The partition coefficient and the Boltzmann distribution.
The basic thermodynamic knowlege for understanding hemoglobin thermodynamics.
The Boltzmann distribution law on the change in the internal energy of gas(⊿E=⊿U) and the ratio of the occupancy probability of the energy level(N₁/N₂).
Quantitatively, the ratio N₁/N₂ of occupancy probabilities of the two energy levels E₁and E₂ at temperature T follows the Boltzmann distribution law.
N₁/N₂ ＝ exp(ーΔE/kB T)
where,
N₁ and N₂ are the probability of occupying the molecules of E₁ and E₂ respectively (the sum total of the probability is 1.),
ΔE ＝ E₂ー E₁、
kB = 1.381x10^-23j / K is the Boltzmann's constant.
Taking the natural logarithm of both sides of the equation, the following equations are obtained.
ln( N₁／N₂) ＝ -ΔE/kB T
ΔE ＝ -kB T ln( N₁/N₂).
The both sides of the above equation multiplied by NA(Avogadro's constant) is the change of internal energy per mole(⊿E₀).
ΔE₀ ＝ ーRT ln( N₁/N₂) (NＡΔE ＝ ΔE₀ 、NＡ kB ＝ R )
The equation shows the relationship between the diffetence of internal energy level of gases(⊿E=⊿U) and the ratio of occupancy probabilities(N₁/N₂) of each energy level.
The Boltzmann distribution law on the difference of Gibbs energy value (⊿rG) of the solute between org layer and aq layer and the partition coefficient of the solute (P=Porg/Paq) between org layer and aq layer.
The relationship between (⊿rG) and (P=Porg/Paq) follows the Boltzmann distribution law.
P = Porg/Paq = exp(-ΔrG/kB T)
ln(Porg/Paq) = -ΔrG/kB T
ΔrG = - kB T ln(Porg/Paq) = - kB T lnP
⊿rG is the difference of Gibbs energy value of the solute between org layer and aq laye,
P=Porg/Paq is the partition coefficient of the solute between org layer and aq layer,
kB = 1.381x10^-23j / K is the Boltzmann's constant.
ΔrG₀ = NＡΔrG = -NＡ kB T lnP = -RT lnP
ΔrG₀ = -RT lnP = - 2.3 RT logP
NA is Avogadro's constant,
ΔrG₀ is the change of Gibbs energy value per mole.
