428
10 The Statistical Basis of Bioenergetics
Appendix F shows how this expression leads to the famous Planck’s distribution for
light from a hot body, like ourselves or the Sun.
10.4 Classical Limit of the Particle Distributions
From a macroscopic view, the discrete nature of energy levels is usually not evident.
Classical macrosystems have particles widely spread out over a set of available
microscopic energy levels with many quantum states near each level. (So g i >>
n i >> 1.) A simple example is an ideal gas at a finite temperature, wherein there
are many more kinetic energy quantum states available in the range of energies
carried by the molecules than there are molecules occupying them.
The particles in such sparse (dilute) systems do not come close to filling the
available quantum states. The Pauli exclusion principle has little impact on the
thermodynamics of these systems. The distinction between bosons and fermions
no longer affects the macroscopic measurements of such systems near equilibrium.
For sparse systems, i.e. systems for which the degeneracy, g i , is much greater
than the occupation number n i (so g i >> n i >> 1), 10 we will have the
approximation, for both W bosons and W fermions ,
W boltzons
r
i=1
g
n i
i
n i !
.
(10.8)
The number W boltzons differs from W Boltzmann because Boltzmann naturally assumed
that the particles in the distribution were distinguishable. His work came before the
full development of quantum theory. Quantum theory makes particles of the same
type indistinguishable.
At or near equilibrium, Eq. (10.8) gives
n i e
α g i e
−ββ i .
(10.9)
This is the distribution of particles found by Boltzmann in 1871, using classical
arguments, and derived even earlier for the special case of an ideal gas by
Maxwell. 11
Biosystems, because they take advantage of complexity, tend to have a dense
array of energy states with occupations of higher energy states beyond the ultimate
10 This condition works for a wide variety of biological systems, including dilute solutions of
macromolecules. An important exception is the system of valence electrons in metals, which do
not satisfy the sparsity condition, and so must be treated as Fermi particles.
11 In 1866, Maxwell showed that the probability of finding a molecule in an ideal gas with a speed
within the range dv x dv y dv z is equal to ((m/(2πk B T )) 3/2 exp (−
1
2 mv 2 /(k B T ))d 3 v). This is the
‘Maxwellian distribution’.
10 The Statistical Basis of Bioenergetics
Appendix F shows how this expression leads to the famous Planck’s distribution for
light from a hot body, like ourselves or the Sun.
10.4 Classical Limit of the Particle Distributions
From a macroscopic view, the discrete nature of energy levels is usually not evident.
Classical macrosystems have particles widely spread out over a set of available
microscopic energy levels with many quantum states near each level. (So g i >>
n i >> 1.) A simple example is an ideal gas at a finite temperature, wherein there
are many more kinetic energy quantum states available in the range of energies
carried by the molecules than there are molecules occupying them.
The particles in such sparse (dilute) systems do not come close to filling the
available quantum states. The Pauli exclusion principle has little impact on the
thermodynamics of these systems. The distinction between bosons and fermions
no longer affects the macroscopic measurements of such systems near equilibrium.
For sparse systems, i.e. systems for which the degeneracy, g i , is much greater
than the occupation number n i (so g i >> n i >> 1), 10 we will have the
approximation, for both W bosons and W fermions ,
W boltzons
r
i=1
g
n i
i
n i !
.
(10.8)
The number W boltzons differs from W Boltzmann because Boltzmann naturally assumed
that the particles in the distribution were distinguishable. His work came before the
full development of quantum theory. Quantum theory makes particles of the same
type indistinguishable.
At or near equilibrium, Eq. (10.8) gives
n i e
α g i e
−ββ i .
(10.9)
This is the distribution of particles found by Boltzmann in 1871, using classical
arguments, and derived even earlier for the special case of an ideal gas by
Maxwell. 11
Biosystems, because they take advantage of complexity, tend to have a dense
array of energy states with occupations of higher energy states beyond the ultimate
10 This condition works for a wide variety of biological systems, including dilute solutions of
macromolecules. An important exception is the system of valence electrons in metals, which do
not satisfy the sparsity condition, and so must be treated as Fermi particles.
11 In 1866, Maxwell showed that the probability of finding a molecule in an ideal gas with a speed
within the range dv x dv y dv z is equal to ((m/(2πk B T )) 3/2 exp (−
1
2 mv 2 /(k B T ))d 3 v). This is the
‘Maxwellian distribution’.