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Gibb's Factor is given by formula:

$Gibb\text{'}sFactor=e^{\left[\frac{-(蔚-n渭)}{kT}\right]}$

where, $k$is Boltzmann's constant, $T$is temperature, $渭$is chemical potential, and $蔚$is energy of state .

As, in hemoglobin molecule, oxygen can have four different states given by:

localid="1647144603783" $$\begin{gathered} {蔚}_o=0eV \\ {蔚}_1=-0.55eV(two\sin glyoccupied) \\ {蔚}_2=-1.3eV \end{gathered}$$

The average number of oxygen molecules is given by,

$$\begin{gathered} \overline{N}=\underset{S}{鈭�}N\left(S\right) \\ =\frac{2e^{{\displaystyle \frac{-({蔚}_1-渭)}{kT}}}}{Z}+\frac{2e^{{\displaystyle \frac{-({蔚}_2-2渭)}{kT}}}}{Z} \end{gathered}$$

Occupancy of system is given by:

$$\begin{gathered} \overline{n}=\frac{\overline{N}}{Z} \\ =\frac{1}{Z}\left(e^{\frac{-({蔚}_1-渭)}{kT}+\frac{-({蔚}_2-2渭)}{kT}}\right) \end{gathered}$$

Chemical Potential formula is :

$渭=-kT脳\ln \left(\frac{V{Z}_{in}}{N{谓}_Q}\right)$

Substitute $\frac{V}{N}=\frac{kT}{P}$in above formula,

$渭=-kT脳\ln \left(\frac{kT{Z}_{in}}{P{谓}_Q}\right)$

$e^{\frac{渭}{kT}}=\frac{P{谓}_Q}{kT{Z}_{in}}$

Gibb's Factor can be written as:

$$\begin{gathered} e^{\left[\frac{-({蔚}_1-渭)}{kT}\right]}=e^{\frac{-{蔚}_1}{kT}}脳\frac{P{谓}_Q}{kT{Z}_{in}} \\ =\frac{P{谓}_Q}{kT{Z}_{in}{e}^{\frac{{蔚}_1}{kT}}} \\ =\frac{P}{P_o} \end{gathered}$$

Oxygen molecule quantum volume is:

${谓}_Q={\left(\frac{h}{\sqrt{2\mathrm{蟺尘办罢}}}\right)}^3$

Substitute the values of the variables in above formula,

$$\begin{gathered} {谓}_Q={\left(\frac{6.63脳{10}^{-34}J路s}{\sqrt{2蟺(32脳1.66脳{10}^{-27}kg)(1.38脳{10}^{-23}J/K)(310K)}}\right)}^3 \\ =5.3脳{10}^{-33}{m}^3 \end{gathered}$$

In equation, $\overline{n}=\frac{1}{Z}\left(e^{\frac{-({蔚}_1-渭)}{kT}+\frac{-({蔚}_2-2渭)}{kT}}\right)$

Substitute $e^{\frac{蔚}{KT}}=1790$ and $P_o=2.03bar$at room temperature,

$\overline{n}=\frac{\left({\displaystyle \frac{P}{2.03}}+1790{\left({\displaystyle \frac{P}{2.03}}\right)}^2\right)}{\left(1+\frac{P}{2.03}+1790{\left(\frac{P}{2.03}\right)}^2\right)}$

Occupancy in this model is nearly $100\%$near lungs given $P=0.20bar$

The table is created between Pressure $P$and fraction of occupied sites:

The graph between fraction of occupied sites and oxygen partial pressure is created as,

We see that the occupancy rises gradually for smaller values of $P$. This shows that with cooperative binding, hemoglobin reacts with oxygen much fast in cells when $P$is small.