Problem 62 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The \(\mathrm{pH}\) of blood stream is maintained by a proper balance of \(\mathrm{H}_{2} \mathrm{CO}_{3}\) and \(\mathrm{NaHCO}_{3}\) concentrations. What volume of \(5 \mathrm{M} \mathrm{NaHCO}_{3}\) solution should be mixed with a \(10 \mathrm{~mL}\) sample of blood which is \(2 \mathrm{M}\) in \(\mathrm{H}_{2} \mathrm{CO}_{3}\) in order to maintain a \(\mathrm{pH}\) of \(7.4 ? K_{a}\) for \(\mathrm{H}_{2} \mathrm{CO}_{3}\) in blood is \(7.8 \times 10^{-7}\).

Step-by-Step Solution

Answer

Mix 78 mL of 5 M NaHCO3 with the blood sample.

Step 1: Understanding the Henderson-Hasselbalch equation

To solve this problem, we'll use the Henderson-Hasselbalch equation for a buffer solution, which is given by:\[\mathrm{pH} = \mathrm{pK_a} + \log\left(\frac{[\mathrm{A}^-]}{[\mathrm{HA}]}\right)\]Here, \( [A^-] \) is the concentration of the base (\( \mathrm{NaHCO}_3 \)) and \( [HA] \) is the concentration of the acid (\( \mathrm{H}_2 \mathrm{CO}_3 \)). The desired pH is 7.4, and the pKa can be calculated from the given \( K_a \) value.

Step 2: Calculate the pKa from Ka

The equation for pKa is:\[\mathrm{pK_a} = -\log(K_a)\]Substituting the given \( K_a = 7.8 \times 10^{-7} \), we find:\[\mathrm{pK_a} = -\log(7.8 \times 10^{-7}) \approx 6.11\]

Step 3: Apply the Henderson-Hasselbalch equation

Insert the known values into the Henderson-Hasselbalch equation:\[7.4 = 6.11 + \log\left(\frac{[\mathrm{NaHCO}_3]}{[\mathrm{H}_2\mathrm{CO}_3]}\right)\]Solving for the fraction:\[7.4 - 6.11 = \log\left(\frac{[\mathrm{NaHCO}_3]}{[\mathrm{H}_2\mathrm{CO}_3]}\right)\]\[1.29 = \log\left(\frac{[\mathrm{NaHCO}_3]}{[\mathrm{H}_2\mathrm{CO}_3]}\right)\]

Step 4: Exponentiate to solve for concentration ratio

Exponentiating both sides of the equation will remove the logarithm:\[10^{1.29} = \frac{[\mathrm{NaHCO}_3]}{[\mathrm{H}_2\mathrm{CO}_3]}\]\[\approx 19.54 = \frac{[\mathrm{NaHCO}_3]}{[\mathrm{H}_2\mathrm{CO}_3]}\]This means the concentration of \( \mathrm{NaHCO}_3 \) must be about 19.54 times the concentration of \( \mathrm{H}_2\mathrm{CO}_3 \).

Step 5: Calculate required volume of NaHCO3

Let \( V \) be the volume in liters of the \( 5\, \mathrm{M} \) \( \mathrm{NaHCO}_3 \) solution required. Since the \( 10\, \mathrm{mL} \) blood sample is \( 2\, \mathrm{M} \) in \( \mathrm{H}_2\mathrm{CO}_3 \). The moles of \( \mathrm{H}_2\mathrm{CO}_3 \) are:\[\text{moles } = 0.01\, \mathrm{L} \times 2\, \mathrm{M} = 0.02\, \text{moles}\]The concentration ratio gives:\[19.54 = \frac{5V}{0.02}\]Solving for \( V \),\[5V = 0.39 \V = \frac{0.39}{5} = 0.078\, \mathrm{L} = 78\, \mathrm{mL}\]Therefore, 78 mL of \( 5\, \mathrm{M} \) \( \mathrm{NaHCO}_3 \) is required.

Key Concepts

Buffer Solutions

Buffer solutions are crucial for maintaining the stability of acid-base systems. They resist changes in their pH when small amounts of acid or base are added. A common example is the direct interplay between carbonic acid (\(\mathrm{H}_{2}\mathrm{CO}_{3}\)) and bicarbonate ion (\(\mathrm{NaHCO}_{3}\)) in blood. This balances the pH to ensure the proper functioning of biological processes.
The Henderson-Hasselbalch equation is your go-to tool. It allows us to predict and adjust the pH of these solutions by understanding the relationship between the concentrations of the acid and its conjugate base. In our example, the blood's desired pH is 7.4, indicating a specific ratio between \(\mathrm{NaHCO}_{3}\) and \(\mathrm{H}_{2}\mathrm{CO}_{3}\) that must be maintained.

• Buffers typically consist of a weak acid and its conjugate base (or vice versa).
• They are essential in biological systems, like blood, because they prevent sudden pH swings.
• The effectiveness of a buffer is influenced by its concentration and the pKa of the acid involved.

Choosing the right combination of weak acid and conjugate base is essential for achieving the required buffer capacity.

Equilibrium Constant

The equilibrium constant, represented as \(K_{a}\) for acids, gives us an insight into the degree of ionization of an acid in solution. It's a measure of the strength of an acid and directly affects the pH. In the case of carbonic acid in blood, \(K_{a}\) is given as \(7.8 \times 10^{-7}\). This relatively small value indicates a weak acid, meaning it does not fully ionize in water.
For buffer solutions, knowing \(K_{a}\) allows one to calculate \(\mathrm{pK}_a\), which is a pivotal value in the Henderson-Hasselbalch equation. The pKa is key to understanding how a buffer will perform at different pH levels.

• Low \(K_{a}\) values indicate weaker acids, which are better suited for buffers.
• Equilibrium constants help predict the position of an equilibrium and how it will shift with changing conditions.

In our solution, the conversion to \(\mathrm{pK}_a\) helps us understand the acid's behavior and tailor the buffer solution appropriately.

Acid-Base Balance

Acid-base balance is a vital aspect of homeostasis within the body, facilitating optimal conditions for enzymes and metabolic processes. A slight shift in pH can have significant physiological effects, hence why the body regulates it tightly through buffer systems.
Blood features one of the most prominent examples of acid-base balancing using buffer solutions, primarily involving carbon dioxide transport. Carbonic acid and bicarbonate ions work together to mitigate changes in \(\mathrm{pH}\).
Using the Henderson-Hasselbalch equation, we find the necessary concentrations to keep pH within this essential range, typically around 7.4 for human blood.

• The presence of carbon dioxide and its conversion to bicarbonate helps regulate the blood's pH.
• Acids and bases enter the system through dietary intake and cellular metabolism, so buffers are constantly working.
• A well-functioning acid-base balance avoids issues like acidosis or alkalosis, maintaining overall health.

Attention to this balance allows for the correct calculation of substances like \(\mathrm{NaHCO}_{3}\) needed in medical scenarios.