Quantum number permitted values

The correct answer is (B). Choice (B) is an impossible configuration because of the value of the 1 quantum number. The value of 1 is only permitted to go from n-1 to 0. Therefore, 1 cannot have the same value as n. 

The permitted values for the other quantum numbers when n = 2 are shown in Table 17-2. The rules governing the assignment of quantum numbers are compared to a more familiar situation in Fig. 17-5. The following examples will illustrate the limitations on the values of the quantum numbers (Table 17-1). 

Table 17-2 Permitted Values of Quantum Numbers when n = 2...

EXAMPLE 17.3. What values of / are permitted for an electron with principal quantum number n = 3 ... 

Depending on the permitted values of the magnetic quantum number m, each subshell is further broken down into units called orbitals. The number of orbitals per subshell depends on the type of subshell but not on the value of n. Each orbital can hold a maximum of two electrons hence, the maximum number of electrons that can occupy a given subshell is determined by the number of orbitals available. These relationships are presented in Table 17-5. The maximum number of electrons in any given energy level is thus determined by the subshells it contains. The first shell can contain 2 electrons the second, 8 electrons the third, 18 electrons the fourth, 32 electrons and so on. 

Quantum number Symbol Parameter specified Permitted values... 

When dealing with atoms possessing more than one electron it is necessary to introduce a fourth quantum number s, the spin quantum number. However, to take into account the intrinsic energy of an electron, the value of 5 is taken to be A. Essentially the intrinsic energy of the electron may interact in a quantized manner with that associated with the angular momentum represented by /, such that the only permitted interactions are l + s and / - s. For atoms possessing more than one electron it is necessary to specify the values of s with respect to an applied magnetic field these are expressed as values of ra of + A or - A. 

A Two, because the only ways of producing unique values of the four quantum numbers (i.e. including ms) are those of the three in the question associated with the two permitted values of + /> and... 

A Six, because there are three possible values of the m( quantum number, all being associated with the two permitted ms values of + A and -A. 

N electrons with the same values of quantum numbers n,7 (LS coupling) or tijljji (jj coupling) are called equivalent. The corresponding configurations will be denoted as nlN (a shell) or nljN (a subshell). A number of permitted states of a shell of equivalent electrons are restricted by the Pauli exclusion principle, which requires antisymmetry of the wave function with respect to permutation of the coordinates of the electrons. 

As discussed in Section 5.1, the structure of many-electron atoms can be understood only by assuming that no more than two electrons can occupy each separate orbital. Taking account of the electron spin allows a deeper interpretation of this fact. One way of expressing the Pauli exclusion principle is no two electrons can have the same values of all four quantum numbers, n, l, m, and ms. As only two values of ms are permitted, it follows that each orbital, specified by a given set of values of n, l, and m, can hold... 

QUANTUM NUMBER SYMOOL PARAMETER SPECIFIED PERMITTED VALUES... 

A. What are the permitted values for the principal quantum number, n ... 

A lithium atom has three electrons. The hrst two of these can have the same sets of quantum numbers as the two electrons of hehum. What should the set of quantum numbers for the third electron be We cannot choose the lowest permitted value for n, which is 1, because C and m would then both be 0. If we choose -j as the value of m, the third electron would have a set of quantum numbers exactly the same as that of one of the hrst two electrons, and if we choose the value = +5, the third electron would have the same set of quantum numbers as the other. Because neither of these situations is permitted by the Pauli principle, n cannot be 1 for the third electron. We must choose the next higher value, = 2. With = 2, the permitted values of are 0 and 1. Because = 0 wUl give a lower value for the sum + , we choose that value for . With = 0, must be 0, and we can choose either 5 or +5 for m. The quantum numbers for the three electrons of the Uthium atom can thus be as follows ... 

The 77 + f rule, the Pauli exclusion principle, and the permitted values of the quantum numbers enable us to determine the order of the electrons in an atom in inereasing energy. 

Even though the rrif and values do not affect the energy of the electron, it is still important to learn about them. The number of combinations of permitted values of these quantum numbers determines the maximum number of electrons in a given type of subshell. For example, in a subshell for which = 2, nif can have five different values ( 2, -1,0, +1, and +2), and can have two different values (—5 and +5). The ten different combinations of and rris allow a maximum of ten electrons in any subshell for which = 2. 

The second electron also can have n = 1, Z =0, and m = 0. Its value of can be either + or — but not the same as that for the first electron. If it were, this second electron would have the same set of four quantum numbers that the first electron has, which is not permitted by the Pauli principle. If we were to try to give the third electron the same values for the first three quantum numbers, we would be stuck when we came to assign the m, value. Both + j and — j have already been used, and we would have a duplicate set of quantum numbers for two electrons, which is not permitted. We cannot use any other values for Z or m with the value of n = 1, and so the third electron must have the next-higher n value, n = 2. The Z values could be 0 or 1, and since 0 will give a lower n + l sum, we choose that value for the third electron. Again the value of m must be 0 since Z = 0, and rris can have a value — (or + j). For the fourth electron, n = 2,1 = 0, nii = 0, and mj = + (or —j if the third were + ). The fifth electron can have n = 2 but not 1=0, since all combinations of n = 2 and 1=0 have been used. Therefore, n = 2,1 =, nii = —, and nis = — are assigned. The rest of the electrons in the aluminum atom are assigned quantum numbers somewhat arbitrarily as shown in Table 4-3. 

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