424
10 The Statistical Basis of Bioenergetics
electron from another, or one hydrogen atom from another, by any intrinsic property.
When the quantum wave for one electron overlaps that of another, the two electrons
can exchange places while we are not looking, without any way to find out that this
happened.
Quantum theory also tells us the spin of particles must be quantized, of
magnitude
√
s(s + 1) ¯
h, and that 2s must be an integer. If 2s is even, the particle
is a boson. Quantum theory allows any number of bosons in one quantum state. If
2s is odd, the particle is a fermion, and quantum theory allows only one fermion in
a quantum state (the ‘Pauli Exclusion Principle). Electrons have s = 1/2, and are
fermions. Lucky for us. Otherwise, atoms would collapse into tight little balls. Life
and even matter as we know it would not exist.
A third quantum-theory property is important to our counting of W : Microsystems with sufficient symmetry will have some energy levels with many quantum
states at or very close to a given energy. For example, isolated hydrogen atoms have
eight different electron quantum states at the second energy level. We will use the
symbol g i to represent the number of quantum states hiding in the energy level i .
The g i is referred to as the ‘degeneracy’ of the energy level i .
10.3 Independent and Indistinguishable Particles
Let’s do a bit of counting, first for classical particles, such as those contemplated by
Ludwig Boltzmann. The number of ways of distributing N distinguishable particles
into r distinct states, with n i in the state with label i, is given by 7
W Boltzmann =
N !
r−1
i=0 n i !
.
(10.3)
This result follows by counting the ways to fill r boxes with N balls, with a given
number n i in the box labeled by ‘i’. There are N(N − 1) · · · (N − n 0 + 1) ways
to select n 0 balls to fill the first box. We divide this number by n 0 ! since the order
of selection is immaterial. For the next box, we have only (N − n 0 ) balls left, so
the number of ways to select n 1 balls, independent of the order selected, is (N −
n 0 ) · · · (N − n 0 − n 1 + 1)/n 1 !. It follows that the number of ways to fill all r boxes
with the given number of balls in each box will be the product given for W Boltzmann
above. 8
realized that in order for the entropy to be an extensive variable of degree one, the constant of
integration in
δQ/T must depend on the number of particles in the system.
7 Boltzmann used W to stand for Wahrscheinlichkeit, which is a measure of the probability of
finding the system in a given state.
8 If the index i is used to label energy levels which have a degeneracy g i , then another factor of
i g
n i
i is needed to account for the number of ways of re-arranging the particles in each energy
state.
10 The Statistical Basis of Bioenergetics
electron from another, or one hydrogen atom from another, by any intrinsic property.
When the quantum wave for one electron overlaps that of another, the two electrons
can exchange places while we are not looking, without any way to find out that this
happened.
Quantum theory also tells us the spin of particles must be quantized, of
magnitude
√
s(s + 1) ¯
h, and that 2s must be an integer. If 2s is even, the particle
is a boson. Quantum theory allows any number of bosons in one quantum state. If
2s is odd, the particle is a fermion, and quantum theory allows only one fermion in
a quantum state (the ‘Pauli Exclusion Principle). Electrons have s = 1/2, and are
fermions. Lucky for us. Otherwise, atoms would collapse into tight little balls. Life
and even matter as we know it would not exist.
A third quantum-theory property is important to our counting of W : Microsystems with sufficient symmetry will have some energy levels with many quantum
states at or very close to a given energy. For example, isolated hydrogen atoms have
eight different electron quantum states at the second energy level. We will use the
symbol g i to represent the number of quantum states hiding in the energy level i .
The g i is referred to as the ‘degeneracy’ of the energy level i .
10.3 Independent and Indistinguishable Particles
Let’s do a bit of counting, first for classical particles, such as those contemplated by
Ludwig Boltzmann. The number of ways of distributing N distinguishable particles
into r distinct states, with n i in the state with label i, is given by 7
W Boltzmann =
N !
r−1
i=0 n i !
.
(10.3)
This result follows by counting the ways to fill r boxes with N balls, with a given
number n i in the box labeled by ‘i’. There are N(N − 1) · · · (N − n 0 + 1) ways
to select n 0 balls to fill the first box. We divide this number by n 0 ! since the order
of selection is immaterial. For the next box, we have only (N − n 0 ) balls left, so
the number of ways to select n 1 balls, independent of the order selected, is (N −
n 0 ) · · · (N − n 0 − n 1 + 1)/n 1 !. It follows that the number of ways to fill all r boxes
with the given number of balls in each box will be the product given for W Boltzmann
above. 8
realized that in order for the entropy to be an extensive variable of degree one, the constant of
integration in
δQ/T must depend on the number of particles in the system.
7 Boltzmann used W to stand for Wahrscheinlichkeit, which is a measure of the probability of
finding the system in a given state.
8 If the index i is used to label energy levels which have a degeneracy g i , then another factor of
i g
n i
i is needed to account for the number of ways of re-arranging the particles in each energy
state.