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

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T h e m o s t c o m m o n l y o c c u r r i n g a n d m o s t i m p o r t a n t c o n t i n u o u s p r o b a b i l i t y d i s t r i b u t i o n
i s t h e G a u s s i a n d i s t r i b u t i o n , g i v e n b y
2 2 2
p ( x ) d x = c e - x
1
a d x
F i n d c , ( x ) , a ; a n d a x .
- O O < X < O O
I
68 Math Chapter B I P R 0 B A B I Ll T Y AN D STAT I S T I C S
S 0 L UTI 0 N : The constant c is determined by normalization:
(B.l5)
If you look in a table of integrals (for example, The CRC Standard Mathematical
Tables or The CRC Handbook of Chemistry and Physics, CRC Press), you won't find
the above integral. However, you will find the integral
e-ax dx = -100 2 ( T{ ) 1/2 o 4a (B.16)
The reason that you won't find the integral with the limts (-oo, oo) is illustrated in
2
Figure B.2(a), where e-ax is plotted against x. Note that the graph is symmetric about
the vertical axis, so that the corresponding areas on the two sides of the axis are equal.
A function that has the mathematical property that f (x) = f (-x) and is called an
even function. For an even function
1A f (x)dx = 21A f (x)dx even even
-A 0
(B.17)
If we recognize that p(x) = ce-x2 12a2 is an even function and use Equation B.l6, then
we find that
(
2) 1/2
= 2c n; = l
f(x) f(x)
(a) (b)
FIGURE 8.2
2 (a) The function f(x) =e-x is an even function, f(x) = f(-x). (b) The function f(x) =
xe-x
2 is an odd function, f(x) = - f( -x).
M a t h C h a p t e r B I P R 0 B A B I L I T Y A N D 5 T A T I 5 T I C 5
T h e m e a n o f x i s g i v e n b y
( x ) = i : x p ( x ) d x = ( 2 n a
2
) -
1 1 2
i : x e - x
2
/ 2 a
2
d x
( B . l 8 )
T h e i n t e g r a n d i n E q u a t i o n B . 1 8 i s p l o t t e d i n F i g u r e B . 2 ( b ) . N o t i c e t h a t t h i s g r a p h i s
a n t i s y m m e t r i c a b o u t t h e v e r t i c a l a x i s a n d t h a t t h e a r e a o n o n e s i d e o f t h e v e r t i c a l a x i s
c a n c e l s t h e c o r r e s p o n d i n g a r e a o n t h e o t h e r s i d e . A f u n c t i o n t h a t h a s t h e m a t h e m a t i c a l
p r o p e r t y t h a t f ( x ) = - f ( - x ) i s c a l l e d a n o d d f u n c t i o n . F o r a n o d d f u n c t i o n ,
/ _ : f o d d ( x ) d x = 0
( B . l 9 )
T h e f u n c t i o n x e - x
2
; z a
2
i s a n o d d f u n c t i o n , a n d s o
1
0 0
2 2
( x ) = ( 2 n a
2
) -
1 1 2
- o o x e - x f Z a d x = 0
T h e s e c o n d m o m e n t o f x i s g i v e n b y
( x 2 ) = ( 2 n a 2 ) - 1 / 2 i : x 2 e - x 2 f 2 a 2 d x
T h e i n t e g r a n d i n t h i s c a s e i s e v e n b e c a u s e y ( x ) = x
2
e - x
2
!
2
&quot;
2
= y ( - x ) . T h e r e f o r e ,
( x 2 ) = 2 ( 2 n a 2 ) - 1 / 2 1 o o x 2 e - x 2 ; z a 2 d x
T h e i n t e g r a l
1
0 0 2 - a x 2 d - 1 ( 7 [ ) 1 / 2
x e x - - -
o 4 a a
c a n b e f o u n d i n i n t e g r a l t a b l e s , a n d s o
( x z ) = 2 ( 2 n a 2 ) l f 2 a 2
( 2 n a 2 ) 1 / 2 2 = a 2
B e c a u s e ( x ) = 0 , a ; = ( x
2
) , a n d s o a x i s g i v e n b y
a = a
X
( B . 2 0 )
T h e s t a n d a r d d e v i a t i o n o f a n o r m a l d i s t r i b u t i o n i s t h e p a r a m e t e r t h a t a p p e a r s i n t h e
e x p o n e n t i a l . T h e s t a n d a r d n o t a t i o n f o r a n o r m a l i z e d G a u s s i a n d i s t r i b u t i o n f u n c t i o n i s
p ( x ) d x = ( 2 n a ; ) -
1
1
2
e - x
2
f
2
a } d x
( B . 2 1 )
6 9
I
70 MathChapter B I P R 0 B A B I L1 T Y AN D STAT I S T I C S
Figure B.3 shows Equation B.21 for various values of ax. Note that the curves become
narrower and taller for smaller values of a .
X
A more general version of a Gaussian distribution is
(B.22)
This expression looks like those in Figure B.3 except that the curves are centered at
x = (x) rather than x = 0. A Gaussian distribution is one of the most important and
commonly used probability distributions in all of science.
FIGURE 8.3
p (x)
('
' \
'
I \
0
A plot of a Gaussian distribution, p(x), (Equation B.21) for three values of ax. The dotted curve
corresponds to ax = 2, the solid curve to ax = 1, and the dash-dotted curve to ax = 0.5.
Problems
B-1. Consider a particle to be constrained to lie along a one-dimensional segment 0 to a. We
will learn in the next chapter that the probability that the particle is found to lie between x
and x + dx is given by
2 2 mrx p(x)dx =-sin --dx
a a
where n = 1, 2, 3, .... First show that p(x) is normalized. Now calculate the average
position of the particle along the line segment. The integrals that you need are (The CRC
Handbook of Chemistry and Physics or The CRC Standard Mathematical Tables, CRC
Press)
and
f 2 x sin2ax sin axdx = 2 -~
f . 2 x 2 x sin 2ax cos 2ax x sm axdx = - - - ---4 4a 8a2
P r o b l e m s
B - 2 . C a l c u l a t e t h e v a r i a n c e a s s o c i a t e d w i t h t h e p r o b a b i l i t y d i s t r i b u t i o n g i v e n i n P r o b l e m B - 1 .
T h e n e c e s s a r y i n t e g r a l i s ( C R C t a b l e s )
f
2
.
2
x
3
( x
2
1 ) . x c o s 2 a x
x S i l l a x d x = - - - - - S i l l 2 a x - - - - ; ; - - - - -
6 4 a 8 a
3
4 a
2
B - 3 . U s i n g t h e p r o b a b i l i t y d i s t r i b u t i o n g i v e n i n P r o b l e m B - 1 , c a l c u l a t e t h e p r o b a b i l i t y t h a t t h e
p a r t i c l e w i l l b e f o u n d b e t w e e n 0 a n d a j 2 . T h e n e c e s s a r y i n t e g r a l i s g i v e n i n P r o b l e m B - 1 .
B - 4 . P r o v e e x p l i c i t l y t h a t
1
0 0 e - o t x 1 d x = 2 r o o e - o t x 1 d x
- o o J o
b y b . r e a k i n g t h e i n t e g r a l f r o m - o o t o o o i n t o o n e f r o m - o o t o 0 a n d a n o t h e r f r o m 0 t o o o .
N o w l e t z = - x i n t h e f i r s t i n t e g r a l a n d z = x i n t h e s e c o n d t o p r o v e t h e a b o v e r e l a t i o n .
B - 5 . B y u s i n g t h e p r o c e d u r e i n P r o b l e m B - 4 , s h o w e x p l i c i t l y t h a t
i : x e - a x
1
d x = 0
B - 6 . W e w i l l l e a r n i n C h a p t e r 2 5 t h a t t h e m o l e c u l e s i n a g a s t r a v e l a t v a r i o u s s p e e d s , a n d t h a t
t h e p r o b a b i l i t y t h a t a m o l e c u l e h a s a s p e e d b e t w e e n v a n d v + d v i s g i v e n b y
p ( v ) d v = 4 r r _ m _ _ v
2
e - m v
2
/
2
k B T d v
(
)
3 / 2
2 n k
8
T
0 : : : 0 V < O O
w h e r e m i s t h e m a s s o f t h e p a r t i c l e , k
8
i s t h e B o l t z m a n n c o n s t a n t ( t h e m o l a r g a s c o n s t a n t R
d i v i d e d b y t h e A v o g a d r o c o n s t a n t ) , a n d T i s t h e K e l v i n t e m p e r a t u r e . T h e p r o b a b i l i t y
d i s t r i b u t i o n o f m o l e c u l a r s p e e d s i s c a l l e d t h e M a x w e l l - B o l t z m a n n d i s t r i b u t i o n . F i r s t s h o w
t h a t p ( v ) i s n o r m a l i z e d , a n d t h e n d e t e r m i n e t h e a v e r a g e s p e e d a s a f u n c t i o n o f t e m p e r a t u r e .
T h e n e c e s s a r y i n t e g r a l s a r e ( C```