10.8 The Partition Function for Ensembles
475
of site with index i. Suppose the first oxygen molecule attaches to a hemoglobin
molecule binds with an energy of 1 and the second one attaches with a binding
energy of 2 . As binding energies, both these values are negative. To make the
second oxygen bind more easily than the first, 2 is more negative than 1 . The
Pauling model uses just these two energies. The Adair model effectively takes into
account four different energies of binding. As our aim is didactic, we will elaborate
on the underlying statistics of the Pauling case, as this model is a little simpler, and
does almost as well as the Adair model compared to experiment.
We can write the energy associated with various attachments of the O 2 molecules
as
(σ 1 , σ 2 , σ 3 , σ 4 ) = 1
i
σ i +
i>j
σ i σ j ,
(10.161)
where 1 < 0 and = 2 − 1 < 0.
The grand partition function for this model that describes these sixteen possible
states of oxygen molecules attached to the hemoglobin becomes
=
1
σ 1 =0
1
σ 2 =0
1
σ 3 =0
1
σ 4 =0
exp
⎛
⎝ β (μ − 1 )
4
i=1
σ i + β(−)
i>j
σ i σ j
⎞
⎠
= 1 + 4λe
−ββ 1 + 6λ
2 e
−2ββ 1 −ββ
+ 4λ
3 e
−3ββ 1 −3ββ
+ λ
4 e
−4ββ 1 −6ββ ,
(10.162)
where λ = exp (βμ).
Aside. Here we see the importance of how the lowest Boltzmann energy is selected,
in that the λ terms relative to unity depends on that choice. We will be using the
balance of chemical potential for oxygen as a gas in the lung with the chemical
potential μ = k B T ln λ in the blood. The lowest Boltzmann energy for the gas (taken
to be ideal here) comes to play. At body temperatures, the accessible quantum states
for the oxygen molecule as a gas in the lung are those with energies clustered within
a few kT ∼ 1/40 eV just above the atomic ground state.
From the partition function, we can find the average number of oxygen molecules
per hemoglobin molecule with
N = λ
∂
∂λ
ln
= 4λκ
1 + 3λκω + 3λ 2 κ 2 ω 3 + λ 3 κ 3 ω 6
1 + 4λκ + 6λ 2 κ 2 ω + 4λ 3 κ 3 ω 3 + λ 4 κ 4 ω 6 ,
(10.163)
wherein, to improve the aesthetics, we have introduced κ ≡ exp (−ββ 1 ) and ω ≡
exp (−ββ).
The saturation level, f of oxygen attached to the hemoglobin will be
475
of site with index i. Suppose the first oxygen molecule attaches to a hemoglobin
molecule binds with an energy of 1 and the second one attaches with a binding
energy of 2 . As binding energies, both these values are negative. To make the
second oxygen bind more easily than the first, 2 is more negative than 1 . The
Pauling model uses just these two energies. The Adair model effectively takes into
account four different energies of binding. As our aim is didactic, we will elaborate
on the underlying statistics of the Pauling case, as this model is a little simpler, and
does almost as well as the Adair model compared to experiment.
We can write the energy associated with various attachments of the O 2 molecules
as
(σ 1 , σ 2 , σ 3 , σ 4 ) = 1
i
σ i +
i>j
σ i σ j ,
(10.161)
where 1 < 0 and = 2 − 1 < 0.
The grand partition function for this model that describes these sixteen possible
states of oxygen molecules attached to the hemoglobin becomes
=
1
σ 1 =0
1
σ 2 =0
1
σ 3 =0
1
σ 4 =0
exp
⎛
⎝ β (μ − 1 )
4
i=1
σ i + β(−)
i>j
σ i σ j
⎞
⎠
= 1 + 4λe
−ββ 1 + 6λ
2 e
−2ββ 1 −ββ
+ 4λ
3 e
−3ββ 1 −3ββ
+ λ
4 e
−4ββ 1 −6ββ ,
(10.162)
where λ = exp (βμ).
Aside. Here we see the importance of how the lowest Boltzmann energy is selected,
in that the λ terms relative to unity depends on that choice. We will be using the
balance of chemical potential for oxygen as a gas in the lung with the chemical
potential μ = k B T ln λ in the blood. The lowest Boltzmann energy for the gas (taken
to be ideal here) comes to play. At body temperatures, the accessible quantum states
for the oxygen molecule as a gas in the lung are those with energies clustered within
a few kT ∼ 1/40 eV just above the atomic ground state.
From the partition function, we can find the average number of oxygen molecules
per hemoglobin molecule with
N = λ
∂
∂λ
ln
= 4λκ
1 + 3λκω + 3λ 2 κ 2 ω 3 + λ 3 κ 3 ω 6
1 + 4λκ + 6λ 2 κ 2 ω + 4λ 3 κ 3 ω 3 + λ 4 κ 4 ω 6 ,
(10.163)
wherein, to improve the aesthetics, we have introduced κ ≡ exp (−ββ 1 ) and ω ≡
exp (−ββ).
The saturation level, f of oxygen attached to the hemoglobin will be