Problem 29 - Chemistry The Central Science

Question

(a) What is the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) in blood of \(\mathrm{pH} 7.4\) ? (b) What is the ratio of \(\mathrm{HCO}_{3}^{-}\) to \(\mathrm{H}_{2} \mathrm{CO}_{3}\) in an exhausted marathon runner whose blood \(\mathrm{pH}\) is \(7.1 ?\)

Step-by-Step Solution

Answer

(a) The ratio of HCO3- to H2CO3 at pH 7.4 is approximately 19.95:1. (b) The ratio of HCO3- to H2CO3 at pH 7.1 is approximately 10:1.

Step 1: Recall the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is given by: \[pH = pK_a + \log \frac{[A^-]}{[HA]}\] In this exercise, \([A^-]\) represents the concentration of bicarbonate ion (HCO3-), and \([HA]\) represents the concentration of carbonic acid (H2CO3).

Step 2: Find pKa for carbonic acid

In order to plug in our pH values into the equation, we need to know the pKa value of carbonic acid. The pKa value for carbonic acid is approximately 6.1.

Step 3: Solve for the ratio at pH 7.4 (Part a)

Plug the pH value of 7.4 and the pKa value of 6.1 into the Henderson-Hasselbalch equation: \[7.4 = 6.1 + \log \frac{[\mathrm{HCO}_3^-]}{[\mathrm{H}_2 \mathrm{CO}_3]}\] Now, solve for the ratio \(\frac{[\mathrm{HCO}_3^-]}{[\mathrm{H}_2 \mathrm{CO}_3]}\): \[7.4 - 6.1 = \log \frac{[\mathrm{HCO}_3^-]}{[\mathrm{H}_2 \mathrm{CO}_3]}\] \[1.3 = \log \frac{[\mathrm{HCO}_3^-]}{[\mathrm{H}_2 \mathrm{CO}_3]}\] To get rid of the logarithm, take 10 to the power of both sides: \[10^{1.3} = \frac{[\mathrm{HCO}_3^-]}{[\mathrm{H}_2 \mathrm{CO}_3]}\] \[19.95 \approx \frac{[\mathrm{HCO}_3^-]}{[\mathrm{H}_2 \mathrm{CO}_3]}\] The ratio of HCO3- to H2CO3 at pH 7.4 is approximately 19.95:1.

Step 4: Solve for the ratio at pH 7.1 (Part b)

Plug the pH value of 7.1 and the pKa value of 6.1 into the Henderson-Hasselbalch equation: \[7.1 = 6.1 + \log \frac{[\mathrm{HCO}_3^-]}{[\mathrm{H}_2 \mathrm{CO}_3]}\] Now, solve for the ratio \(\frac{[\mathrm{HCO}_3^-]}{[\mathrm{H}_2 \mathrm{CO}_3]}\): \[7.1 - 6.1 = \log \frac{[\mathrm{HCO}_3^-]}{[\mathrm{H}_2 \mathrm{CO}_3]}\] \[1.0 = \log \frac{[\mathrm{HCO}_3^-]}{[\mathrm{H}_2 \mathrm{CO}_3]}\] To get rid of the logarithm, take 10 to the power of both sides: \[10^{1.0} = \frac{[\mathrm{HCO}_3^-]}{[\mathrm{H}_2 \mathrm{CO}_3]}\] \[10 \approx \frac{[\mathrm{HCO}_3^-]}{[\mathrm{H}_2 \mathrm{CO}_3]}\] The ratio of HCO3- to H2CO3 at pH 7.1 is approximately 10:1.

Key Concepts

Acid-Base Equilibrium

Acid-base equilibrium is a fundamental concept in chemistry and physiology. It refers to the balance between acids and bases in a solution, which determines the solution's pH level. This balance is crucial in biological systems, like human blood, where it ensures proper functioning of enzymes and metabolic processes.
The Henderson-Hasselbalch equation is a crucial tool for understanding acid-base equilibrium. It helps calculate the pH of a solution based on the acid's pKa (acid dissociation constant) and the ratio of the concentration of the base (A-) to its corresponding acid (HA).
For carbonic acid in the blood, we consider carbonic acid (\(\mathrm{H_2CO_3}\)) and bicarbonate ion (\(\mathrm{HCO_3^-}\)). The equilibrium between these components helps maintain the blood pH within its narrow range. This balance prevents conditions like acidosis or alkalosis, which occur when the pH falls outside the optimal range.

Bicarbonate Ion

The bicarbonate ion (\(\mathrm{HCO_3^-}\)) plays a critical role in maintaining acid-base balance in the body. It acts as a buffer by reacting with hydrogen ions (\(\mathrm{H^+}\)) to form carbonic acid, thus regulating blood pH. This interaction is a key element in respiratory and renal systems, allowing the body to neutralize excess acids formed during metabolism.
In the blood, bicarbonate is primarily controlled by the kidneys. They regulate bicarbonate levels by either reabsorbing it into the bloodstream or excreting it into the urine based on the body's needs.

• An increase in \(\mathrm{HCO_3^-}\) concentration can raise blood pH, leading to alkalosis.
• A decrease can lower blood pH, causing acidosis.

The ratio of bicarbonate ion to carbonic acid directly affects the blood pH, with the Henderson-Hasselbalch equation allowing calculation of this ratio under different physiological conditions.

Carbonic Acid

Carbonic acid (\(\mathrm{H_2CO_3}\)) is a weak acid formed when carbon dioxide (CO2) dissolves in water. It exists in equilibrium with \(\mathrm{CO_2}\) and water in the blood and is a key component of the carbonic acid-bicarbonate buffer system.
In the body, carbonic acid plays several roles:

• It helps buffer pH changes that occur due to metabolism.
• It converts back into CO2, which is expelled from the body through respiration.
• Its conversion to bicarbonate ion allows for efficient hydrogen ion neutralization.

This equilibrium is vital, as it allows for quick responses to physiological changes in blood pH, maintaining homeostasis.

Blood pH

Blood pH is a measure of the acidity or alkalinity of blood. Its normal range is tightly regulated between 7.35 and 7.45. Any deviation from this range can significantly impact health and physiological functions.
An acid-base imbalance can lead to either acidosis (pH < 7.35) or alkalosis (pH > 7.45). These conditions can disrupt cellular processes and enzyme activities, leading to various symptoms and potentially severe health issues.
The body uses several mechanisms to maintain blood pH, including:

• The carbonic acid-bicarbonate buffer system, which adjusts quickly to changes in hydrogen ion concentration.
• Renal regulation of bicarbonate ion reabsorption or excretion.
• Respiratory control of carbon dioxide removal.

Understanding blood pH and its regulation through the Henderson-Hasselbalch equation aids in diagnosing and managing acid-base disorders.