 90% probability contour. 

An atomic orbital is specified by three quantum numbers. One is related to the orbital s size, another to its shape, and the third to its orientation in space. The quantum numbers have a hierarchical relationship the size-related number limits the shape-related number, which limits the orientation-related number. Let s examine this hierarchy and then look at the shapes and orientations. 

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. 

The magnetic quantum number (mi) is an integer from -I through 0 to +1. It prescribes the orientation of the orbital in the space around the nucleus and is sometimes called the orbital-orientation quantum number. The possible values of an orbital s magnetic quantum number are set by its angular momentum quantum number that is, I sets the possible values of m,. An orbital with I = 0 can have only ni/ = 0. However, an orbital with / = 1 can have any one of three m/ values, - 1, 0, or -fl thus, there are three possible orbitals with / = 1, each with its own orientation. Note that the number of possible m/ values equals the number of orbitals, which is 2/ -f 1 for a given I value. 

Check Table 7.2 shows that we are correct. The total number of orbitals for a given n value is n, and for n = 3, = 9. 

SAMPLE PROBLEM 7.4 Determining Quantum Numbers for an Energy Level 

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If the n quantum number of an atomic orbital is 4, what are the possible values of 11 If the I quantum number is 3, what are the possible values of m ... 

Which of the following are allowable sets of quantum numbers for an atomic orbital Explain your answer in each case. 

Alagnetic quantum number ntf) (6.4) One of the three quantum numbers identifying an atomic orbital. Specifies the spatial orientation of an orbital. Allowed values of m( are related to the value of the secondary quantum number ( ). 

For some given values for the L and S quantum numbers of an atomic state, what is the largest possible spin-orbit interaction energy Express this in terms of the parameter y in Equation 10.24. [Flint Consider how / in nation 10.24 is related to the L and S values and attempt to maximize Espin orbit J... 

The hierarchy of shells, subshells, and orbitals is summarized in Fig. 1.30 and Table 1.3. Each possible combination of the three quantum numbers specifies an individual orbital. For example, an electron in the ground state of a hydrogen atom has the specification n = 1, / = 0, nij = 0. Because 1=0, the ground-state wavefunction is an example of an s-orbital and is denoted Is. Each... 

It is possible to distinguish atoms by writing sets of quantum numbers for each of their electrons. However, writing quantum numbers for an atom such as uranium, which has 92 electrons, would be mind-bogglingly tedious. Fortunately, chemists have developed a shortcut to represent the number and orbital arrangements of electrons in each atom. As you will see shortly, these electron configurations, as they are called, are intimately connected to the structure and logic of the periodic table. 

Principal quantum number, n An integer that specifies the quantized energy level of an atomic orbital. 

Angular momentum quantum number ( ) the quantum number relating to the shape of an atomic orbital, which can assume any integral value from 0 to n — 1 for each value of n. (12.9) Anion a negative ion. (2.7)... 

The quantum number n is primarily responsible for determining the overall energy of an atomic orbital the other quantum numbers have smaller effects on the energy. The quantum number I determines the angular momentum of the orbital or shape of the orbital and has a smaller effect on the energy. The quantum number m/ determines the orientation of the angular momentum vector in a magnetic field, or the position of the orbital in space, as shown in Table 2-3. The quantum number ntg determines the orientation of the electron... 

The electron s wave function (iK atomic orbital) is a mathematical description of the electron s wavelike behavior in an atom. Each wave function is associated with one of the atom s allowed energy states. The probability density of finding the electron at a particular location is represented by An electron density diagram and a radial probability distribution plot show how the electron occupies the space near the nucleus for a particular energy level. Three features of an atomic orbital are described by quantum numbers size (n), shape (/), and orientation (m/). Orbitals with the same n and / values constitute a sublevel sublevels with the same n value constitute an energy level. A sublevel with / = 0 has a spherical (s) orbital a sublevel with / = 1 has three, two-lobed (p) orbitals and a sublevel with / = 2 has five, multi-lobed (d) orbitals. In the special case of the H atom, the energy levels depend on the n value only. 

Recall from Chapter 7 that the three quantum numbers n, I, and m, describe the size (energy), shape, and orientation, respectively, of an atomic orbital. However, an additional quantum number is needed to describe a property of the electron itself, called spin, which is not a property of the orbital. Electron spin becomes important when more than one electron is present. 

Table 1.4 summarizes the relationship between quantum numbers and hydrogenlike atomic orbitals. When Z = 0, (2/ + 1) = 1 and there is only one value of m/, so we have an s orbital. When / = 1, (2/ + 1) = 3, so there are three values of m , giving rise to three p orbitals, labeled p, Py, and p. When / = 2, (2Z + 1) = 5, so there are five values of mi, and the corresponding five d orbitals are labeled with more elaborate subscripts. In the following sections we discuss the s, p, and d orbitals separately. 

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