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Chapter 15: Problem 102

The protein hemoglobin (Hb) transports \(\mathrm{O}_{2}\) in mammalian blood. Each Hb can bind \(4 \mathrm{O}_{2}\) molecules. The equilibrium constant for the \(\mathrm{O}_{2}\) binding reaction is higher in fetal hemoglobin than in adult hemoglobin. In discussing protein oxygen-binding capacity, biochemists use a measure called the \(P 50\) value, defined as the partial pressure of oxygen at which \(50 \%\) of the protein is saturated. Fetal hemoglobin has a \(\mathrm{P} 50\) value of \(2.53 \mathrm{kPa},\) and adult hemoglobin has a P50 value of \(3.57 \mathrm{kPa}\). Use these data to estimate how much larger \(K_{c}\) is for the aqueous reaction $4 \mathrm{O}_{2}(g)+\mathrm{Hb}(a q) \rightleftharpoons\left[\mathrm{Hb}\left(\mathrm{O}_{2}\right)_{4}(a q)\right]\( in a fetus, compared to \)K_{c}$ for the same reaction in an adult.

Short Answer

Expert verified
The Kc value for the aqueous reaction of O2 binding to Hb in a fetus is approximately 2.977 times larger than the Kc value for the same reaction in an adult. This indicates that fetal hemoglobin has a higher affinity for oxygen compared to adult hemoglobin.

Step by step solution

01

Write the reaction equation

The reaction of O2 binding to Hb can be written as: \[ 4\, O_2(g) + Hb(aq) \rightleftharpoons [Hb(O_2)_4](aq) \]
02

Express the reaction in terms of equilibrium constant Kc

The Kc of this reaction can be written as: \[ K_c = \frac{[Hb(O_2)_4]}{[Hb] [O_2]^4} \]
03

Utilize the P50 values at 50% saturation

At 50% saturation, the concentrations of Hb and [Hb(O2)4] are equal. Therefore, we can write the Kc expressions for fetal and adult hemoglobin as: - Fetal hemoglobin: \[ K_{c,f} = \frac{[Hb(O_2)_4]}{[Hb] \times (2.53\,kPa)^4} \] - Adult hemoglobin: \[ K_{c,a} = \frac{[Hb(O_2)_4]}{[Hb] \times (3.57\,kPa)^4} \]
04

Compare Kc values for fetal and adult hemoglobin

To compare the Kc values for fetal and adult hemoglobin, we will take the ratio of the two Kc expressions. Cancel out equal terms (in this case [Hb(O2)4]) in both numerator and denominator, and then simplify: \[ \frac{K_{c,f}}{K_{c,a}} = \frac{[Hb] \times (3.57\,kPa)^4}{[Hb] \times (2.53\,kPa)^4} \] Since the [Hb] terms are the same in both expressions, they cancel out, and the ratio simplifies to: \[ \frac{K_{c,f}}{K_{c,a}} = \frac{(3.57\,kPa)^4}{(2.53\,kPa)^4} \]
05

Calculate the numerical value of the Kc ratio

Evaluate the expression to determine the numerical value of the Kc ratio: \[ \frac{K_{c,f}}{K_{c,a}} = \frac{(3.57)^4}{(2.53)^4} \approx 2.977 \] #Conclusion# The Kc value for the aqueous reaction of O2 binding to Hb in a fetus is approximately 2.977 times larger than the Kc value for the same reaction in an adult. This indicates that fetal hemoglobin has a higher affinity for oxygen compared to adult hemoglobin.

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Most popular questions from this chapter

When \(2.00 \mathrm{~mol}\) of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is placed in a 5.00 -Lflaskat \(310 \mathrm{~K}\), \(40 \%\) of the $\mathrm{SO}_{2} \mathrm{Cl}_{2}\( decomposes to \)\mathrm{SO}_{2}\( and \)\mathrm{Cl}_{2}$ : $$\mathrm{SO}_{2} \mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g)$$ (a) Calculate \(K_{c}\) for this reaction at this temperature. (b) Calculate \(K_{P}\) for this reaction at \(310 \mathrm{~K}\). (c) According to Le Châtelier's principle, would the percent of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) that decomposes increase, decrease or stay the same if the mixture was transferred to a 1.00 -L vessel? (d) Use the equilibrium constant you calculated above to determine the percentage of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) that decomposes when 2.00 mol of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is placed in a \(1.00-\mathrm{L}\) vessel at \(310 \mathrm{~K}\).

A mixture of \(0.140 \mathrm{~mol}\) of \(\mathrm{NO}, 0.060 \mathrm{~mol}\) of \(\mathrm{H}_{2},\) and 0.260 mol of \(\mathrm{H}_{2} \mathrm{O}\) is placed in a \(2.0-\mathrm{L}\) vessel at \(330 \mathrm{~K}\). Assume that the following equilibrium is established: $$2 \mathrm{NO}(g)+2 \mathrm{H}_{2}(g) \rightleftharpoons \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g)$$ At equilibrium \(\left[\mathrm{H}_{2}\right]=0.010 \mathrm{M}\). (a) Calculate the equilibrium concentrations of \(\mathrm{NO}, \mathrm{N}_{2},\) and \(\mathrm{H}_{2} \mathrm{O} .\) (b) Calculate \(K_{c}\).

At \(2000^{\circ} \mathrm{C}\), the equilibrium constant for the reaction $2 \mathrm{NO}(g) \rightleftharpoons \mathrm{N}_{2}(g)+\mathrm{O}_{2}(g)$ is \(K_{c}=2.4 \times 10^{3} .\) If the initial concentration of NO is \(0.250 \mathrm{M}\), what are the equilibrium concentrations of \(\mathrm{NO}, \mathrm{N}_{2}\), and \(\mathrm{O}_{2}\) ?

Write the expressions for \(K_{c}\) for the following reactions. In each case indicate whether the reaction is homogeneous or heterogeneous. (a) \(\mathrm{O}_{2}(g) \rightleftharpoons 2 \mathrm{O}(g)\) (b) $\mathrm{Si}(s)+2 \mathrm{Cl}_{2}(g) \rightleftharpoons \mathrm{SiCl}_{4}(g)$ (c) $\mathrm{H}_{2}(g)+\mathrm{Cl}_{2}(g) \rightleftharpoons 2 \mathrm{HCl}(g)$ (d) $\mathrm{O}_{2}(g)+2 \mathrm{CO}(g) \rightleftharpoons 2 \mathrm{CO}_{2}(g)$ (e) $\mathrm{HCO}_{3}^{-}(a q) \rightleftharpoons \mathrm{CO}_{3}^{2-}(a q)+\mathrm{H}^{+}(a q)$ (f) $\mathrm{Fe}^{2+}(a q)+\mathrm{Ce}^{4+}(a q) \rightleftharpoons \mathrm{Fe}^{3+}(a q)+\mathrm{Ce}^{3+}(a q)$ (g) $\mathrm{CaCO}_{3}(s) \rightleftharpoons \mathrm{CaO}(s)+\mathrm{CO}_{2}(g)$

For the equilibrium $$2 \operatorname{IBr}(g) \rightleftharpoons \mathrm{I}_{2}(g)+\operatorname{Br}_{2}(g)$$ \(K_{p}=8.5 \times 10^{-3}\) at \(150^{\circ} \mathrm{C}\). If \(5.07 \mathrm{kPa}\) of IBr is placed in a 10.0 -L container, what is the partial pressure of all substances after equilibrium is reached?
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