Problem: The protein hemoglobin (Hb) transports O2 in mammalian blood. Each Hb can bind 4 O2 molecules. The equilibrium constant for the O2-binding reaction is higher in fetal hemoglobin than in adult hemoglobin. In discussing protein oxygen-binding capacity, biochemists use a measure called the  P50 value, defined as the partial pressure of oxygen at which 50% of the protein is saturated. Fetal hemoglobin has a P50 value of 19 torr, and adult hemoglobin has a P50 value of 26.8 torr. Use these data to estimate how much larger Kc is for the aqueous reaction 4 O2(g) + Hb(aq) ⇌ [Hb(O2)4](aq) in a fetus, comparedto Kc for the same reaction in an adult.

FREE Expert Solution

4 O₂(g) + Hb(aq) ⇌ [Hb(O₂)₄](aq)

• K_p → equilibrium units in terms of pressure of gaseous species

$$\begin{gathered} {\mathbf{K}}_{\mathbf{p}}\mathbf{=}\frac{\mathbf{products}}{\mathbf{reactants}} \\ {\mathbf{K}}_{\mathbf{p}}\mathbf{=}\frac{\mathbf{1}}{{\left(P_{O_2}\right)}^{\mathbf{4}}} \end{gathered}$$

Fetal hemoglobin:

${\mathbf{P}}_{\mathbf{fetal}}\mathbf{=}\mathbf{19}\mathbf{ }\cancel{\mathbf{torr}}\mathbf{\times }\frac{\mathbf{1}\mathbf{ }\mathbf{atm}}{\mathbf{760}\mathbf{ }\cancel{\mathbf{torr}}}$

P_fetal = 0.025 atm

${\mathbf{K}}_{\mathbf{p}\mathbf{,}\mathbf{ }\mathbf{fetal}}\mathbf{=}\frac{\mathbf{1}}{{(0.025)}^{\mathbf{4}}}$

K_{p, fetal} = 2.56x10⁶

Adult hemoglobin:

${\mathbf{P}}_{\mathbf{adult}}\mathbf{=}\mathbf{26}\mathbf{.}\mathbf{8}\mathbf{ }\cancel{\mathbf{torr}}\mathbf{\times }\frac{\mathbf{1}\mathbf{ }\mathbf{atm}}{\mathbf{760}\mathbf{ }\cancel{\mathbf{torr}}}$

P_adult = 0.035 atm

${\mathbf{K}}_{\mathbf{p}\mathbf{,}\mathbf{ }\mathbf{adult}}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{(}\mathbf{0}\mathbf{.}\mathbf{0353}\mathbf{)}}^{\mathbf{4}}}$

K_{p, adult} = 6.44x10⁵

Kp and Kc are related to one another by the following equation below: 

$\boxed{{\mathbf{K}}_{\mathbf{p}}\mathbf{=}{\mathbf{K}}_{\mathbf{c}}{\left(\mathbf{RT}\right)}^{\mathbf{∆}\mathbf{n}}}$

R = gas constant = 0.08206 (L∙atm)/(mol∙K)
T = temperature, K
Δn = moles of gas products – moles of gas reactants

Δn = 0 moles gas – 4 moles gas
Δn = –4 mol

$$\begin{gathered} \boxed{{\mathbf{K}}_{\mathbf{p}}\mathbf{=}{\mathbf{K}}_{\mathbf{c}}{\left(\mathbf{RT}\right)}^{\mathbf{∆}\mathbf{n}}} \\ \frac{{\mathbf{K}}_{\mathbf{p}}}{{\left(\mathbf{RT}\right)}^{\mathbf{∆}\mathbf{n}}}\mathbf{=}\frac{{\mathbf{K}}_{\mathbf{c}}\cancel{{\left(\mathbf{RT}\right)}^{\mathbf{∆}\mathbf{n}}}}{\cancel{{\left(\mathbf{RT}\right)}^{\mathbf{∆}\mathbf{n}}}} \\ \boxed{{\mathbf{K}}_{\mathbf{c}}\mathbf{=}\frac{{\mathbf{K}}_{\mathbf{p}}}{{\left(\mathbf{RT}\right)}^{\mathbf{∆}\mathbf{n}}}} \end{gathered}$$

T = body temperature = 37°C + 273.15 = 310.15 K