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Of course, other groups are also of interest. Carbon, silicon, and germanium, for example, have similar chemistries and are in Group 4 (Group IV). Carbon, in particular, is extraordinary in its ability to form many types of bonds and to be part of long chains, such as inorganic molecules. The large group of what are called transitional elements is characterized by the filling of the d size 12{d} {} subshells and crossing of energy levels. Heavier groups, such as the lanthanide series, are more complex—their shells do not fill in simple order. But the groups recognized by chemists such as Mendeleev have an explanation in the substructure of atoms.

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Section summary

  • The state of a system is completely described by a complete set of quantum numbers. This set is written as n, l, m l , m s .
  • The Pauli exclusion principle says that no two electrons can have the same set of quantum numbers; that is, no two electrons can be in the same state.
  • This exclusion limits the number of electrons in atomic shells and subshells. Each value of n size 12{n} {} corresponds to a shell, and each value of l size 12{l} {} corresponds to a subshell.
  • The maximum number of electrons that can be in a subshell is 2 2 l + 1 size 12{2 left (2l+1 right )} {} .
  • The maximum number of electrons that can be in a shell is 2 n 2 size 12{2n rSup { size 8{2} } } {} .

Conceptual questions

Identify the shell, subshell, and number of electrons for the following: (a) 2 p 3 size 12{2p rSup { size 8{3} } } {} . (b) 4 d 9 size 12{4d rSup { size 8{9} } } {} . (c) 3 s 1 size 12{3s rSup { size 8{1} } } {} . (d) 5 g 16 size 12{5g rSup { size 8{"16"} } } {} .

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Which of the following are not allowed? State which rule is violated for any that are not allowed. (a) 1 p 3 size 12{1p rSup { size 8{3} } } {} (b) 2 p 8 size 12{2p rSup { size 8{8} } } {} (c) 3 g 11 size 12{3g rSup { size 8{"11"} } } {} (d) 4 f 2 size 12{4f rSup { size 8{2} } } {}

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Problem exercises

(a) How many electrons can be in the n = 4 size 12{n=4} {} shell?

(b) What are its subshells, and how many electrons can be in each?

(a) 32. (b) 2 in s , 6 in p , 10 in d , and 14 in f size 12{f} {} , for a total of 32.

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(a) What is the minimum value of 1 for a subshell that has 11 electrons in it?

(b) If this subshell is in the n = 5 shell, what is the spectroscopic notation for this atom?

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(a) If one subshell of an atom has 9 electrons in it, what is the minimum value of l size 12{l} {} ? (b) What is the spectroscopic notation for this atom, if this subshell is part of the n = 3 size 12{n=3} {} shell?

(a) 2

(b) 3 d 9 size 12{3d rSup { size 8{9} } } {}

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(a) List all possible sets of quantum numbers n , l , m l , m s for the n = 3 shell, and determine the number of electrons that can be in the shell and each of its subshells.

(b) Show that the number of electrons in the shell equals 2 n 2 size 12{2n rSup { size 8{2} } } {} and that the number in each subshell is 2 2 l + 1 size 12{2 left (2l+1 right )} {} .

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Which of the following spectroscopic notations are not allowed? (a) 5 s 1 (b) 1 d 1 (c) 4 s 3 (d) 3 p 7 (e) 5 g 15 . State which rule is violated for each that is not allowed.

(b) n l is violated,

(c) cannot have 3 electrons in s subshell since 3 > ( 2 l + 1 ) = 2

(d) cannot have 7 electrons in p subshell since 7 > ( 2 l + 1 ) = 2 ( 2 + 1 ) = 6

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Which of the following spectroscopic notations are allowed (that is, which violate none of the rules regarding values of quantum numbers)? (a) 1 s 1 size 12{1s rSup { size 8{1} } } {} (b) 1 d 3 size 12{1d rSup { size 8{3} } } {} (c) 4 s 2 size 12{4s rSup { size 8{2} } } {} (d) 3 p 7 size 12{3p rSup { size 8{7} } } {} (e) 6 h 20 size 12{6h rSup { size 8{"20"} } } {}

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(a) Using the Pauli exclusion principle and the rules relating the allowed values of the quantum numbers n , l , m l , m s size 12{ left (n,`l,`m rSub { size 8{l} } ,`m rSub { size 8{s} } right )} {} , prove that the maximum number of electrons in a subshell is 2 n 2 size 12{2n rSup { size 8{2} } } {} .

(b) In a similar manner, prove that the maximum number of electrons in a shell is 2 n 2 .

(a) The number of different values of m l size 12{m rSub { size 8{l} } } {} is ± l , ± ( l 1 ) , ..., 0 for each l > 0 size 12{l>0} {} and one for l = 0 ( 2 l + 1 ) . size 12{l=0 drarrow \( 2l+1 \) "." } {} Also an overall factor of 2 since each m l size 12{m rSub { size 8{l} } } {} can have m s size 12{m rSub { size 8{s} } } {} equal to either + 1 / 2 size 12{+1/2} {} or 1 / 2 2 ( 2 l + 1 ) size 12{ - 1/2 drarrow 2 \( 2l+1 \) } {} .

(b) for each value of l size 12{l} {} , you get 2 ( 2 l + 1 ) size 12{2 \( 2l+1 \) } {}

= 0, 1, 2, ..., ( n –1 ) 2 ( 2 ) ( 0 ) + 1 + ( 2 ) ( 1 ) + 1 + . . . . + ( 2 ) ( n 1 ) + 1 = 2 1 + 3 + . . . + ( 2 n 3 ) + ( 2 n 1 ) n terms size 12{ {}=0, 1," 2, " "." "." "." ", " \( "n–1" \) drarrow 2 left lbrace left [ \( 2 \) \( 0 \) +1 right ]+ left [ \( 2 \) \( 1 \) +1 right ]+ "." "." "." "." + left [ \( 2 \) \( n - 1 \) +1 right ] right rbrace = {2 left [1+3+ "." "." "." + \( 2n - 3 \) + \( 2n - 1 \) right ]} underbrace { size 8{n" terms"} } } {} to see that the expression in the box is = n 2 , imagine taking ( n 1 ) size 12{ \( n - 1 \) } {} from the last term and adding it to first term = 2 1 + ( n –1 ) + 3 + . . . + ( 2 n 3 ) + ( 2 n 1 ) ( n 1 ) = 2 n + 3 + . . . . + ( 2 n 3 ) + n . size 12{ {}=2 left [1+ \( n"–1" \) +3+ "." "." "." + \( 2n - 3 \) + \( 2n - 1 \) – \( n - 1 \) right ]=2 left [n+3+ "." "." "." "." + \( 2n - 3 \) +n right ]"." } {} Now take ( n 3 ) size 12{ \( n - 3 \) } {} from penultimate term and add to the second term 2 n + n + . . . + n + n n terms = 2 n 2 size 12{2 { left [n+n+ "." "." "." +n+n right ]} underbrace { size 8{n" terms"} } =2n rSup { size 8{2} } } {} .

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Questions & Answers

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Ganesh Reply
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Timothy Reply
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Timothy
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Timothy
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Timothy
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Mbutene
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Stephen
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MITHRA Reply
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what's critical angle
Mahmud Reply
The Critical Angle Derivation So the critical angle is defined as the angle of incidence that provides an angle of refraction of 90-degrees. Make particular note that the critical angle is an angle of incidence value. For the water-air boundary, the critical angle is 48.6-degrees.
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Chidalu
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Temiloluwa
the ratio of the density of a substance to the density of a standard, usually water for a liquid or solid, and air for a gas.
Chidalu
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aliyu Reply
mass ×velocity
Chidalu
it is the product of mass ×velocity of an object
Chidalu
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Sean Reply
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Practice Key Terms 4

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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