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Macroscopic Thermodynamics and the Description of Growth and Product Formation in Microorganisms J. A. ROELS Delft University of Technology, Netherlands Central Organization for Applied Scientific Research, T.N.O., P.O. Box 108, 3700 A C ZEIST, The Netherlands
From the point of view of macroscopic thermodynamics living organisms are energy transducers c o n v e r t i n g a source of energy,e.g. chemical substances or photons, i n t o other forms o f energy. As such they are subject to the c o n s t r a i n t s posed by the first and second laws o f thermodynamics. As micro organisms are open systems and as such e x i s t i n a s t a t e o u t s i d e e q u i l i b r i u m , n o n - e q u i l i b r i u m thermo dynamics provide the p e r f e c t v e h i c l e f o r a first approach to the d e s c r i p t i o n of t h e i r behaviour. The concept o f the thermodynamic e f f i c i e n c y of growth i s developed and it is shown t h a t , as a r u l e of thumb, the maximum observed e f f i c i e n c i e s are about 0.65 i r r e s p e c t i v e the nature of the energy supplying process. A number of notable exceptions are shown to be most probably caused by l i m i t a t i o n s other than a v a i l a b l e energy. The nature o f growth and product formation i s discussed i n terms o f the c o u p l i n g o f the transformation of a given amount of substrate energy i n t o biomass energy to the energy obtained from a flow of e l e c t r o n s to a l e v e l of high to a l e v e l of low energy. The treatment i s shown to r e s u l t i n a r e l i a b l e r u l e of thumb f o r a first estimate o f the order of magnitude of the growth y i e l d o f an organism feeding on a given energy supplying t r a n s f o r m a t i o n process.
The b i o s p h e r e o n e a r t h i s , t h e r m o d y n a m i c a l l y s p e a k i n g , i n a c e r t a i n s e n s e an o p e n s y s t e m . I t r e c e i v e s e n e r g y f r o m t h e s u n i n t h e f o r m o f r a d i a t i o n . The e n e r g y o f t h e p h o t o n s r e a c h i n g e a r t h i s , i n p a r t , converted t o chemical energy i n a process c a l l e d 0097-6156/83/0207Ό295$08.00/0 © 1983 American Chemical Society
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
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296
BIOCHEMICAL
ENGINEERING
p h o t o s y n t h e s i s . F o r the l a r g e r p a r t t h i s energy takes the form o f c a r b o h y d r a t e s , e.g. s u g a r s , s t a r c h e s and c e l l u l o s i c s . The components o f t h e b i o m a s s o f t h e p r i m a r y p r o d u c e r s a r e t h e s t a r t i n g p o i n t o f a wide v a r i e t y o f t r a n s f o r m a t i o n s , which takes p l a c e under c a t a l y t i c a c t i o n o f l i v i n g organisms. As a f i n a l r e s u l t t h e s o l a r e n e r g y i s c o n v e r t e d t o l e s s energy r i c h forms o f r a d i a t i o n w h i c h t r a n s p o r t energy t o o u t e r space o r i s used t o d e c r e a s e t h e e n t r o p y o f t h e b i o s p h e r e . The r e a s o n i n g d e v e l o p e d above shows t h a t t h e b i o s p h e r e i s a s y s t e m , w h i c h i s s u b j e c t t o a f l o w o f e n e r g y . The e n e r g y e n t e r s t h e s y s t e m a t a l o w e n t r o p y l e v e l and l e a v e s i t a t a s u b s t a n t i a l l y h i g h e r e n t r o p y l e v e l . As s u c h t h e b i o s p h e r e c a n m a i n t a i n a s t a t e w i t h a n e n t r o p y l o w e r t h a n t h e maximum c o r r e s p o n d i n g t o t h e r m o d y n a m i c e q u i l i b r i u m and p r o c e s s e s known as " l i f e " r e s u l t (J_). A s i n g l e organism o r a s p e c i e s f e e d i n g on a g i v e n energy s u p p l y i n g p r o c e s s e x i s t s i n much t h e same p o s i t i o n as t h e b i o s p h e r e as a w h o l e . I t i s an open s y s t e m t h r o u g h w h i c h e n e r g y f l o w s f r o m a low e n t r o p y s t a t e , e.g. c h e m i c a l e n e r g y s t o r e d i n compounds more r e d u c e d t h a n C 0 , t o a h i g h e n t r o p y s t a t e , e.g. h e a t a t a l o w t e m p e r a t u r e l e v e l . As a r e s u l t t h e o r g a n i s m s c a n m a i n t a i n a s t a t e o u t s i d e t h e r m o d y n a m i c e q u i l i b r i u m and c a n continue performing the processes c h a r a c t e r i s t i c f o r t h e i r " l i f e " . As o r g a n i s m s a r e s y s t e m s w h i c h e x i s t o u t s i d e t h e r m o d y n a m i c e q u i l i b r i u m and i r r e v e r s i b l e p r o c e s s e s a r e t a k i n g p l a c e , t h e f o r m a l i s m o f thermodynamics of i r r e v e r s i b l e p r o c e s s e s c o n s t i t u t e s the l o g i c a l v e h i c l e t o t r e a t t h e i r b e h a v i o u r . I n the p r e s e n t a r t i c l e t h e f o r m a l i s m w i l l be b r i e f l y s u m m a r i z e d f o r t h e p u r p o s e o f i t s a p p l i c a t i o n t o m i c r o o r g a n i s m s engaged i n g r o w t h and p r o d u c t f o r m a t i o n . F o r a more t h o r o u g h t r e a t m e n t o f t h e b a s i c f o r m a l i s m t h e r e a d e r i s r e f e r r e d t o t h e s t a n d a r d t e x t s ( 2 - 4 ) and e a r l i e r w o r k o f t h e p r e s e n t a u t h o r ( 5 , 6^). 2
Macroscopic
t h e r m o d y n a m i c s and p r o c e s s e s
i n open s y s t e m s .
05, 6)
For t h e purpose o f the present a n a l y s i s o f m i c r o b i a l metabolism, a g i v e n amount o f m i c r o o r g a n i s m s i s c o n s i d e r e d t o be an e n e r g y t r a n s d u c e r . I t i s s c h e m a t i c a l l y r e p r e s e n t e d i n f i g . 1. The s y s t e m e x c h a n g e s c h e m i c a l e n e r g y and h e a t w i t h t h e e n v i r o n m e n t . F o r s i m p l i c i t y ' s s a k e t h e c a s e o f p r o c e s s e s i n v o l v i n g radiâtional e n e r g y i s e x c l u d e d . The b a s i c f o r m a l i s m , h o w e v e r , c a n be e a s i l y extended t o i n c l u d e these s i t u a t i o n s . The s t a t e o f t h e s y s t e m c a n be c h a r a c t e r i z e d by a number o f e x t e n s i v e q u a n t i t i e s ; t h e s e s p e c i f y t h e amount o f t h e v a r i o u s c h e m i c a l s u b s t a n c e s and t h e amount o f e n e r g y p r e s e n t i n t h e s y s t e m . F o r e a c h e x t e n s i v e q u a n t i t y , w h i c h c a n be a t t r i b u t e d t o t h e s y s t e m , a b a l a n c e e q u a t i o n c a n be f o r m u l a t e d ; i t e x p r e s s e s t h e a c c u m u l a t i o n o f t h e q u a n t i t y i n s i d e t h e s y s t e m as t h e sum o f t h e c h a n g e s o f i t s amount due t o t r a n s f o r m a t i o n and t r a n s p o r t p r o c e s s e s r e s p e c t i v e l y . M a t h e m a t i c a l l y t h i s c a n be e x p r e s s e d a s :
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
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13.
ROELS
Macroscopic
Thermodynamics
and
Growth
\ HEAT
Figure 1. An open system for macroscopic analysis. It exchanges chemical sub stances and heat with the environment. The flow of chemical substances, Φι, is characterized by the elemental composition and the partial enthalpy of the chemical substances.
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
297
298
BIOCHEMICAL
ENGINEERING
(1) V
V
s
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I n t h e f o r m u l a t i o n o f eqn. (1) i t i s assumed t h a t t h e volume and t h e s u r f a c e a r e a o f t h e s y s t e m do n o t change. I n t h e p r e s e n t a r t i c l e t h e d i s c u s s i o n w i l l be r e s t r i c t e d t o s y s t e m s i n a s t a t i o n a r y s t a t e , i . e . systems f o r which the time d e r i v a t i v e a p p e a r i n g a t t h e l e f t hand s i d e o f eqn. (1) has become z e r o . I n s u c h a c a s e eqn. (1) c a n be s i m p l i f i e d t o : (2) V W i t h r e s p e c t t o the t r a n s f o r m a t i o n p r o c e s s e s open to a g i v e n n o n - e q u i l i b r i u m system the e x t e n s i v e q u a n t i t i e s c h a r a c t e r i z i n g t h e s y s t e m can be d i s t i n g u i s h e d i n t o two g r o u p s : c o n s e r v e d and non-conserved q u a n t i t i e s . S o - c a l l e d conserved q u a n t i t i e s cannot be p r o d u c e d o r consumed i n t h e t r a n s f o r m a t i o n p r o c e s s e s o p e n t o a g i v e n s y s t e m . T h e r e f o r e , t h e f i r s t t e r m a t t h e r i g h t hand s i d e of eqn. (1) i s n e c e s s a r i l y z e r o and eqn. (2) c a n be s i m p l y w r i t t e n as : (3)
E q u a t i o n (2) e x p r e s s e s t h e f a c t t h a t f o r e a c h c o n s e r v e d q u a n t i t y t h e t r a n s p o r t t o a s y s t e m i n s t a t i o n a r y s t a t e must e x a c t l y match t r a n s p o r t from t h a t system. For a non-conserved q u a n t i t y such s i m p l i f i c a t i o n i s not p o s s i b l e . The a p p l i c a t i o n o f t h e f o r m a l m a c r o s c o p i c t h e o r y t o t r a n s f o r m a t i o n p r o c e s s e s i n o p e n s y s t e m s i s b a s e d on t h e f o r m u l a t i o n o f b a l a n c e e q u a t i o n s f o r a number o f c o n s e r v e d q u a n t i t i e s and an a d d i t i o n a l t h e r m o d y n a m i c c o n s t r a i n t a l l o w i n g t h e f o r m u l a t i o n of a u s e f u l e f f i c i e n c y measure. The e l e m e n t a l b a l a n c e e q u a t i o n s . In m i c r o b i a l conversion p r o c e s s e s t h e amounts o f t h e v a r i o u s a t o m i c s p e c i e s a r e c o n s e r v e d . T h i s o b s e r v a t i o n r e s u l t s i n the f o r m u l a t i o n of e l e m e n t a l balance e q u a t i o n s . U s i n g eqn. (3) t h e e l e m e n t a l b a l a n c e e q u a t i o n f o r a t o m i c s p e c i e s j c a n , a g a i n a s s u m i n g a s t a t i o n a r y s t a t e , be e x p r e s s e d as ( 5 , 6 ) : η Σ Φ.«*. .= 0 i-i 1
(4)
1 J
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
13.
Macroscopic
ROELS
Thermodynamics
and
Growth
299
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I n e q n . ( 4 ) Φ. s t a n d s f o r t h e n e t m o l a r f l o w o f compound i t o t h e s y s t e m , e. . s t a n d s f o r t h e number o f m o l e s o f a t o m i c s p e c i e s j i n one m J l e o f compound i . E q u a t i o n s o f t h e t y p e o f eqn. ( 4 ) p r o v i d e one c o n s t r a i n t t o t h e n e t exchange f l o w s f o r e a c h atomic species considered. The thermodynamic c o n s t r a i n t s . The a p p l i c a t i o n o f n o n - e q u i l i b r i u m thermodynamics t o t r a n s f o r m a t i o n p r o c e s s e s i s based on t h e f o r m u l a t i o n o f two b a s i c b a l a n c e e q u a t i o n s . The f i r s t o n e , a b a l a n c e e q u a t i o n f o r e n e r g y , c a n , by v i r t u e o f t h e f a c t t h a t the f i r s t l a w o f t h e r m o d y n a m i c s a s s u r e s e n e r g y t o be a c o n s e r v e d q u a n t i t y i n any s y s t e m , f o r a s y s t e m i n s t a t i o n a r y s t a t e be expressed as: Φ
Ε
= 0
(5)
i n w h i c h Φ^ i s t h e n e t f l o w o f e n e r g y t o w a r d s t h e s y s t e m . Thermodynamics show t h a t , f o r a n open s y s t e m o n w h i c h no work i s p e r f o r m e d by e x t e r n a l f o r c e f i e l d s , t h e e n e r g y f l o w t o w a r d s t h a t s y s t e m c a n be e x p r e s s e d as f o l l o w s : Φ
Φ
Ε - Η
+
Σ
Φ
Α
(
6
)
1
I n t h i s e q u a t i o n Φ^ i s t h e s o - c a l l e d h e a t f l o w o f P r i g o g i n e ( 2 ) and t h e h. a r e t h e p a r t i a l m o l a r e n t h a l p i e s o f t h e compounds exchanged w i t h t h e environment. I f eqns. ( 5 ) and ( 6 ) a r e combined t h e f a m i l i a r b a l a n c e equation f o r enthalpy i s obtained. I t allows the c a l c u l a t i o n of the heat exchanged w i t h t h e environment from t h e f o l l o w i n g equation: Φ
Η
= - Σ Φ h.
(7)
1
Equation ( 7 ) i n t r o d u c e s one new unknown f l o w , t h e h e a t f l o w Φ^, and one a d d i t i o n a l c o n s t r a i n t , hence t h e t o t a l number o f unknown f l o w s does n o t change by t h e a p p l i c a t i o n o f t h e f i r s t l a w o f thermodynamics. A second r e s t r i c t i v e e q u a t i o n r e s u l t s from t h e b a l a n c e e q u a t i o n f o r e n t r o p y . F o r a system i n s t a t i o n a r y s t a t e the b a l a n c e equation f o r entropy r e s u l t s i n the f o l l o w i n g expression:
i n w h i c h Tig i s t h e t o t a l e n t r o p y p r o d u c t i o n i n t h e s y s t e m , Φ 0
δ
ENGINEERING
(9)
Furthermore, be w r i t t e n as :
t h e r m o d y n a m i c s show t h a t t h e e n t r o p y f l o w c a n
0 i
(12)
ί
The p a r t i a l m o l a r g
i
=
h
i "
T
s
f r e e e n t h a l p y i s d e f i n e d by: ( 1 3 )
i
E q u a t i o n (12) w i l l be shown t o a l l o w t h e f o r m u l a t i o n o f an e f f i c i e n c y m e a s u r e , w h i c h c a n be used t o a n a l y s e g r o w t h and p r o d u c t f o r m a t i o n i n m i c r o o r g a n i s m s , i t s d e v e l o p m e n t w i l l be undertaken i n the next s e c t i o n . The t h e r m o d y n a m i c
efficiency.
The e n e r g y and e n t r o p y c o n t e n t o f c h e m i c a l s u b s t a n c e s . The t h e r m o d y n a m i c t h e o r y o u t l i n e d above c a n , i n p r i n c i p l e , be s t r a i g h t f o r w a r d l y a p p l i e d t o t h e d e s c r i p t i o n o f m i c r o b i a l growth and p r o d u c t f o r m a t i o n . I n o r d e r t o p e r f o r m s u c h an a n a l y s i s , t h e r m o d y n a m i c d a t a a r e needed r e g a r d i n g t h e compounds w h i c h a r e exchanged w i t h t h e environment, i . e . t h e p a r t i a l molar e n t h a l p i e s
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
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13.
ROELS
Macroscopic
Thermodynamics
and
Growth
301
and e n t r o p i e s o f t h e compounds i n v o l v e d i n t h e p r o c e s s e s i n t h e s y s t e m need t o be known. I n p r i n c i p l e t h e p a r t i a l m o l a r q u a n t i t i e s , e n t h a l p y a s w e l l as e n t r o p y , a r e f u n c t i o n s o f t h e c o n c e n t r a t i o n s o f e a c h and e v e r y c h e m i c a l compound p r e s e n t , i . e . d e f i n i t e v a l u e s c a n n o t be a t t r i b u t e d t o a s i n g l e compound. I n the approximation t o the e n e r g e t i c s o f m i c r o b i a l growth presented h e r e , i d e a l i t y o f t h e m i x t u r e o f compounds i n v o l v e d w i l l be assumed. T h i s i m p l i e s t h e p a r t i a l m o l a r t h e r m o d y n a m i c q u a n t i t i e s to be e q u a l t o t h e s p e c i f i c m o l a r q u a n t i t i e s , t h e s e l a t t e r q u a n t i t i e s depend o n i n t e n s i v e v a r i a b l e s l i k e t e m p e r a t u r e and p r e s s u r e and t h e c o n c e n t r a t i o n o f t h e compound c o n s i d e r e d o n l y . To a good d e g r e e o f a p p r o x i m a t i o n , e n t h a l p y c a n , a t a g i v e n t e m p e r a t u r e and p r e s s u r e , be assumed i n d e p e n d e n t o f t h e c o n c e n t r a t i o n o f t h e compound u n d e r c o n s i d e r a t i o n . The f r e e e n t h a l p y i s , h o w e v e r , by v i r t u e o f t h e e n t r o p y c o n t r i b u t i o n t o t h a t q u a n t i t y ( e q n . 1 3 ) , d e f i n i t e l y c o n c e n t r a t i o n d e p e n d e n t . The f o l l o w i n g relationship holds: g
L
= g? + R T l n C
(14)
i
i n w h i c h g? i s t h e f r e e e n t h a l p y a t a g i v e n t e m p e r a t u r e and p r e s s u r e and u n i t c o n c e n t r a t i o n o f t h e compound, w h i c h i s c a l l e d standard free enthalpy. C. i s t h e c o n c e n t r a t i o n o f compound i . As a f i r s t a p p r o a c h s t a n d a r d q u a n t i t i e s , i . e . a s s u m i n g u n i t c o n c e n t r a t i o n s , w i l l be u s e d i n t h e p r e s e n t e v a l u a t i o n o f g r o w t h and product formation. One f u r t h e r c o n v e n i e n t c o n v e n t i o n needs t o be d i s c u s s e d . E n e r g y , and hence a l s o d e r i v e d q u a n t i t i e s l i k e e n t h a l p y and f r e e e n t h a l p y , c a n n o t be a t t r i b u t e d a d e f i n i t e v a l u e ; i t s m a g n i t u d e c a n o n l y be d e f i n e d w i t h r e s p e c t t o a g i v e n r e f e r e n c e s t a t e w h i c h i s a t t r i b u t e d a zero energy l e v e l . A convenient r e f e r e n c e s t a t e f o r t h e e v a l u a t i o n o f g r o w t h and p r o d u c t f o r m a t i o n i n m i c r o o r g a n i s m s i s o b t a i n e d i f C 0 , H 0 , 0 and N a r e a s s i g n e d a z e r o e n e r g y l e v e l . The e n e r g y o f a compound t h u s becomes e q u a l t o i t s e n e r g y o f c o m b u s t i o n t o C 0 , H 0 and N . T h i s c o n v e n t i o n c a n be m o t i v a t e d by t h e f a c t t h a t m i c r o o r g a n i s m s c a n u n d e r no c i r c u m stances d e r i v e u s e f u l energy from processes i n which o n l y C 0 , H 0 , 0 and N a r e i n v o l v e d . The m o l a r f r e e e n t h a l p i e s and e n t h a l p i e s _ o f c o m b u s t i o n a t s t a n d a r d c o n d i t i o n s w i l l be termed Ag°. and A h , r e s p e c t i v e l y . ^ From e q n . (7) i t i s c l e a r t h a t t h e e n f n a l p y o f c o m b u s t i o n , A h ° ^ , equals t h e heat o f combustion. The f r e e e n t h a l p y o f c o m b u s t i o n , A g . , i s m a r k e d l y d e p e n d e n t on t h e c o n c e n t r a t i o n s o f t h e r e a c t a n t s i n v o l v e d ( e q n . ( 1 4 ) ) and most b i o l o g i c a l p r o c e s s e s t a k e p l a c e i n aquous s o l u t i o n s a t a h y d r o g e n i o n c o n c e n t r a t i o n c o r r e s p o n d i n g t o a pH o f 7 r a t h e r t h a n a t u n i t c o n c e n t r a t i o n o f t h e H - i o n , c o r r e s p o n d i n g t o a pH o f z e r o . The f r e e e n t h a l p i e s o f c o m b u s t i o n t o l i q u i d w a t e r , t h e HC0 i o n (the p r e d o m i n a n t f o r m i n w h i c h C 0 e x i s t s a t a pH o f 7) and N a t a pH o f 7 c a n t h u s be c o n s i d e r e d more r e l e v a n t t o b i o l o g i c a l 9
2
2
2
2
2
2
2
2
2
2
2
0
C 1
0
3
2
2
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
302
BIOCHEMICAL
ENGINEERING
t r a n s f o r m a t i o n s . The f r e e e n t h a l p y o f c o m b u s t i o n u n d e r t h e s e c o n d i t i o n s w i l l be i n d i c a t e d A g ! . The e n t h a l p y o f c o m b u s t i o n i s h a r d l y a f f e c t e d by t h e pH. T^e f r e e e n t h a l p i e s and e n t h a l p i e s o f c o m b u s t i o n a r e known t o obey r e g u l a r i t i e s ( 6 , 8 ) . T h e s e c a n be t r e a t e d u s i n g t h e c o n c e p t o f t h e d e g r e e o f r e d u c t i o n a s i n t r o d u c e d by M i n k e v i c h and E r o s h i n (_7) and e x t e n d e d and g e n e r a l i z e d by t h e p r e s e n t a u t h o r ( 5 , 6 ) . The g e n e r a l i z e d d e g r e e o f r e d u c t i o n , γ., o f a compound w i t h r e s p e c t t o m o l e c u l a r n i t r o g e n i s d e f i n e d by: Downloaded by UNIV OF MISSOURI COLUMBIA on April 15, 2013 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch013
0
Ύ
=
ί
4
+
a
i"
2
b
( 1 5 )
i
I n w h i c h a. and b. a r e t h e number o f m o l e s o f Η and 0 p r e s e n t i n one C-mole ( b e i n g t h e amount c o n t a i n i n g 12 grams o f c a r b o n ) o f coumpound i . F o r a component i t h e e n t h a l p i e s and f r e e e n t h a l p i e s o f c o m b u s t i o n p e r C-mole, t o be i n d i c a t e d A h ^ and Ag°. r e s p e c t i v e l y , a r e , t o a f i r s t a p p r o x i m a t i o n , a u n i q u e f u n c t i o n o f t h e d e g r e e o f r e d u c t i o n , γ., as i n t r o d u c e d i n e q n . ( 1 5 ) . A s t a t i s t i c a l a n a l y s i s o f d a t a f o r some 60 o r g a n i c compounds o f b i o l o g i c a l s i g n i f i c a n c e revealed the existence of the f o l l o w i n g r e g u l a r i t i e s : 0
0
Ah .
= 115γ.
CL
Ag°.
(16)
' 1
= 94.4γ. + 86.6
(17)
The r e s i d u a l e r r o r o f t h e e s t i m a t e i s 18 k J f o r b o t h e q n s . ( 1 6 ) and ( 1 7 ) . E q u a t i o n (16) s t a t e s t h a t t h e h e a t o f c o m b u s t i o n p e r C-mole i s more o r l e s s d i r e c t l y p r o p o r t i o n a l t o t h e d e g r e e o f r e d u c t i o n . As i s a p p a r e n t f r o m eqn. (17) s u c h a s i m p l e p r o p o r t i o n a l i t y r e l a t i o n does n o t a p p l y t o f r e e e n t h a l p i e s o f combustion. E q u a t i o n s (16) and (17) show t h a t s y s t e m a t i c d e v i a t i o n s b e t w e e n f r e e e n t h a l p i e s and h e a t s o f c o m b u s t i o n must e x i s t . F o r s u b s t r a t e s o f a low degree o f r e d u c t i o n the f r e e e n t h a l p i e s o f combustion exceed t h e heats o f combustion, f o r s u b s t r a t e s of a h i g h d e g r e e o f r e d u c t i o n t h e r e v e r s e a p p l i e s . T h i s phenomenon c a n be i l l u s t r a t e d i f t h e e n t r o p y c o n t r i b u t i o n t o t h e f r e e enthalpy o f combustion, T A s . , i s c a l c u l a t e d . I t i s obtained from the equation: 0
Ag°. 6
C1
0
0
CI
CI
= Ah . - TAs .
(18)
I n f i g . 2 t h e r e l a t i o n i s shown b e t w e e n t h e s a i d e n t r o p y c o n t r i b u t i o n and t h e d e g r e e o f r e d u c t i o n u s i n g d a t a f o r a v a r i e t y o f o r g a n i c compounds. A d e f i n i t e t r e n d c a n i n d e e d be shown t o e x i s t ( a p a r t f r o m an i n c i d e n t a l o u t l y e r ) : The e n t r o p y c o n t r i b u t i o n i n c r e a s e s w i t h i n c r e a s i n g degree o f r e d u c t i o n . I n v i e w o f the o b s e r v a t i o n t h a t f r e e e n t h a l p y changes a t a pH o f 7 may w e l l be more r e l e v a n t i n m i c r o b i o l o g i c a l p r o c e s s e s , t h e e n t r o p y c o n t r i b u -
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
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13.
Macroscopic
ROELS
501 ο
Thermodynamics
ι 2.5
and
303
Growth
ι 5
ί 7.5
Γ Figure 2. The entropy contribution, T A s ° (kJ/C-mole), to the free enthalpy of combustion at standard conditions, as a function of the degree of reduction, y, of the compounds considered, for acids (%), carbohydrates (A), alkanes (Ο), ethene and ethyne (O), alcohols (%), acetone (|Λ aldehydes (A), and amino acids (*). c
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
304
BIOCHEMICAL
ENGINEERING
0 f
t i o n t o t h e f r e e e n t h a l p y o f c o m b u s t i o n a t a pH o f 7, T A s , was a l s o c a l c u l a t e d . T h e s e v a l u e s a r e p l o t t e d as a f u n c t i o n o? t h e d e g r e e o f r e d u c t i o n i n f i g . 3. The f e a t u r e s o f f i g s . 2 and 3 a r e s e e n t o be v e r y much a l i k e , e x c e p t f o r t h e a c i d s , w h i c h have a m a r k e d l y h i g h e r T A s a t a pH o f 7, i . e . t h e i r f r e e e n t h a l p y o f c o m b u s t i o n i s îower a t t h a t pH. T h i s d i f f e r e n c e b e t w e e n s t a n d a r d c o n d i t i o n s and a s i t u a t i o n i n w h i c h t h e c o n c e n t r a t i o n s d i f f e r f r o m u n i t y , e.g. a t a pH o f 7, i s c h a r a c t e r i s t i c f o r a l i m i t a t i o n o f t h e use o f s t a n d a r d f r e e e n t h a l p y changes i n t h e a n a l y s i s o f p r o c e s s e s , where t h e c o n c e n t r a t i o n s a t t h e l o c a l e o f t h e p r o c e s s may d i f f e r f r o m u n i t y : t h e Ag° v a l u e s c a n o n l y be a p p l i e d t o an a p p r o x i m a t e a n a l y s i s . F o r d e t a i l e d c o n s i d e r a t i o n s the A g v a l u e s at the c o n c e n t r a t i o n s at the l o c a l e o f e n e r g y g e n e r a t i o n a r e n e e d e d . S t i l l , as w i l l be shown, t h e approximate analyses g r e a t l y c o n t r i b u t e to the understanding of m i c r o b i a l energetics.
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0 f
Q
T h e r e a l s o e x i s t compounds o f w h i c h t h e e n e r g y r e l e a s e d on t h e i r c o m b u s t i o n does d e f i n i t e l y n o t f o l l o w t h e t r e n d s i n d i c a t e d by e q n s . (16) and ( 1 7 ) . E x a m p l e s a r e o x y g e n and n i t r i c a c i d w h i c h have a " h e a t o f c o m b u s t i o n " w h i c h i s c l o s e t o z e r o and a d e g r e e o f r e d u c t i o n o f -4 and -5 r e s p e c t i v e l y . By v i r t u e o f t h i s f e a t u r e t h e s e compounds can s e r v e as v e r y e f f i c i e n t " e l e c t r o n a c c e p t o r s " . The t h e r m o d y n a m i c e f f i c i e n c y . The t h e r m o d y n a m i c t h e o r y d e v e l o p e d e a r l i e r was shown t o r e s u l t i n eqn. ( 1 2 ) , a r e s t r i c t i v e e q u a t i o n r e g a r d i n g the f l o w s o f m a t t e r exchanged w i t h the e n v i r o n ment by an open s y s t e m . On eqn. (12) a d e f i n i t i o n o f t h e t h e r m o d y n a m i c e f f i c i e n c y c a n be b a s e d i f t h e d i s s i p a t i o n , ΤΠ , i s compared t o t h e t o t a l o f t h e f l o w s o f f r e e e n t h a l p y e n t e r i n g t h e s y s t e m . However, a p r o b l e m w h i c h was a l r e a d y i n d i c a t e d above has t o be t a c k l e d . The amount o f e n e r g y c a n n o t be s p e c i f i e d i n a u n i q u e way and i t can o n l y be d e f i n e d w i t h r e f e r e n c e t o a b a s e l e v e l , w h i c h i s a r b i t r a r i l y a t t r i b u t e d z e r o e n e r g y c o n t e n t . As s o o n as s u c h a r e f e r e n c e s t a t e has been c h o s e n , t h e t h e r m o d y n a m i c e f f i c i e n c y i s e a s i l y c a l c u l a t e d . The p r o c e d u r e i s i l l u s t r a t e d i n f i g . 4. The t h e r m o d y n a m i c e f f i c i e n c y , fL,» i s d e f i n e d e q u a l t o t h e r a t i o o f t h e f r e e e n t h a l p y g a i n e d i f t h e compounds l e a v i n g t h e s y s t e m were t r a n s f o r m e d t o t h e r e f e r e n c e s t a t e , t o t h a t , w h i c h w o u l d be o b t a i n e d i f t h i s p r o c e d u r e were a p p l i e d t o t h e compounds e n t e r i n g t h e s y s t e m . I t i s e a s i l y u n d e r s t o o d t h a t η , i s c o n s t r a i n e d between z e r o , i f a l l t h e f r e e e n t h a l p y e n t e r i n g trie s y s t e m i s d i s s i p a t e d and u n i t y i f Ag!j e q u a l s Ag^ , i . e . i f t h e d i s s i p a t i o n equals zero. I t i s important to r e a l i z e t h a t the f o r m e r c o n s t r a i n t s t r o n g l y depends on t h e c o r r e c t c h o i c e o f t h e reference state. A p p l i c a t i o n s of the
theory
A e r o b i c g r o w t h w i t h o u t p r o d u c t f o r m a t i o n . The a p p l i c a t i o n of the theory to a e r o b i c growth without f o r m a t i o n of products
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
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13.
Macroscopic
ROELS
Thermodynamics
and
305
Growth
k J/C-mole
100
50
Π Ο
Θ
Δ*
*
-50
*
*
*
** *
7.5
2.5 Υ
Figure 3. The entropy contribution, T A S (kJ/C-mole), to the free enthalpy of combustion at a pH of 7, as a function of the degree of reduction, y, of the com pounds considered; symbols as in Figure 2. C
0 /
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
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306
BIOCHEMICAL
ENGINEERING
w i l l now be shown. F i g u r e 5 shows t h e s y s t e m and t h e exchange f l o w s f o r t h i s c a s e . The e l e m e n t a r y c o m p o s i t i o n s and t h e h e a t s o f c o m b u s t i o n o f t h e compounds exchanged w i t h t h e e n v i r o n m e n t a r e i n d i c a t e d . A e r o b i c g r o w t h i s f o r t h e p r e s e n t a n a l y s i s assumed t o i n v o l v e u p t a k e o f a n i t r o g e n s o u r c e , a c a r b o n s o u r c e and o x y g e n and t r a n s p o r t t o t h e e n v i r o n m e n t o f c a r b o n d i o x i d e , w a t e r and new b i o m a s s . The b i o m a s s i s assumed t o be one compound, w h i c h c a n be c h a r a c t e r i z e d f u l l y by i t s e l e m e n t a l c o m p o s i t i o n . As t h e e n t h a l p y as w e l l as t h e f r e e e n t h a l p y o f c o m b u s t i o n , i . e . t h e e n t h a l p y and f r e e e n t h a l p y c o n t e n t w i t h r e s p e c t t o t h e r e f e r e n c e s t a t e adopted i n t h e p r e s e n t a n a l y s i s , i s zero f o r oxygen, carbon d i o x i d e and w a t e r , i t i s e a s i l y u n d e r s t o o d t h a t t h e thermodynamic e f f i c i e n c y c a n be d e f i n e d a s f o l l o w s : Φ Ag° n'th . u
-
—
^
—
(19)
^
cN i n which Φ , and Φ^ a r e t h e f l o w s o f b i o m a s s ( C - m o l e / h r ) , s u b s t r a t e ^ C - m o l e / h r ) and n i t r o g e n s o u r c e ( m o l e / h r ) t o o r f r o m t h e s y s t e m ( a s i n d i c a t e d i n f i g u r e 5 ) . Ag° and A g ° a r e t h e f r e e e n t h a l p i e s _ o f c o m b u s t i o n o f a C-mole o i b i o m a s s and s u b s t r a t e r e s p e c t i v e l y . A g ^ i s t h e f r e e e n t h a l p y o f combustion o f a mole of t h e n i t r o g e n source. E q u a t i o n (19) c a n a l s o be w r i t t e n a s : g
η,.
Ag° — A o / r g
s
+
(20) ( /c )Ai» C )
4
N
In the formulation o f t h i s equation a balance f o r atomic n i t r o g e n i s u s e d t o r e l a t e t h e f l o w s Φ , and Φ . Y i sa yield factor • Ν χ sx · f o r b i o m a s s o n s u b s t r a t e o n a p e r C-mole b a s e ; C-moles o f b i o m a s s p r o d u c e d p e r C-mole o f s u b s t r a t e consumed, t h u s i t i s d e f i n e d by: f f
%
γ " = φ /φ SX X s
(21)
A n o t h e r u s e f u l e f f i c i e n c y measure i s t h e s o - c a l l e d e n t h a l p y e f f i c i e n c y o f g r o w t h , η^, i t i s by a n a l o g y o b t a i n e d i f t h e Ag°. i n eqn. (20) a r e r e p l a c e d by t h e r e s p e c t i v e A h . . As was alreaày i n d i c a t e d t h e e n t h a l p y e f f i c i e n c y does n o t have t h e f u n d a m e n t a l p r o p e r t i e s o f t h e thermodynamic e f f i c i e n c y a s t h e r e s t r i c t i o n f o l l o w i n g from t h e a p p l i c a t i o n o f t h e second law o f thermodynamics ( s e e e q n . ( 1 1 ) ) d o e s n o t pose an u p p e r l i m i t t o Φ^. H e n c e , p r o c e s s e s f o r w h i c h η„ e x c e e d s u n i t y c a n by no means be e x c l u d e d . I t c a n e a s i l y b e shown t h a t t h e e f f i c i e n c y m e a s u r e s d e v e l o p e d a b o v e , i . e . η^, and η^, a l l o w t h e f o r m u l a t i o n o f e x p r e s s i o n s f o r t h e d i s s i p a t i o n ΤΠ and t h e h e a t p r o d u c t i o n (-Φ„). The f o l l o w i n g S n 0
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
13.
ROELS
FREE
Macroscopic
Thermodynamics
and
307
Growth
ENTHALPY
ENTERING
SYSTEM
FREE
ENTHALPY
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LEAVING
FREE
ENTHALPY
OF
REFERENCE
SYSTEM
STATE
Figure 4. The thermodynamic efficiency, ηα, of a process. A g / and A g / are the amounts of free enthalpy gained when the compounds entering and leaving the system, respectively, are transformed to the reference state.
c
_\h°-0
co
2
_H 0 2
cs
W
Figure 5.
W
CHa
b
Nc
i° i i
System and flows for thermodynamic analysis of aerobic growth formation of products.
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
without
BIOCHEMICAL
308 equations
ENGINEERING
hold: (22)
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(23) I n eqn. (22) D e q u a l s t h e d i s s i p a t i o n ΤΠ . For a e r o b i c growth w i t h o u t product f o r m a t i o n the a b s o l u t e m a g n i t u d e second t e r m a p p e a r i n g a t t h e l e f t hand s i d e o f eqn. (11) c a n be shown t o be s m a l l as compared t o Φ^/Τ, i . e . t h e exchange o f h e a t l a r g e l y e x c e e d s t h e exchange o f " c h e m i c a l e n t r o p y " ( 5 , 6^. T h i s i m p l i e s t h a t D and a r e , t o a good a p p r o x i m a t i o n e q u a l and t h i s , o f c o u r s e , a l s o a p p l i e s t o η * Π^· I n t h i s c a s e an a n a l y s i s b a s e d on an e n t h a l p y e f f i c i e n c y o f g r o w t h i s more o r l e s s v a l i d and hence a l s o t h e i n t e r p r e t a t i o n of t h e s e c o n d law as t o f o r b i d p r o c e s s e s w i t h u p t a k e o f h e a t . As has been shown e a r l i e r (6) t h e v e r y f a c t t h a t eqn. (16) i s more o r l e s s v a l i d i m p l i e s t h a t h e a t p r o d u c t i o n and oxygen consumption, Φ , are p r o p o r t i o n a l a c c o r d i n g to the f o l l o w i n g equation : a n a
Φ„ = 460
Φ
(24)
ο
The a p p r o x i m a t e v a l i d i t y o f t h i s e q u a t i o n i s e a s i l y u n d e r s t o o d as f o l l o w s . The d e g r e e o f r e d u c t i o n γ. i n d i c a t e s t h e number o f moles o f e l e c t r o n s a v a i l a b l e f o r t r a n s f e r t o o x y g e n on c o m p l e t e c o m b u s t i o n o f a C-mole o f a compound t o C0 , H 0 and N . On t r a n s f e r t o oxygen t h e e n e r g y o f t h e e l e c t r o n s i s d i s s i p a t e d , r e s u l t i n g i n a h e a t p r o d u c t i o n o f 115 k J p e r mole o f e l e c t r o n s . On a e r o b i c g r o w t h p a r t o f t h e e n e r g y c o n t e n t o f t h e s u b s t r a t e and the n i t r o g e n source i s conserved i n the form o f newly s y n t h e s i z e d biomass, the e l e c t r o n s c o r r e s p o n d i n g to the remainder a r e t r a n s f e r r e d t o o x y g e n and p r o v i d e d i s s i p a t i o n . As f o u r moles o f e l e c t r o n s a r e a c c e p t e d by one m o l e o f o x y g e n eqn. (16) shows 460 k J t o be g e n e r a t e d f o r e a c h m o l e o f oxygen consumed. The v a l i d i t y o f eqn. (24) shows t h a t a t r e a t m e n t can a l s o be b a s e d on o x y g e n e f f i c i e n c y o f g r o w t h C5"\7) · A s u b s t a n t i a l body o f d a t a on a e r o b i c g r o w t h w i t h o u t p r o d u c t f o r m a t i o n s u p p o r t e d by ammonia as a n i t r o g e n s o u r c e has r e c e n t l y been r e v i e w e d ( 9 ) . From t h e s e d a t a t h e v a l u e s o f η (- η ) and t h e d i s s i p a t i o n p e r u n i t b i o m a s s p r o d u c e d (D/Φ , k J / C - m o l e ) were c a l c u l a t e d . The r e s u l t s were a v e r a g e d f o r e a c h o f t h e c a r b o n s o u r c e s c o n s i d e r e d and a r e shown i n f i g s . 6 and 7. A l t h o u g h , not u n e x p e c t e d l y , i t i s c l e a r t h a t c o n s i d e r a b l e s c a t t e r i s e x i s t e n t i n b o t h f i g s . 6 and 7,some g l o b a l r e g u l a r i t i e s seem t o be p r e s e n t , w h i c h a r e i n d i c a t e d i n t h e f i g u r e s . F o r s u b s t r a t e s w i t h a d e g r e e o f r e d u c t i o n l o w e r t h a n about 5, t h e thermo dynamic e f f i c i e n c y a v e r a g e s 0.58 ( t h e o n l y s i g n i f i c a n t o u t l y e r b e i n g t h e v e r y low e f f i c i e n c y o b s e r v e d f o r g r o w t h s u p p o r t e d by 2
2
2
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
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13.
ROELS
Macroscopic
Thermodynamics
and
Growth
309
0.25
Figure 6. Thermodynamic efficiency, η , of aerobic growth with NH as the nitrogen source, plotted as a function of the degree of reduction, γ, of the substrate. Theoretical limits due to the second law and C-limitation. Shown are averages of experimental data for methane (%), n-alkanes (A), methanol (J^), ethanol (*ψ), glycerol (®), mannitol (O), acetic acid (Δλ lactic acid glucose (*), formalde hyde (VA gluconic acid (S), succinic acid (@), citric acid (®), malic acid (®), formic acid (®), oxalic acid (A)ιη
S
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
310
BIOCHEMICAL
Î
1270
1240
I·
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A
ENGINEERING
Figure 7. Dissipation of aerobic growth (kJ/C-mole of biomass produced) for aerobic growth with ΝΗ as a nitrogen source as a function of the degree of reduction of substrate. Theoretical limits due to second law and C-limitation. Average of experimental data for various substrates, symbols as in Figure 6. Ά
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
Downloaded by UNIV OF MISSOURI COLUMBIA on April 15, 2013 | http://pubs.acs.org Publication Date: January 18, 1983 | doi: 10.1021/bk-1983-0207.ch013
13.
ROELS
Macroscopic
Thermodynamics
and
311
Growth
o x a l i c a c i d ) , t h i s corresponds to a d i s s i p a t i o n of roughly 400 k J / m o l e o f b i o m a s s p r o d u c e d . F o r s u b s t r a t e s w i t h a d e g r e e o f r e d u c t i o n e x c e e d i n g 5 t h e thermodynamic e f f i c i e n c y p r o g r e s s i v e l y d e c r e a s e s and t h e d i s s i p a t i o n i n c r e a s e s up t o 1400 kJ/C-mole f o r t h e c a s e o f g r o w t h s u p p o r t e d by methane. I n f i g s . 6 and 7 t h e r e s t r i c t i o n s o f u n i t e f f i c i e n c y and z e r o d i s s i p a t i o n r e s p e c t i v e l y , a s imposed by t h e s e c o n d law a r e i n d i c a t e d as w e l l as a l i m i t imposed by a l i m i t a t i o n o f a d i f f e r e n t n a t u r e , i . e . c a r b o n l i m i t a t i o n . The l a t t e r l i m i t m e r i t s a more t h o r o u g h d i s c u s s i o n . The d e g r e e o f r e d u c t i o n , γ , o f b i o mass f r o m a v a r i e t y o f s o u r c e s a v e r a g e s 4.8 (5,6,10) and hence by v i r t u e o f eqn. (16) i t s e n e r g y c o n t e n t i s a b o u t 550 k J / C - m o l e . I f g r o w t h i s s u p p o r t e d by a s u b s t r a t e o f a d e g r e e o f r e d u c t i o n e x c e e d i n g 4.8 i t s e n e r g y c o n t e n t w i l l e x c e e d t h e s a i d 550 k J / C - m o l e and h e n c e , even i f a l l s u b s t r a t e c a r b o n were f i x e d i n b i o m a s s , a thermodynamic e f f i c i e n c y l o w e r t h a n u n i t y w o u l d be o b t a i n e d . A c o m p l e t e e x p l o i t a t i o n o f t h e e n e r g y p r e s e n t i n t h e s u b s t r a t e w o u l d r e q u i r e f i x a t i o n o f a d d i t i o n a l low e n e r g y c a r b o n e.g. f r o m C0 . T h i s i s , however, e x c l u d e d , as o n l y one c a r b o n s o u r c e i s assumed t o be s u p p l i e d . On o b s e r v a t i o n o f f i g . 6 i t becomes c l e a r t h a t t h e t r e n d s o b s e r v e d i n t h e e x p e r i m e n t a l v a l u e s o f t h e thermodynamic e f f i c i e n c y mimic t h e shape o f t h e t h e o r e t i c a l r e s t r i c t i o n s a t a l e v e l o f a b o u t 60%. The d e v i a t i o n between t h e t h e o r e t i c a l l i m i t and t h e v a l u e s a c t u a l l y o b s e r v e d i s q u i t e e a s i l y u n d e r s t o o d i n g e n e r a l terms i n t h e r e g i o n o f low d e g r e e s o f r e d u c t i o n , where the energy a v a i l a b l e i n the s u b s t r a t e r a t h e r than i t s carbon c o n t e n t l i m i t s the v a l u e s o f Y . The t h e o r e t i c a l l i m i t o f u n i t y sχ · c a n n e v e r be r e a c h e d as any p r o c e s s needs a n o n - z e r o d i s s i p a t i o n t o proceed at a non-zero r a t e (2-4). In f a c t the r a t e at which a p r o c e s s p r o c e e d s , e.g. t h e r a t e o f g r o w t h o f t h e amount o f biomass, i s to a c e r t a i n extent i n c r e a s i n g w i t h i n c r e a s i n g d i s s i p a t i o n . V a r i o u s o p t i m a l i t y p r i n c i p l e s ( 1 1 ) , may d i c t a t e an o p t i m a l thermodynamic e f f i c i e n c y o f r o u g h l y t h e m a g n i t u d e o b s e r v e d h e r e ( 1 1 , 1 2 ) . The b e h a v i o u r a t h i g h d e g r e e s o f r e d u c t i o n i s l e s s e a s i l y u n d e r s t o o d on f u n d a m e n t a l g r o u n d s . The t h e r m o dynamic e f f i c i e n c y c o u l d w e l l a p p r o a c h t h e t h e o r e t i c a l l i m i t c l o s e r with a s u f f i c i e n t l y l a r g e d i s s i p a t i o n . A p p a r e n t l y o t h e r phenomena t o w h i c h t h e f o r m a l m a c r o s c o p i c t r e a t m e n t p r o v i d e s no c l u e s , l i m i t t h e maximum c o n s e r v a t i o n o f s u b s t r a t e c a r b o n i n b i o m a s s t o a b o u t 2/3 o f t h e maximum. O b v i o u s l y one has t o r e s o r t t o b i o c h e m i c a l t h e o r y t o o b t a i n c l u e s t o the n a t u r e o f t h e mechanisms u n d e r l y i n g t h e phenomenon. T h e r e e x i s t s s t i l l a n o t h e r u s e f u l q u a n t i t y , w h i c h has been used t o s y s t e m a t i z e t h e e x p e r i m e n t a l d a t a on t h e e f f i c i e n c y o f g r o w t h s u p p o r t e d by v a r i o u s c a r b o n s o u r c e s ; i t i s P a y n e s y i e l d on a v a i l a b l e e l e c t r o n s , Υ , ( 1 3 ) . I t i s d e f i n e d as t h e amount o f b i o m a s s p r o d u c e d p e r mole o f e l e c t r o n s a v a i l a b l e f o r t r a n s f e r t o o x y g e n on c o m p l e t e c o m b u s t i o n . N u m e r i c a l l y t h e number o f m o l e s o f e l e c t r o n s a v a i l a b l e f o r t r a n s f e r t o o x y g e n on 2
M
1
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
312
BIOCHEMICAL
ENGINEERING
c o m b u s t i o n o f a C-mole o f a g i v e n s u b s t r a t e i s e q u a l t o t h e d e g r e e o f r e d u c t i o n , γ., as d e f i n e d by eqn. ( 1 5 ) . As a f i r s t a p p r o x i m a t i o n , t a k i n g i n t o account the f a c t t h a t biomass from a«-·• vVW a. rAi.^ eW tWy^ νo fΛ. I mUX i.V c.rLVUo b iJLaj. a l sIo7uWUr1c.eU^t7 s can beVJ t.w eW lI— lJ. J- r e p r e s e n t e d by t h e COU c a n be c o n s i d e r e d composition formula C H O . 5 N 0 . 2 » / p r o p o r t i o n a l t o X]^ ( o r r a t h e r n ) (§, 6 ) Y
l e 8
C
0
e
H
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Υ
, av/e
= 5.85
η., th
(25)
F i g u r e 8 shows t h e d a t a d e p i c t e d i n f i g s . 6 and 7 i n a p l o t o f Y . v e r s u s t h e d e g r e e o f r e d u c t i o n γ. The t r e n d s and t h e o r e t i c a l l i m i t s a r e a l s o shown. Growth w i t h e l e c t r o n a c c e p t o r s o t h e r than oxygen. In the p r e c e e d i n g s e c t i o n a c a s e was t r e a t e d i n w h i c h o x y g e n was t h e e l e c t r o n a c c e p t i n g m o i e t y . Oxygen i s , h o w e v e r , by no means t h e o n l y compound, w h i c h c a n p e r f o r m s u c h f u n c t i o n . O t h e r examples a r e n i t r a t e and s u l p h a t e . The c h a r a c t e r i s t i c f e a t u r e o f an e l e c t r o n a c c e p t o r i s t h a t f r e e e n t h a l p y i s gained i n the o v e r a l l p r o c e s s o f t r a n s f e r o f an e l e c t r o n f r o m a g i v e n s o u r c e (a s u b s t r a t e ) to the acceptor. T h i s f r e e enthalpy i s p a r t l y d i s s i p a t e d t o p r o v i d e t h e n e c e s s a r y i r r e v e r s i b i l i t y and i t c a n be u s e d t o t r a n s f e r c a r b o n ( - d i o x i d e ) f r o m a low e n e r g y l e v e l t o a h i g h e n e r g y one. I n t a b l e I a summary i s p r o v i d e d o f t h e e n e r g y b e c o m i n g a v a i l a b l e on t r a n s f e r o f one m o l e o f e l e c t r o n s f r o m g l u c o s e t o a g i v e n e l e c t r o n a c c e p t o r engaged i n a g i v e n reduction process. Table
I.
F r e e e n t h a l p y g a i n e d a t pH = 7 and s t a n d a r d c o n d i t i o n s on t r a n s f e r o f one m o l e o f e l e c t r o n s f r o m g l u c o s e t o a g i v e n e l e c t r o n a c c e p t o r (14)
e l e c t r o n acceptor + * (NH ) N +
4
2
S
moles of e l e c t r o n s accepted 6
(HS~) 2-
SO, 4
(HS
)
0
1
Ag ^ ^ kJ 79.1
^Sai/^ kjfmole 13.2
2
27.7
13.9
8
150.7
18.8
6
171.9
28.7
9
S 0
3 N0 ~ 2
(HS~) +
6
435.5
72.6
+
598.4
74.8
(NH ) 4
N0 "
(NH )
8
N0 "
(N ) 2
5
559.5
111.9
(H 0)
4
473.9
118.5
3
3
°2 *
4
2
I n b r a c k e t s reduced
form of e l e c t r o n a c c e p t o r
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
Macroscopic
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ROELS
Thermodynamics
and Growth
313
'av/e g DM/mole 6h
SECOND LAW
4H
2h
1h
Figure 8. Yield on electrons available for transfer to oxygen, Yav/e, (g DM/mole av/e) for aerobic growth with NH as a nitrogen source. Theoretical limits due to the second law and C-limitation. Average of experimental data for various substrates, symbols as in Figure 6. 3
In Foundations of Biochemical Engineering; Blanch, H., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1983.
BIOCHEMICAL
314
ENGINEERING
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The v a l u e s o f t h e f r e e e n t h a l p y g a i n e d p e r m o l e o f e l e c t r o n s t r a n s f e r r e d a r e a r r a n g e d i n o r d e r o f i n c r e a s i n g m a g n i t u d e . The " e f f i c i e n c y " o f an e l e c t r o n a c c e p t o r t h u s i n c r e a s e s f r o m t h e t o p to t h e b o t t o m o f t h e t a b l e and f a i t h i n l i f e ' s u n i v e r s a l s t r i v i n g f o r more o p t i m a l g r o w t h w o u l d assume p r e f e r e n c e f o r t h e e l e c t r o n a c c e p t o r s a t t h e b o t t o m o f t h e t a b l e i f two o r more e l e c t r o n acceptors are simultaneously a v a i l a b l e . Growth w i t h o u t e x t e r n a l l y s u p p l i e d e l e c t r o n a c c e p t o r s . I n a number o f c a s e s g r o w t h i s o b s e r v e d t o t a k e p l a c e w i t h o u t an i d e n t i f i a b l e seperate e l e c t r o n a c c e p t o r b e i n g p r e s e n t . I n such case t h e s u b s t r a t e o r a s u b s t r a t e d e r i v e d moiety i s both donor and a c c e p t o r o f e l e c t r o n s . F o r s i m p l i c i t y ' s s a k e o n l y t h e c a s e o f a n a e r o b i c g r o w t h on a s i n g l e c a r b o n s o u r c e w i t h f o r m a t i o n o f a s i n g l e p r o d u c t and NH^ as t h e n i t r o g e n s o u r c e w i l l be t r e a t e d . I t i s e a s i l y u n d e r s t o o d t h a t an a n a l y s i s o f a n a e r o b i c g r o w t h c a n be b a s e d o n a b a l a n c e o n t h e d e g r e e o f r e d u c t i o n as d e f i n e d by eqn. ( 1 6 ) . T a b l e I I shows t h e f l o w s o f t h e v a r i o u s compounds and t h e i r c o n t e n t o f e l e c t r o n s a v a i l a b l e f o r t r a n s f e r t o oxygen on f o r m a t i o n o f Φ C-moles of b i o m a s s . χ Table I I .
A degree o f r e d u c t i o n balance f o r anaerobic growth w i t h o u t e x t e r n a l e l e c t r o n a c c e p t o r s
molecular species
degree of reduction
substrate N-source
Y
(NH^)
contribution to degree o f reduction balance
flow
γ Φ /Y" 's X SX 3ο.Φ 1 χ
Φ /Υ" χ
s
X
3
C φ
Y
Φ
SX
1 χ B i o m a s s (CH ,0, N -) a1 b1 c1 4
x
- γ Φ χ χ
χ
Product
- γ φ Ρ Ρ
φ Y
P
Ρ
2
0
Φ /Υ" --Φ -φ χ sx χ ρ
0
H 0
0
Φ
0
co 2
W
By v i r t u e o f t h e f a c t t h a t no e x t e r n a l e l e c t r o n a c c e p t o r s are present the c o n t r i b u t i o n s t o the degree o f r e d u c t i o n balance as shown i n t h e l a s t c o l u m n o f t a b l e 2 must add up t o z e r o , i t follows : Φ
Φ
