1.3.4 Derivation of Generation Rate of Entropy Production
Evidence that rate of entropy production is shown as Φ ¼
P
J i X i is explained as
follows. When general expression of conservation law is converted on flowing
coordinate and equation of continuity is used, φ s becomes internal production rate
of entropy. Therefore, equation showing flow of entropy of system in
non-equilibrium state, ρ
ds
dt ¼
Φ
T À ∇j s is obtained when φ s ¼
Φ
T . On the other hand,
Gibbs’s equation, du ¼ Tds À pdV þ
P
i μ i dx i is converted to ρ
ds
dt ¼
ρ
T
du
dt À
ρ
T
P μ i
dx i
dt
by use of v ¼
1
ρ .
Following equations of differential of time are replaced in the Gibb’s equation.
Internal energy À ρ
du
dt
¼ σ : ∇v À ∇q À ∇
X
j i h i þ
X
j i x i
Pressure
dρ
dt
¼ Àρ∇v
Mole fraction of component ρ
dx i
dt
¼ À∇j i þ ρ
X
k
ν
0
ik
dζ k
dt
And the equation is classified into 4 terms by use of j s ¼
q
T À
P
j s i s , and they are
viscous flow of unit volume gravity center, chemical reaction in system, diffusion of
substance in isothermal system and entropy production rate by heat flow. From
comparison of equations, meaning of each term is obtained as follows. Force of
velocity gradient of gravity center of part makes flux of stress and static pressure.
Force of affinity makes flux of chemical reaction. Force of chemical potential
gradient makes flux of diffusion. Force of temperature gradient makes flux of heat
flow. These relations are supposed as product of flux and force. Therefore, Φ ¼ ∑ J i X i
is obtained.
References
1. M.O. Dayhoff, E.R. Lippincott, R.V. Eck, Science 146, 1461 (1964)
2. A.C. Butlerov, Comptes Rendus, vol 53 (1861), p. 145
3. J. Oró, A.P. Kimball, Arch. Biochem. Biophys. 96, 293 (1962)
4. S.L. Miller, Science 117, 528 (1953)
5. N.H. Horrowitz, Proc. NAS 31, 153 (1945)
6. L. Michaelis, M. Menten, Biochem. Z. 49, 369 (1913)
7. G.E. Briggs, J.B.S. Haldane, Biochem. J. 19, 339 (1925)
8. H.J. Morowits, Energy Flow in Biology (Ox Bow Press, Woodbridge, 1979)
9. P. Glansdorff, I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations
(Willey-Interscience, London, 1971)
References
17
Evidence that rate of entropy production is shown as Φ ¼
P
J i X i is explained as
follows. When general expression of conservation law is converted on flowing
coordinate and equation of continuity is used, φ s becomes internal production rate
of entropy. Therefore, equation showing flow of entropy of system in
non-equilibrium state, ρ
ds
dt ¼
Φ
T À ∇j s is obtained when φ s ¼
Φ
T . On the other hand,
Gibbs’s equation, du ¼ Tds À pdV þ
P
i μ i dx i is converted to ρ
ds
dt ¼
ρ
T
du
dt À
ρ
T
P μ i
dx i
dt
by use of v ¼
1
ρ .
Following equations of differential of time are replaced in the Gibb’s equation.
Internal energy À ρ
du
dt
¼ σ : ∇v À ∇q À ∇
X
j i h i þ
X
j i x i
Pressure
dρ
dt
¼ Àρ∇v
Mole fraction of component ρ
dx i
dt
¼ À∇j i þ ρ
X
k
ν
0
ik
dζ k
dt
And the equation is classified into 4 terms by use of j s ¼
q
T À
P
j s i s , and they are
viscous flow of unit volume gravity center, chemical reaction in system, diffusion of
substance in isothermal system and entropy production rate by heat flow. From
comparison of equations, meaning of each term is obtained as follows. Force of
velocity gradient of gravity center of part makes flux of stress and static pressure.
Force of affinity makes flux of chemical reaction. Force of chemical potential
gradient makes flux of diffusion. Force of temperature gradient makes flux of heat
flow. These relations are supposed as product of flux and force. Therefore, Φ ¼ ∑ J i X i
is obtained.
References
1. M.O. Dayhoff, E.R. Lippincott, R.V. Eck, Science 146, 1461 (1964)
2. A.C. Butlerov, Comptes Rendus, vol 53 (1861), p. 145
3. J. Oró, A.P. Kimball, Arch. Biochem. Biophys. 96, 293 (1962)
4. S.L. Miller, Science 117, 528 (1953)
5. N.H. Horrowitz, Proc. NAS 31, 153 (1945)
6. L. Michaelis, M. Menten, Biochem. Z. 49, 369 (1913)
7. G.E. Briggs, J.B.S. Haldane, Biochem. J. 19, 339 (1925)
8. H.J. Morowits, Energy Flow in Biology (Ox Bow Press, Woodbridge, 1979)
9. P. Glansdorff, I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations
(Willey-Interscience, London, 1971)
References
17