Question

Carbon monoxide is toxic because it binds more strongly to the iron in hemoglobin (Hb) than does \(\mathrm{O}_{2}\) , as indicated by these approximate standard free-energy changes in blood: $$\begin{array}{cl}{\mathrm{Hb}+\mathrm{O}_{2} \longrightarrow \mathrm{HbO}_{2}} & {\Delta G^{\circ}=-70 \mathrm{kJ}} \\\ {\mathrm{Hb}+\mathrm{CO} \longrightarrow \mathrm{HbCO}} & {\Delta G^{\circ}=-80 \mathrm{kJ}}\end{array}$$ Using these data, estimate the equilibrium constant at 298 K for the equilibrium $$\mathrm{HbO}_{2}+\mathrm{CO} \rightleftharpoons \mathrm{HbCO}+\mathrm{O}_{2}$$

Short answer

The equilibrium constant (K) for the reaction HbO2 + CO ⇌ HbCO + O2 at 298 K can be calculated using the given standard free-energy changes and the thermodynamic relationship between ΔG° and K. After manipulating the given reactions and calculating ΔG° for the target reaction, we find that K ≈ 4.58 at 298 K.

Step-by-step solution

Write down the given information

The given approximate standard free-energy changes of the reactions are as follows: 1. Hb + O2 -> HbO2, ΔG° = -70 kJ 2. Hb + CO -> HbCO, ΔG° = -80 kJ We need to find the equilibrium constant (K) for the reaction: HbO2 + CO ⇌ HbCO + O2

Use the given reactions to set up the target reaction

We will now manipulate the given reactions to form the target reaction: 1. Hb + O2 -> HbO2, ΔG° = -70 kJ 2. Hb + CO -> HbCO, ΔG° = -80 kJ Subtract Reaction 1 from Reaction 2: (Hb + CO -> HbCO) - (Hb + O2 -> HbO2) = HbO2 + CO -> HbCO + O2 Now, we have the target reaction. Since we have subtracted Reaction 1 from Reaction 2, we should also subtract the standard free-energy change of Reaction 1 from that of Reaction 2: ΔG°(HbO2 + CO -> HbCO + O2) = ΔG°(Hb + CO -> HbCO) - ΔG°(Hb + O2 -> HbO2)

Calculate the standard free-energy change for the target reaction

Now, we can calculate the standard free-energy change for the target reaction: ΔG°(HbO2 + CO -> HbCO + O2) = -80 kJ - (-70 kJ) = -10 kJ

Use the relationship between standard free-energy change and the equilibrium constant to find K

We know the thermodynamic relationship between the standard free-energy change (ΔG°) and the equilibrium constant (K) is given by: ΔG° = -RT ln(K) Where R is the gas constant (8.314 J/(mol·K)) and T is the temperature (298 K in this case). We can rearrange this equation to solve for K: K = exp(-ΔG° / (RT)) Now, substitute the values: K = exp(-(-10000 J/mol) / (8.314 J/(mol·K) * 298 K)) (Note: We have converted -10 kJ to -10000 J, as 1 kJ = 1000 J)

Compute the equilibrium constant

Calculate the value of K using the given values: K = exp(10000 / (8.314 * 298)) ≈ 4.58 Hence, the equilibrium constant (K) for the reaction HbO2 + CO ⇌ HbCO + O2 at 298 K is approximately 4.58.

Key concepts

Standard Free-Energy Change

The concept of standard free-energy change (\r\(\r\Delta G^\r\circ\r\)) is crucial for understanding the spontaneity of chemical reactions. In the realm of biochemistry and gas binding with hemoglobin, this concept becomes particularly vital. It's a measure of the amount of free energy that is released or absorbed when a reaction occurs under standard conditions (1 bar of pressure, and 298K in temperature for biochemical reactions).

The free-energy change provides insight into the direction in which a reaction naturally tends to proceed. A negative value of \r\(\r\Delta G^\r\circ\r\) indicates that the reaction occurs spontaneously, releasing energy to the surroundings, as seen in the binding of oxygen (\r\(\r\mathrm{O}_2\r\)) and carbon monoxide (CO) to hemoglobin. Specifically, in the given problem, the smaller negative value for HbO2 formation (\r\(\r\Delta G^\r\circ = -70 \r\mathrm{kJ}\r\)) compared to HbCO formation (\r\(\r\Delta G^\r\circ = -80 \r\mathrm{kJ}\r\)) signifies that CO binds more strongly to hemoglobin than oxygen does, under standard conditions.

Gibbs Free Energy

Gibbs free energy, named after Josiah Willard Gibbs, is a thermodynamic quantity represented by the symbol \r\(G\r\). It is the maximum amount of non-expansion work that can be extracted from a closed system; this maximum can be attained only in a completely reversible process.

Gibbs free energy connects the concepts of energy, entropy, and temperature to predict the direction of chemical reactions. In a simple formula, the free energy is expressed as \r\(G = H - TS\r\), where \r\(H\r\) is the enthalpy (the total heat content), \r\(T\r\) is the temperature, and \r\(S\r\) is the entropy (a measure of disorder). For a reaction to be spontaneous at constant temperature and pressure, \r\(\r\Delta G\r\) must be negative.

The usefulness of Gibbs free energy in biochemical contexts, like the binding of gases to hemoglobin, lies in its ability to identify which reactions can occur spontaneously and how equilibrium is achieved in such reactions.

Equilibrium in Chemical Reactions

Equilibrium in chemical reactions refers to the state in which the rate of the forward reaction equals the rate of the reverse reaction. As a result, the concentrations of the reactants and products remain constant over time. This concept is central to understanding reactions like the binding and releasing of gases by hemoglobin.

In the context of the reaction between hemoglobin, oxygen, and carbon monoxide, equilibrium is achieved when the rate at which hemoglobin binds to oxygen (\r\(\r\mathrm{HbO_2}\r\)) and carbon monoxide (\r\(\r\mathrm{HbCO}\r\)) is equal to the rate at which they dissociate. The equilibrium constant (K) is a ratio that quantifies the concentrations of the products to the reactants at equilibrium. A larger value of K indicates a greater extent of the reaction proceeding towards the products, whereas a smaller K implies a reaction favoring the reactants.

Hemoglobin and Gas Binding

Hemoglobin is a protein found in red blood cells and is critical for transporting oxygen from the lungs to the tissues and facilitating the return of carbon dioxide for exhalation. Hemoglobin's affinity for different gases is a subject of great interest, as it directly impacts its efficiency in oxygen transport and delivery.

The binding of gases to hemoglobin is a reversible process and follows the principles of chemical equilibrium. Oxygen and carbon monoxide compete for the same binding sites on hemoglobin. The strength of these bindings is reflected in the standard free-energy changes. Carbon monoxide binds more strongly to hemoglobin with a more negative \r\(\r\Delta G^\r\circ\r\) value, which can be problematic, as it can displace oxygen and reduce the oxygen-carrying capacity of hemoglobin, leading to tissue hypoxia and other complications. Understanding the equilibrium constants of these reactions enables researchers and medical professionals to predict and quantify the extent of gas binding under various conditions and the potential impacts on human health.