[Donald A. McQuarrie, John D. Simon] Physical Chem(BookZZ.org)
1279 pág.

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


DisciplinaFísico-química I6.497 materiais97.703 seguidores
Pré-visualização50 páginas
p e c i f i e d S i m u l t a n e o u s l y 
o r 
I E = R
0 0 
= 1 0 9 7 3 7 c m -
1 
= 2 . 1 7 8 7 x w -
1 8 
1 
= 1 3 . 5 9 8 e V = 1 3 1 2 . 0 k J · m o l -
1 
N o t e t h a t w e h a v e e x p r e s s e d t h e e n e r g y i n u n i t s o f w a v e n u m b e r s ( c m -
1 
) . T h i s u n i t i s 
n o t s t r i c t l y a u n i t o f e n e r g y , b u t b e c a u s e o f t h e s i m p l e r e l a t i o n b e t w e e n w a v e n u m b e r 
a n d e n e r g y , E = h c v , e n e r g y i s o f t e n e x p r e s s e d i n t h i s w a y ( c f . P r o b l e m 1 - 1 ) . 
D e s p i t e a n u m b e r o f s u c c e s s e s a n d t h e b e a u t i f u l s i m p l i c i t y o f t h e B o h r t h e o r y , 
t h e t h e o r y c o u l d n o t b e e x t e n d e d s u c c e s s f u l l y e v e n t o a t w o - e l e c t r o n s y s t e m s u c h a s 
h e l i u m . F u r t h e r m o r e , e v e n f o r a s i m p l e s y s t e m s u c h a s h y d r o g e n , i t c o u l d n o t e x p l a i n 
t h e s p e c t r a t h a t a r i s e w h e n a m a g n e t i c f i e l d i s a p p l i e d t o t h e s y s t e m , n o r c o u l d i t p r e d i c t 
t h e i n t e n s i t i e s o f t h e s p e c t r a l l i n e s . 
1 - 9 . T h e H e i s e n b e r g U n c e r t a i n t y P r i n c i p l e S t a t e s T h a t t h e P o s i t i o n 
a n d t h e M o m e n t u m o f a P a r t i c l e C a n n o t B e S p e c i f i e d 
S i m u l t a n e o u s l y w i t h U n l i m i t e d P r e c i s i o n 
W e n o w k n o w t h a t w e m u s t c o n s i d e r l i g h t a n d m a t t e r a s h a v i n g t h e c h a r a c t e r i s t i c s o f 
b o t h w a v e s a n d p a r t i c l e s . L e t ' s c o n s i d e r a m e a s u r e m e n t o f t h e p o s i t i o n o f a n e l e c t r o n . 
I f w e w i s h t o l o c a t e t h e e l e c t r o n w i t h i n a d i s t a n c e ~x, t h e n w e m u s t u s e a m e a s u r i n g 
d e v i c e t h a t h a s a s p a t i a l r e s o l u t i o n l e s s t h a n ~x. O n e w a y t o a c h i e v e t h i s r e s o l u t i o n i s 
t o u s e l i g h t w i t h a w a v e l e n g t h o n t h e o r d e r o f A ~ ~x. F o r t h e e l e c t r o n t o b e " s e e n " , 
a p h o t o n m u s t i n t e r a c t o r c o l l i d e i n s o m e w a y w i t h t h e e l e c t r o n , f o r o t h e r w i s e t h e 
p h o t o n w i l l j u s t p a s s r i g h t b y a n d t h e e l e c t r o n w i l l a p p e a r t r a n s p a r e n t . T h e p h o t o n 
h a s a m o m e n t u m p = h j A , a n d d u r i n g t h e c o l l i s i o n , s o m e o f t h i s m o m e n t u m w i l l b e 
t r a n s f e r r e d t o t h e e l e c t r o n . T h e v e r y a c t o f l o c a t i n g t h e e l e c t r o n l e a d s t o a c h a n g e i n 
i t s m o m e n t u m . I f w e w i s h t o l o c a t e t h e e l e c t r o n m o r e a c c u r a t e l y , w e m u s t u s e l i g h t 
w i t h a s m a l l e r w a v e l e n g t h . C o n s e q u e n t l y , t h e p h o t o n s i n t h e l i g h t b e a m w i l l h a v e 
g r e a t e r m o m e n t u m b e c a u s e o f t h e r e l a t i o n p = h j A . B e c a u s e s o m e o f t h e p h o t o n ' s 
m o m e n t u m m u s t b e t r a n s f e r r e d t o t h e e l e c t r o n i n t h e p r o c e s s o f l o c a t i n g i t , t h e m o -
m e n t u m c h a n g e o f t h e e l e c t r o n b e c o m e s g r e a t e r . A c a r e f u l a n a l y s i s o f t h i s p r o c e s s 
w a s c a r r i e d o u t i n t h e m i d - 1 9 2 0 s b y t h e G e r m a n p h y s i c i s t W e r n e r H e i s e n b e r g , w h o 
s h o w e d t h a t i t i s n o t p o s s i b l e t o d e t e r m i n e e x a c t l y h o w m u c h m o m e n t u m i s t r a n s f e r r e d 
t o t h e e l e c t r o n . T h i s d i f f i c u l t y m e a n s t h a t i f w e w i s h t o l o c a t e a n e l e c t r o n t o w i t h i n 
a r e g i o n ~x, t h e r e w i l l b e a n u n c e r t a i n t y i n t h e m o m e n t u m o f t h e e l e c t r o n . H e i s e n -
b e r g w a s a b l e t o s h o w t h a t i f ~p i s t h e u n c e r t a i n t y i n t h e m o m e n t u m o f t h e e l e c t r o n , 
t h e n 
~x~p 2 : h 
( 1 . 2 6 ) 
2 3 
24 Chapter 1 I The Dawn of the Quantum Theory 
Equation 1.26 is called Heisenberg's Uncertainty Principle and is a fundamental 
principle of nature. The Uncertainty Principle states that if we wish to locate any 
particle to within a distance b..x, then we automatically introduce an uncertainty in the 
momentum of the particle and that the uncertainty is given by Equation 1.26. Note that 
this uncertainty does not stem from poor measurement or experimental technique but is 
a fundamental property of the act of measurement itself. The following two examples 
demonstrate the numerical consequences of the Uncertainty Principle. 
EXAMPLE 1-10 
Calculate the uncertainty in the position of a baseball thrown at 90 mph if we measure 
its momentum to a millionth of 1.0%. 
S 0 L UTI 0 N : According to Example 1-6, a baseball traveling at 90 mph has a 
momentum of5.6 kg·m·s- 1\u2022 A millionth of 1.0% ofthis value is 5.6 x 10-8 kg·m·s- 1 , 
so 
!:,.p = 5.6 X 10-8 kg·m·s- 1 
The minimum uncertainty in the position of the baseball is 
h 6.626xl0-34 J·s 
!:.x = - = ----;;---------,-
t:.p 5.6 X 10-8kg·m·S-1 
= 1.2 X 10-26 m 
a completely inconsequential distance. 
EXAMPLE 1-11 
What is the uncertainty in momentum if we wish to locate an electron within an atom, 
say, so that t:.x is approximately 50 pm? 
SOLUTION: h 6.626x10-
34 J·s 
!:,.p = !:.x = 50 X 10-12 m 
= 1.3 X 10-23 kg·m·s- 1 
Because p = mv and the mass of an electron is 9.11 x 10-31 kg, this value of t:.p 
corresponds to 
!:.p 1.3 X 10-23 kg·m·S- 1 
t:.v = - = ------'~--
me 9.11 X 10-31 kg 
= 1.4 x 107 m-s- 1 
which is a very large uncertainty in the speed. 
P r o b l e m s 
T h e s e t w o e x a m p l e s s h o w t h a t a l t h o u g h t h e H e i s e n b e r g U n c e r t a i n t y P r i n c i p l e i s o f 
n o c o n s e q u e n c e f o r e v e r y d a y , m a c r o s c o p i c b o d i e s , i t h a s v e r y i m p o r t a n t c o n s e q u e n c e s 
i n d e a l i n g w i t h a t o m i c a n d s u b a t o m i c p a r t i c l e s . T h i s c o n c l u s i o n i s s i m i l a r t o t h e o n e 
t h a t w e d r e w f o r t h e a p p l i c a t i o n o f t h e d e B r o g l i e r e l a t i o n b e t w e e n w a v e l e n g t h a n d 
m o m e n t u m . T h e U n c e r t a i n t y P r i n c i p l e l e d t o a n a w k w a r d r e s u l t . I t t u r n s o u t t h a t t h e 
B o h r t h e o r y i s i n c o n s i s t e n t w i t h t h e U n c e r t a i n t y P r i n c i p l e . F o r t u n a t e l y , a n e w , m o r e 
g e n e r a l q u a n t u m t h e o r y w a s s o o n p r e s e n t e d t h a t i s c o n s i s t e n t w i t h t h e U n c e r t a i n t y 
P r i n c i p l e . W e w i l l s e e t h a t t h i s t h e o r y i s a p p l i c a b l e t o a l l a t o m s a n d m o l e c u l e s a n d 
f o r m s t h e b a s i s f o r o u r u n d e r s t a n d i n g o f a t o m i c a n d m o l e c u l a r s t r u c t u r e . T h i s t h e o r y 
w a s f o r m u l a t e d b y t h e A u s t r i a n p h y s i c i s t E r w i n S c h r o d i n g e r a n d w i l l b e d i s c u s s e d i n 
C h a