Physical Chemistry Third Edition

756 17 The Electronic States of Atoms. I. The Hydrogen Atom

Solution

We can obtain this quantity from the expectation value of the kinetic energy:

〈K〉−E 1  2. 18 × 10 −^18 J

〈v^2 〉

2 〈K〉

m



2(2. 18 × 10 −^18 kg m^2 s−^2 )

9. 11 × 10 −^31 kg

 4. 69 × 1012 m^2 s−^2

vrms〈ν^2 〉{^1 /^2 } 2. 16 × 106 ms−^1

which is smaller than the speed of light by a factor of roughly 1/100. For elements past the

middle of the periodic table, the electron speeds approach the speed of light. For accurate

work with these elements, relativistic corrections must be included or the Dirac equation must

be used.

Exercise 17.15

The kinetic energy of 1selectron in a hydrogen-like atom is proportional toZ^2. Find the root-

mean-square speed of a 1selectron in a uranium atom withZ92. Compare this with the speed

of light.

Thezcomponent of the intrinsic angular momentum is found experimentally to

have one of only two possible values,h/ ̄ 2 and−h/ ̄ 2. We denote the intrinsic angular

momentum bySand write

Sz±

h ̄

2

(17.7-1)

We assign a new quantum number,ms, for thezcomponent of the intrinsic angular

momentum, with the values

ms±

1

2

(17.7-2)

so that:

Szmsh ̄ (17.7-3)

The values ofmsfollow the standard pattern of angular momentum quantum num-

bers if we assign a quantum numbers 1 /2 for the square of the intrinsic angular

momentum. The square of the intrinsic angular momentum has the fixed value

S^2 h ̄^2

1

2

(

1

2

+ 1

)



3

4

h ̄^2 (17.7-4)

The orbital angular momentum and the intrinsic angular momentum are different

from each other in three ways. First, the orbital angular momentum occurred naturally

in the solution of the Schrödinger equation, whereas the intrinsic angular momen-

tum is arbitrarily added to the Schrödinger theory. Second, the magnitude of the orbital

angular momentum takes on various quantized magnitudes, whereas the intrinsic angu-

lar momentum has only one fixed magnitude. Third, the intrinsic angular momentum

corresponds to quantum numbers that are half-integers instead of integers.