Question

One facet of the central field approximation for many-electron atoms is that inner-shell electrons screen the nuclear charge. To understand how this works quantitatively, first note that the probability distributions for electrons in different shells generally do not overlap much. For instance, the electrons in the M shell (n=3) are almost always farther from the nucleus than the electrons of the K (n=1) and L (n=2) shells. Thus, it is a good approximation to assume that the inner shells completely screen the nucleus from the outer shells. For example, if there are ten electrons altogether in the K and L shells of an atom, then the electrons in the M shell experience force from a charge of roughly Z−10, where Z is the charge on the nucleus as an integer multiple of e, the magnitude of the charge on an electron. This is called the effective nuclear charge Zeff. Part A In a beryllium atom (Z=4), how many electrons are in the K shell? Part B In xenon (Z=54), what is the effective charge Zeff experienced by an electron in the M (n=3) shell? Part C How many electrons are there altogether in the K, L, and M shells of xenon? Recall that for n=3, the orbital quantum number l must be zero, one, or two and that ml can take any value between postive and negative l. Part D The energy for the 5p valence electron in indium (Z=49) is −5.79 electron volts. What is the effective nuclear charge Zeff experienced by this electron?

Answer 1

Part A

In a beryllium atom (Z=4), how many electrons are in the K shell?

Electronic configuration of Be (At.No. 4) : 1s2 2s2

K shell is the innermost i.e. first shell form nucleus and here there are 2 electrons in K-shell.

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Part B

Let us write the electronic configuration of Xe (Z=54),

E.C. of Xe (Z=54): 1s2,2s2,2p6,3s2,3p6,4s2,3d10,4p6,4d10,5s2,5p6.

K and L shell electrons i.e. total of 10 electrons completely screen the nucleus from the outer M shell.

Hence,

Zeff = Z – S

Z = Number of protons in an atom (atomic number)

S= Number of inner shell electrons shielding the given outer shell electrons.

In the given case,

Z = 54 atomic number of Xe,

S = 10 i.e. electrons in K and L shell.

Hence for Zeff experienced by an electron in M shell ia,

Zeff = 54-10

Zeff = 44.

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Part C

Form the electronic configuration for Xe its clear that K shell (n=1) contains 2 electrons, L shell (n=2) contains 8 electrons and M shell (n=3) contains 18 electrons.

Hence altogether in K,L,M shell there are 2+8+18 = 28 electrons.

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Part D

Given data: for 5p valence shell n= 5, and E = -5.79 eV.

E = -5.79 eV

E = -5.79 x 96.485 kJ/mol ……. (since, 1eV = 96.485 kJ/mol)

E = -558.65 kJ/mol.

Z_eff =?, for n=5,

The relation between Zeff. And Energy is given as,

E = -(Z_eff²/n²) x (1310) kJ/mol

Let us put all known values and solve it for Zeff.

-555.65 kJ/mol= -(Z_eff²/5²) x 1310 kJ/mol

555.65 = (Z_eff²/25) x 1310 kJ/mol.

Z²_eff = (555.65x25) / 1310

Z²_eff = 10.60

Z_eff = 3.256

Hence Zeff for 5p electrons is 3.256.

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