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: $$ \begin{array}{clrl} \mathrm{Hb}+\mathrm{O}_{2} \longrightarrow \mathrm{HbO}_{2} & & \Delta G^{\circ}=-70 \mathrm{~kJ} \\ \mathrm{Hb}+\mathrm{CO} \longrightarrow \mathrm{HbCO} & \Delta G^{\circ}=-80 \mathrm{~kJ} \end{array} $$ Using these data, estimate the equilibrium constant at \(298 \mathrm{~K}\) for the equilibrium $$ \mathrm{HbO}_{2}+\mathrm{CO} \rightleftharpoons \mathrm{HbCO}+\mathrm{O}_{2} $$

Short answer

The equilibrium constant (K) for the given reaction HbO₂ + CO ⇌ HbCO + O₂ is approximately 4.77 at 298 K.

Step-by-step solution

Write down the given data

We are given the standard free-energy changes for the following reactions: 1. Hb + O₂ → HbO₂, ΔG° = -70 kJ 2. Hb + CO → HbCO, ΔG° = -80 kJ The equilibrium reaction we want to find the equilibrium constant for is: HbO₂ + CO ⇌ HbCO + O₂

Combining the given reactions to form the equilibrium reaction

To obtain the equilibrium reaction, we need to perform the following steps: 1. Reverse the first reaction. 2. Add the reversed first reaction to the second reaction. After reversing the first reaction, we get: HbO₂ → Hb + O₂, ΔG° = 70 kJ Now, add the reversed first reaction and the second reaction: HbO₂ → Hb + O₂ Hb + CO → HbCO This results in the equilibrium reaction: HbO₂ + CO ⇌ HbCO + O₂

Calculating the standard free-energy change for the equilibrium reaction

To find the standard free-energy change (ΔG°) for the equilibrium reaction, we simply sum the standard free-energy changes of the combined reactions: ΔG° = ΔG°₁ + ΔG°₂ = 70 kJ + (-80 kJ) = -10 kJ

Calculate the equilibrium constant using the Gibbs free energy equation

We can calculate the equilibrium constant (K) using the relationship between the standard free-energy change and the equilibrium constant: ΔG° = -RT * ln(K) Where: ΔG° = standard free-energy change R = gas constant = 8.314 J/(mol*K) T = temperature = 298 K K = equilibrium constant Solve for K: K = exp(-ΔG°/(RT)) = exp(10,000 J/mol / (8.314 J/(mol*K) * 298 K)) K ≈ 4.77 #Conclusion# The equilibrium constant (K) for the given reaction HbO₂ + CO ⇌ HbCO + O₂ is approximately 4.77 at 298 K.

Key concepts

Standard Free-Energy Change

The concept of 'standard free-energy change', denoted as \( \Delta G^\circ \), is fundamental in thermodynamics, particularly in the context of chemical reactions. It measures the spontaneity of a reaction under standard conditions, which typically include a temperature of 298 K, 1 atm pressure, and 1 M concentration for all the reactants and products. A negative value of \( \Delta G^\circ \) indicates a spontaneous reaction, while a positive value suggests that the reaction is not spontaneous under standard conditions.

In our exercise, we have two reactions involving hemoglobin (Hb) with oxygen and carbon monoxide, respectively. The given standard free-energy changes enable us to understand the affinity Hb has for each gas. This information is crucial when we delve into the toxic effects of carbon monoxide; its higher binding affinity to Hb as compared to \( \mathrm{O}_{2} \) is captured through a more negative \( \Delta G^\circ \).

To emphasize the importance of this concept, let's delve further into the numerical values provided in the exercise. The value of -70 kJ for the binding of Hb with \( \mathrm{O}_{2} \) is already an indication of a strong and spontaneously favorable interaction. However, the more negative \( \Delta G^\circ \) of -80 kJ for the binding of Hb with CO points to an even more spontaneous process, thereby hinting at why CO can effectively displace \( \mathrm{O}_{2} \) from Hb, leading to toxic consequences.

Gibbs Free Energy Equation

The 'Gibbs free energy equation' is a powerful tool that bridges thermodynamics with chemical equilibrium. Represented as \( \Delta G^\circ = -RT \ln(K) \), it relates the standard free-energy change \( \Delta G^\circ \) of a reaction to its equilibrium constant \( K \), temperature \( T \), and the universal gas constant \( R \).

Understanding this equation allows us to predict the direction in which a reaction will proceed and the extent to which the reactants will be converted to products. When the temperature is set at 298 K (approximately room temperature) and the standard free-energy change is known, we can calculate the equilibrium constant for the reaction. This quantitative relationship arms us with the ability to forecast how much a reaction will favor the formation of products under those standard conditions.

For instance, in our textbook solution, by applying the Gibbs free energy equation to the calculated value of \( \Delta G^\circ \) for the Hb binding reaction with CO and O₂, the resulting equilibrium constant is derived to be approximately 4.77 at 298 K. This implies that at equilibrium, the concentration ratio of products to reactants is such that Hb prefers CO over \( \mathrm{O}_{2} \) by that factor, highlighting CO's dangerous predilection for hemoglobin.

Hemoglobin and Carbon Monoxide

The interaction between 'hemoglobin and carbon monoxide' is a topic of critical importance in biochemistry and medicine. Hemoglobin is a protein in red blood cells that transports oxygen throughout the body. Its function is vital for life as it supplies oxygen to tissues and organs. Carbon monoxide, however, is a colorless, odorless gas produced from the combustion of carbon-containing materials.

When CO is inhaled, it binds with the iron in hemoglobin with a much greater affinity than oxygen. This binding is so strong that it inhibits the ability of hemoglobin to transport oxygen, leading to a condition known as carbon monoxide poisoning. The high affinity of hemoglobin for CO is evidenced by the more negative standard free-energy change value in the reactions provided in the exercise.

The danger of CO stems from its deceptive nature; since it binds more strongly than oxygen, it can quickly accumulate in the bloodstream without noticeable initial symptoms. It's critical for individuals to be aware of potential CO sources and ensure proper ventilation to prevent poisoning. Understanding the molecular basis of CO's affinity for hemoglobin reinforces the grave consequences of its exposure and underscores the importance of measures to detect and avoid CO inhalation.