[Donald A. McQuarrie, John D. Simon] Physical Chem(BookZZ.org)
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[Donald A. McQuarrie, John D. Simon] Physical Chem(BookZZ.org)


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h 6.626 X 10-34 J·S 
A-- - = 2.43 X 10-10 m 
- p - 2.73 X 10-24 kg·m·S-l 
= 243 pm 
This wavelength is of atomic dimensions. 
The wavelength of the electron calculated in Example 1-7 corresponds to the 
wavelength ofX-rays. Thus, although Equation 1.12 is of no consequence for a macro-
scopic object such as a baseball, it predicts that electrons can be observed to act like 
X-rays. The wavelengths of some other moving objects are given in Table 1.2. 
1-7. de Broglie Waves Are Observed Experimentally 
When a beam of X rays is directed at a crystalline substance, the beam is scattered in a 
definite manner characteristic of the atomic structure of the crystalline substance. This 
phenomenon is called X-ray diffraction and occurs because the interatomic spacings in 
T A B l E 1 . 2 
T h e d e B r o g l i e w a v e l e n g t h s o f v a r i o u s m o v i n g o b j e c t s . 
P a r t i c l e 
M a s s / k g 
S p e e d l m · s -
1 
W a v e l e n g t h / p m 
E l e c t r o n a c c e l e r a t e d 
t h r o u g h 1 0 0 V 
9 . 1 1 x 1 0 -
3 1 
5 . 9 X 1 0
6 
1 2 0 
E l e c t r o n a c c e l e r a t e d 
t h r o u g h 1 0 , 0 0 0 V 
9 . 2 9 x 1 0 -
3 1 
5 . 9 X 1 0
7 
1 2 
a p a r t i c l e e j e c t e d 
f r o m r a d i u m 
6 . 6 8 x w - n 
1 . 5 X 1 0
7 
6 . 6 x 1 0 -
3 
2 2 - c a l i b e r r i f l e b u l l e t 
1 . 9 x 1 0 -
3 
3 . 2 X 1 0
2 
1 . 1 x 1 0 -
2 1 
G o 1 f b a l l 0 . 0 4 5 3 0 
4 . 9 x 1 0 -
2 2 
t h e c r y s t a l a r e a b o u t t h e s a m e a s t h e w a v e l e n g t h o f t h e X - r a y s . T h e X - r a y d i f f r a c t i o n 
p a t t e r n f r o m a l u m i n u m f o i l i s s h o w n i n F i g u r e 1 . 8 a . T h e X - r a y s s c a t t e r f r o m t h e f o i l 
i n r i n g s o f d i f f e r e n t d i a m e t e r s . T h e d i s t a n c e s b e t w e e n t h e r i n g s a r e d e t e r m i n e d b y t h e 
i n t e r a t o m i c s p a c i n g i n t h e m e t a l f o i l . F i g u r e 1 . 8 b s h o w s a n e l e c t r o n d i f f r a c t i o n p a t t e r n 
f r o m a l u m i n u m f o i l t h a t r e s u l t s w h e n a b e a m o f e l e c t r o n s i s s i m i l a r l y d i r e c t e d . T h e 
( a ) 
( b ) 
F I G U R E 1 . 8 
( a ) T h e X - r a y d i f f r a c t i o n p a t t e r n o f a l u m i n u m f o i l . ( b ) T h e e l e c t r o n d i f f r a c t i o n p a t t e r n o f 
a l u m i n u m f o i l . T h e s i m i l a r i t y o f t h e s e t w o p a t t e r n s s h o w s t h a t e l e c t r o n s c a n b e h a v e l i k e X - r a y s 
a n d d i s p l a y w a v e l i k e p r o p e r t i e s . 
1 7 
18 Chapter 1 I The Dawn of the Quantum Theory 
similarity of the two patterns shows that both X-rays and electrons do indeed behave 
analogously in these experiments. 
The wavelike property of electrons is used in electron microscopes. The wave-
lengths of the electrons can be controlled through an applied voltage, and the small 
de Broglie wavelengths attainable offer a more precise probe than an ordinary light 
microscope. In addition, in contrast to electromagnetic radiation of similar wavelengths 
(X-rays and ultraviolet), the electron beam can be readily focused by using electric and 
magnetic fields, generating sharper images. Electron microscopes are used routinely in 
chemistry and biology to investigate atomic and molecular structures. 
An interesting historical aside in the concept of the wave-particle duality of matter 
is that the first person to show that the electron was a subatomic particle was the English 
physicist Sir Joseph J. Thomson in 1895 and then his son Sir George P. Thomson was 
among the first to show experimentally in 1926 that the electron could act as a wave. 
The father won a Nobel Prize in 1906 for showing that the electron is a particle and the 
son won a Nobel Prize in 1937 for showing that it is a wave. 
1-8. The Bohr Theory of the Hydrogen Atom Can Be Used to Derive 
the Rydberg Formula 
In 1911, the Danish physicist Niels Bohr presented a theory of the hydrogen atom that 
gave a beautifully simple explanation of the hydrogen atomic spectrum. We present 
here a brief discussion of the Bohr theory. 
According to the nuclear model of the atom, the hydrogen atom can be pictured as 
a central, rather massive nucleus with one associated electron. Because the nucleus is 
so much more massive than the electron, we can consider the nucleus to be fixed and 
the electron to be revolving about it. The force holding the electron in a circular orbit 
is supplied by the coulombic force of attraction between the proton and the electron 
(Coulomb's law): 
where r is the radius of the orbit, e is the charge on the electron, and s0 = 8.85419 x 
10-12 C2 -N-1 -m-2 is the permittivity of free space. The occurrence of the factor 4rrs0 
in Coulomb's law is a result of using SI units. The coulombic force is balanced by the 
centrifugal force (see Problem 1-41) 
m v2 J=-e-
r 
(1.13) 
where me and v are the mass and the speed of the electron, respectively. If we equate 
the coulombic force and the centrifugal force, then we obtain 
(1.14) 
r 
1 - 8 . T h e B o h r T h e o r y o f t h e H y d r o g e n A t o m C a n B e U s e d t o D e r i v e t h e R y d b e r g F o r m u l a 
W e a r e t a c i t l y a s s u m i n g h e r e t h a t t h e e l e c t r o n i s r e v o l v i n g a r o u n d t h e f i x e d n u c l e u s 
i n a c i r c u l a r o r b i t o f r a d i u s r . C l a s s i c a l l y , h o w e v e r , b e c a u s e t h e e l e c t r o n i s c o n s t a n t l y 
b e i n g a c c e l e r a t e d a c c o r d i n g t o E q u a t i o n 1 . 1 3 ( P r o b l e m 1 - 4 1 ) , i t s h o u l d e m i t e l e c t r o -
m a g n e t i c r a d i a t i o n a n d l o s e e n e r g y . C o n s e q u e n t l y , c l a s s i c a l p h y s i c s p r e d i c t s t h a t a n 
e l e c t r o n r e v o l v i n g a r o u n d a n u c l e u s w i l l l o s e e n e r g y a n d s p i r a l i n t o t h e n u c l e u s , a n d s o 
a s t a b l e o r b i t f o r t h e e l e c t r o n i s c l a s s i c a l l y f o r b i d d e n . B o h r ' s g r e a t c o n t r i b u t i o n w a s t o 
m a k e t w o n o n c l a s s i c a l a s s u m p t i o n s . T h e f i r s t w a s t o a s s u m e t~~-~xistence o f s t a t i o n a r y 
e l e c t r o n o r b i t s , i n d e f i a n c e o f c l a s s i c a l p h y s i c s . H e t h e n s p e c i f i e d t h e s e o r b i t s b y t h e 
e q u i v a l e n t o f a s s u m i n g t h a t t h e d e B r o g l i e w a v e s o f t h e o r b i t i n g e l e c t r o n m u s t " m a t c h " 
o r b e i n p h a s e , a s t h e e l e c t r o n m a k e s o n e c o m p l e t e r e v o l u t i o n . W i t h o u t s u c h m a t c h i n g , 
c a n c e l l a t i o n o f s o m e a m p l i t u d e o c c u r s d u r i n g e a c h r e v o l u t i o n , a n d t h e w a v e w i l l d i s -
a p p e a r ( s e e F i g u r e 1 . 9 ) . F o r t h e w a v e p a t t e r n a r o u n d a n o r b i t t o b e s t a b l e , w e a r e l e d 
t o t h e c o n d i t i o n t h a t a n i n t e g r a l n u m b e r o f c o m p l e t e w a v e l e n g t h s m u s t f i t a r o u n d t h e 
c i r c u m f e r e n c e o f t h e o r b i t . B e c a u s e t h e c i r c u m f e r e n c e o f a c i r c l e i s 2 n : r , w e h a v e t h e 
q u a n t u m c o n d i t i o n 
2 n : r = n ' A 
n = 1 , 2 , 3 , . . . 
( 1 . 1 5 ) 
I