Question

The pH of human blood is steady at a value of approximately 7.4 owing to the following equilibrium reactions: $$ \mathrm{CO}_{2}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}_{2} \mathrm{CO}_{3}(a q) \rightleftharpoons \mathrm{HCO}_{3}^{-}(a q)+\mathrm{H}^{+}(a q) $$ Acids formed during normal cellular respiration react with the \(\mathrm{HCO}_{3}^{-}\) to form carbonic acid, which is in equilibrium with \(\mathrm{CO}_{2}(a q)\) and \(\mathrm{H}_{2} \mathrm{O}(l) .\) During vigorous exercise, a person's \(\mathrm{H}_{2} \mathrm{CO}_{3}\) blood levels were \(26.3 \mathrm{mM},\) whereas his \(\mathrm{CO}_{2}\) levels were 1.63 \(\mathrm{mM}\) . On resting, the \(\mathrm{H}_{2} \mathrm{CO}_{3}\) levels declined to 24.9 \(\mathrm{m} M\) . What was the \(\mathrm{CO}_{2}\) blood level at rest?

Short answer

The \(\mathrm{CO}_{2}\) blood concentration at rest is approximately \(\boxed{1.54\ \mathrm{mM}}\).

Step-by-step solution

Write the equilibrium constant expression for the reactions

\[ K=\frac{[\mathrm{H}_{2}\mathrm{CO}_{3}(a q)]}{[\mathrm{CO}_{2}(a q)][\mathrm{H}_{2}\mathrm{O}(l)]} \] Let's assume that the concentration of water does not change significantly during the reaction. So, the concentration of water will remain constant. Therefore, we can write a new equilibrium constant, \(K_1\), which includes the concentration of \(\mathrm{H}_{2}\mathrm{O}(l)\): \[ K_1 = K[\mathrm{H}_{2}\mathrm{O}(l)] = \frac{[\mathrm{H}_{2}\mathrm{CO}_{3}(a q)]}{[\mathrm{CO}_{2}(a q)]} \] Now we will use the given concentrations during exercise to find the equilibrium constant \(K_1\).

Calculate the equilibrium constant \(K_1\) during exercise

Given the concentrations during exercise: \([\mathrm{H}_{2}\mathrm{CO}_{3}(a q)] = 26.3\ \mathrm{mM}\) and \([\mathrm{CO}_{2}(a q)] = 1.63\ \mathrm{mM}\). Let's substitute these values into the \(K_1\) expression: \[ K_1 = \frac{26.3\ \mathrm{mM}}{1.63\ \mathrm{mM}} \approx 16.13 \] Now we have the equilibrium constant \(K_1\). We will use this and the given \(\mathrm{H}_{2}\mathrm{CO}_{3}\) concentration at rest to find the \(\mathrm{CO}_{2}\) concentration at rest.

Calculate the \(\mathrm{CO}_{2}\) concentration at rest

We are given that the concentration of \(\mathrm{H}_{2}\mathrm{CO}_{3}\) at rest is \([\mathrm{H}_{2}\mathrm{CO}_{3}(a q)] = 24.9\ \mathrm{mM}\). We will use the equilibrium constant \(K_1\) to find the \([\mathrm{CO}_{2}(a q)]\) at rest: \[ 16.13 = \frac{24.9\ \mathrm{mM}}{[\mathrm{CO}_{2}(a q)]} \] Solving for \([\mathrm{CO}_{2}(a q)]\): \[ [\mathrm{CO}_{2}(a q)] = \frac{24.9\ \mathrm{mM}}{16.13} \approx 1.54\ \mathrm{mM} \] So, the \(\mathrm{CO}_{2}\) blood concentration at rest is approximately \(\boxed{1.54\ \mathrm{mM}}\).

Key concepts

Carbonic Acid Equilibrium

In the human body, carbonic acid equilibrium plays a vital role in maintaining a stable blood pH level. This balance involves a series of reversible reactions that help regulate acidity. The primary reaction involves carbon dioxide (\(\mathrm{CO}_2\)) combining with water (\(\mathrm{H}_2\mathrm{O}\)) to form carbonic acid (\(\mathrm{H}_2\mathrm{CO}_3\)). This carbonic acid can further dissociate into bicarbonate ions (\(\mathrm{HCO}_3^-\)) and hydrogen ions (\(\mathrm{H}^+\)).

• When added to the blood, CO2 reacts with water to temporarily increase carbonic acid levels.
• Carbonic acid can release hydrogen ions, which could lower blood pH and make it more acidic.
• The equilibrium shifts to balance the formation and dissociation of carbonic acid depending on CO2 and other reactant concentrations.

This chemical equilibrium ensures that even when CO2 levels fluctuate, as during exercises, the body's pH level is regulated by compensating shifts between carbonic acid and bicarbonate.

Cellular Respiration

Cellular respiration is a life-sustaining process that produces energy by breaking down glucose and other molecules. This biological process occurs in the mitochondria of cells, where glucose is broken down with oxygen to produce carbon dioxide, water, and ATP, the energy currency of the cell.

• During strenuous exercise, cellular respiration increases, leading to higher levels of carbon dioxide as a byproduct.
• CO2 is transported via the bloodstream, where it can enter into equilibrium reactions to form carbonic acid.
• The removal of CO2 generated through respiration is vital for maintaining pH balance and preventing acidosis.

Without a well-regulated system in place, the excess CO2 from heightened cellular respiration could lead to undesirable changes in blood pH.

pH Balance in Blood

Maintaining a stable pH in the blood is critical for normal physiological functions. Human blood operates optimally within a tight pH range of 7.35 to 7.45. This balance is achieved through the bicarbonate buffer system, a dynamic composition of bicarbonate ions and carbonic acid.

• When blood pH drops below the normal range, the condition is termed acidosis. Conversely, when it rises, it is called alkalosis.
• The body utilizes the carbonic acid-bicarbonate equilibrium system to resist drastic changes in pH.
• Kidneys and lungs function together to remove excess hydrogen ions and carbon dioxide, aiding pH balance.

The continuous adjustment of these elements helps maintain the finely-tuned environment necessary for enzymes and biological processes to operate effectively.