For the system hemoglobin \(\cdot \mathrm{O}_{2}(a q)+\mathrm{CO}(g) \rightleftharpoons\) hemoglobin \(\cdot \mathrm{CO}(a q)+\mathrm{O}_{2}(g)\) \(K=2.0 \times 10^{2} .\) What must be the ratio of \(P_{\mathrm{Co}} / P_{\mathrm{O}_{2}}\) if \(12.0 \%\) of the hemoglobin in the bloodstream is converted to the CO complex?

The equilibrium constant, represented as K, is a fundamental concept in chemistry, conveying the ratio of concentrations of products to reactants at equilibrium. It’s crucial to understand that K is constant only at a particular temperature and it reflects the relative concentration of the substances involved in a reversible chemical reaction when it is no longer changing – a state known as chemical equilibrium.

In the context of the problem, the equilibrium expression for the reaction between hemoglobin, oxygen, and carbon monoxide is defined as
\t$$K = \frac{[H \cdot CO][O_{2}]}{[H \cdot O_{2}][CO]}$$
\t where square brackets denote concentrations. This equation indicates that the product of the concentrations of hemoglobin bound to carbon monoxide and oxygen gas divided by the product of the concentrations of hemoglobin bound to oxygen (in its aquatic form) and carbon monoxide gas is constant at equilibrium.

It is essential to understand that the equilibrium constant tells us the extent of a reaction; a large K value suggests that the reaction tends to produce more products, whereas a small K value indicates a reaction that favors the reactants. In the given exercise, by using the value of K and the percentage of hemoglobin converted, we can calculate the desired ratio of partial pressures.

Hemoglobin is a protein found in red blood cells that is crucial for transporting oxygen from the lungs to tissues throughout the body. It can bind to various gases, including oxygen (O2) and carbon monoxide (CO). Hemoglobin's binding to these gases is reversible, allowing it to pick up and release oxygen efficiently. However, the binding of these gases is also competitive, meaning high concentrations of CO can prevent O2 from binding to hemoglobin.

Understanding the competitive nature of hemoglobin binding is vital, particularly in cases of CO poisoning, where CO outcompetes O2 for the same binding sites on hemoglobin. In the provided problem, the hemoglobin binding dynamics are represented in the equilibrium reaction, and we are asked to determine the influence of gas partial pressures on this competition.

In our case, since 12% of hemoglobin has bound to CO, which is a significant amount given the affinity hemoglobin has for CO, it implies that there must be defined conditions – dictated by the equilibrium constant and partial pressures – where this percentage is stable. By calculating the ratio of partial pressures, we can infer the environmental conditions conducive to this level of hemoglobin conversion.

Gas partial pressure is a concept in chemistry that describes the pressure a single gas in a mixture of gases would exert if it alone occupied the entire volume of the mixture. Dalton’s Law of Partial Pressures states that the total pressure of a gas mixture is the sum of the partial pressures of each individual gas.

The partial pressure of a gas is directly proportional to its concentration, meaning that a higher partial pressure corresponds to a higher concentration of that gas. In biological systems, like the human body, the partial pressure of oxygen and carbon monoxide significantly affects the binding of these gases to hemoglobin.

For instance, in the exercise, we use this proportional relationship to equate the ratio of gas concentrations to the ratio of their partial pressures to solve for the needed ratio of partial pressures given the equilibrium condition. By understanding the interplay between gas partial pressure and hemoglobin binding, one can comprehend the delicate balance necessary for proper bodily function, such as the transport of oxygen throughout the body.