’ž1 8 . ( a ) I n a n a d i a b a t i c p r o c e s s Q = 0 . T h i s c a n b e d o n e b y p l a c i n g t h e g a s i n a t h e r m a l l y i n s u l a t e d
c o n t a i n e r w h o s e v o l u m e c a n b e a d j u s t e d ( s a y , b y m e a n s o f a m o v a b l e p i s t o n ) . I f t h e v o l u m e i s
s l o w l y i n c r e a s e d f r o m V i t o V x , t h e n t h e p r o c e s s i s r e v e r s i b l e . T o r e a l i z e t h e r e v e r s i b l e , c o n s t a n t -
v o l u m e p r o c e s s f r o m x t o f , w e w o u l d p l a c e t h e g a s i n a r i g i d c o n t a i n e r , w h i c h h a s a f i x e d v o l u m e
V f a n d i s i n t h e r m a l c o n t a c t w i t h a h e a t r e s e r v o i r . I f w e g r a d u a l l y i n c r e a s e t h e t e m p e r a t u r e o f t h e
r e s e r v o i r f r o m T x t o T f , t h e g a s w i l l u n d e r g o t h e d e s i r e d r e v e r s i b l e p r o c e s s f r o m x t o f .
³
( b ) F o r t h e t w o s t a t e s i a n d x w e h a v e p i V i / T i = p x V x / T x a n d p i V i ³ = p x V x . W e e l i m i n a t e p i a n d p x
f r o m t h e s e e q u a t i o n s t o o b t a i n
³- 1
T x V i
= .
T i V x
F o r m o n a t o m i c i d e a l g a s e s ³ = 5 / 3 ( s e e § 2 0 - 8 a n d § 2 0 - 1 1 ) , s o ³ - 1 = 2 / 3 . A l s o V x = V f .
S u b s t i t u t i n g t h e s e i n t o t h e e q u a t i o n a b o v e , w e o b t a i n T x = T i ( V i / V f ) 2 / 3 .
( c ) F o r a n i d e a l g a s u n d e r g o i n g a n i s o t h e r m a l p r o c e s s , E q . 2 0 - 4 5 i m p l i e s "E i n t = 0 . A n d E q . 2 0 - 1 4
g i v e s W = n R T l n ( V f / V i ) f o r s u c h a p r o c e s s . T h e r e f o r e , t h e f i r s t l a w o f t h e r m o d y n a m i c s l e a d s t o
V f
Q p a t h I = "E i n t I + W I = n R T i l n .
V i
A n d f o r p a t h I I , w e h a v e
Q p a t h I I = Q a d i a b a t + Q c o n s t v o l = 0 + n C V "T .
3
B u t C V = R ( s e e E q . 2 0 - 4 3 ) , s o w e o b t a i n
2
3
Q p a t h I I = n R ( T f - T x ) .
2
W e s e e t h a t Q p a t h I = Q p a t h I I .
( d ) S i n c e t h e f i r s t p a r t o f p a t h I I i s r e v e r s i b l y a d i a b a t i c , t h e n t h e e n t r o p y c h a n g e s o n l y d u r i n g t h e
s e c o n d , c o n s t a n t - v o l u m e , p a r t o f t h e p a t h :
T f
n C V d T T f 3 T f
"S = = n C V l n = n R l n .
T T x 2 T x
T x
E n t r o p y i s a f u n c t i o n o f w h e r e y o u a r e o n t h e p V d i a g r a m , n o t h o w y o u g o t t h e r e . S i n c e t h e
b e g i n n i n g a n d e n d i n g p o i n t o f p a t h I a r e t h e s a m e a s t h o s e o f p a t h I I , t h e n "S i s t h e s a m e f o r
b o t h .
1
( e ) U s i n g t h e r e s u l t i n p a r t ( b ) w i t h V i / V f = a n d T i = 5 0 0 K , w e f i n d
2
2 / 3
1
T x = ( 5 0 0 K ) = 3 1 5 K .
2
F o r p a t h I , E q . 2 1 - 2 g i v e s Q I = ( "S ) T w h e r e T = T i = T f a n d "S i s t h e e x p r e s s i o n c a l c u l a t e d i n
t h e p a r t ( d ) . T h u s ,
3 T f
Q I = n R T f l n
2 T x
w h i c h c a n b e a l t e r n a t i v e l y d e r i v e d f r o m E q . 2 0 - 1 4 a n d t h e f i r s t l a w o f t h e r m o d y n a m i c s . W i t h
n = 1 m o l , T f = T i = 5 0 0 K , w e f i n d
3 5 0 0
Q I = ( 1 ) ( 8 . 3 1 ) ( 5 0 0 ) l n = 2 8 8 0 J .
2 3 1 5
F o r p a t h I I , Q = Q c o n s t a n t v o l u m e = n C V "T a n d w e o b t a i n
3
Q I I = ( 1 ) ( 8 . 3 1 ) ( 5 0 0 - 3 1 5 ) = 2 3 0 6 J .
2
Wyszukiwarka
Podobne podstrony:
P218file8885quin?81101129081 oeb?9 r1Blac?80440337935 oeb?8 r1P0008P21 8Pala85515839 oeb toc r1mari?81440608889 oeb?9 r1Pala85515839 oeb?6 r1Thom?80553904765 oeb?4 r1file8872knig?81440601187 oeb fm3 r1page8p21byer?81101110454 oeb?2 r1knig?81440601187 oeb?0 r1wiÄcej podobnych podstron