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Chapter 5: Q14P (page 218)

The ground state of dysprosium (element 66, in the 6th row of the Periodic Table)

is listed as Is5. What are the total spin, total orbital, and grand total angular

momentum quantum numbers? Suggest a likely electron configuration for

dysprosium.

Short Answer

Expert verified

The total spin, orbital spin and total angular momentum are:

S=2;L=6;J=8.

Electronic configuration:

(1s)2(2s)2(2p)6(3s)2(3p)6(3d)10(4s)2(4p)6⏟definite(36electrons)(4d)10(5s)2(5p)6(4f)10(6s)2⏟likely(30electrons)

Step by step solution

01

Definition of total spin, total orbital and total angular momentum quantum number

Spin quantum number, the fourth quantum number introduced to explain the rotation and direction of the electron spin in space. By combining a particle's orbital and intrinsic angular momentum, the total angular momentum quantum number in quantum mechanics parameterizes the total angular momentum of a particular particle.

02

Determining the total spin, total orbital and grand total angular momentum quantum numbers

Ground state of dysprosium is 5l8. That means 2s + 1 = 5 and J = 8. So total spin , orbital spin and total angular momentum are:

S = 2, L = 6, J = 8.

Electronic configuration :

(1s)2(2s)2(2p)6(3s)2(3p)6(3d)10(4s)2(4p)6⏟definite (36 electrons)(4d)10(5s)2(5p)6(4f)10(6s)2⏟likely (30 electrons)

30 electrons which fill all orbits from4d10 to6s2 are likely there.

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Most popular questions from this chapter

Certain cold stars (called white dwarfs) are stabilized against gravitational collapse by the degeneracy pressure of their electrons (Equation 5.57). Assuming constant density, the radius R of such an object can be calculated as follows:

P=23EtotV=23h2kF510π2m=(3π2)2/3h25mp5/3(5.57)

(a) Write the total electron energy (Equation 5.56) in terms of the radius, the number of nucleons (protons and neutrons) N, the number of electrons per nucleon d, and the mass of the electron m. Beware: In this problem we are recycling the letters N and d for a slightly different purpose than in the text.

Etot=h2V2π2m∫0kFK4dk=h2kF5V10π2m=h2(3π2Nd)5/310π2mV-2/3(5.56)

(b) Look up, or calculate, the gravitational energy of a uniformly dense sphere. Express your answer in terms of G (the constant of universal gravitation), R, N, and M (the mass of a nucleon). Note that the gravitational energy is negative.

(c) Find the radius for which the total energy, (a) plus (b), is a minimum.

R=(9π4)2/3h2d5/3GmM2N1/3

(Note that the radius decreases as the total mass increases!) Put in the actual numbers, for everything except , using d=1/2 (actually, decreases a bit as the atomic number increases, but this is close enough for our purposes). Answer:

(d) Determine the radius, in kilometers, of a white dwarf with the mass of the sun.

(e) Determine the Fermi energy, in electron volts, for the white dwarf in (d), and compare it with the rest energy of an electron. Note that this system is getting dangerously relativistic (seeProblem 5.36).

Derive the Stefan-Boltzmann formula for the total energy density in blackbody radiation

EV=(π2kB415ħ3C3)T4=7.57×10-16Jm-3K-4T4

Chlorine has two naturally occurring isotopes,CI35and CI37. Show that

the vibrational spectrum of HCIshould consist of closely spaced doublets,

with a splitting given by ∆v=7.51×10-4v, where v is the frequency of the

emitted photon. Hint: Think of it as a harmonic oscillator, with ω=k/μ, where

μis the reduced mass (Equation 5.8 ) and k is presumably the same for both isotopes.

(a) Suppose you put both electrons in a helium atom into the n=2state;

what would the energy of the emitted electron be?

(b) Describe (quantitatively) the spectrum of the helium ion,He+.

(a) Construct the completely anti symmetric wave function ψ(xA,xB,xC)for three identical fermions, one in the state ψ5, one in the state ψ7,and one in the state ψ17

(b)Construct the completely symmetric wave function ψ(xA,xB,xC)for three identical bosons (i) if all are in state ψ11(ii) if two are in state ψ19and another one is role="math" localid="1658224351718" ψ1c) one in the state ψ5, one in the state ψ7,and one in the stateψ17
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