Problem 87 - Chemistry An Atoms First Approach

Question

Phosphate buffers are important in regulating the \(\mathrm{pH}\) of intracellular fluids at \(\mathrm{pH}\) values generally between 7.1 and 7.2 a. What is the concentration ratio of \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) to \(\mathrm{HPO}_{4}^{2-}\) in intracellular fluid at \(\mathrm{pH}=7.15 ?\) \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}(a q) \rightleftharpoons \mathrm{HPO}_{4}^{2-}(a q)+\mathrm{H}^{+}(a q) \quad K_{\mathrm{a}}=6.2 \times 10^{-8}\) b. Why is a buffer composed of \(\mathrm{H}_{3} \mathrm{PO}_{4}\) and \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) ineffective in buffering the \(\mathrm{pH}\) of intracellular fluid? \(\mathrm{H}_{3} \mathrm{PO}_{4}(a q) \rightleftharpoons \mathrm{H}_{2} \mathrm{PO}_{4}^{-}(a q)+\mathrm{H}^{+}(a q) \quad K_{\mathrm{a}}=7.5 \times 10^{-3}\)

Step-by-Step Solution

Answer

In summary, the concentration ratio of \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) to \(\mathrm{HPO}_{4}^{2-}\) in intracellular fluid at \(\mathrm{pH}=7.15\) can be calculated using the Henderson-Hasselbalch equation, giving a value of \(10^{(7.15 - (-\log(6.2 \times 10^{-8}))}\). As for the buffer composed of \(\mathrm{H}_{3} \mathrm{PO}_{4}\) and \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\), it is ineffective in buffering the pH of intracellular fluid because its pKa value is not within 1 pH unit of the pH range of interest (7.1-7.2).

Step 1: Understanding the Buffer and the Henderson-Hasselbalch Equation

Before we can attempt any calculations, it's important to understand the basic concept of a buffer. In biological systems, buffers maintain pH within a specific range. The equilibrium established between the weak acid and its conjugate base allows for the absorption of protons (H+ ions) when pH gets too low, and the donation of protons when pH gets too high. If we look at part (a), it gives us a buffer system - a weak acid, H2PO4-, and its conjugate base, HPO42-. The Henderson-Hasselbalch equation provides a mathematical relation for the ratio of the concentrations of the weak acid and its conjugate base to pH of the solution and the acid dissociation constant, Ka. It is given by: \[ pH = pKa + \log \left( \frac{{[\text{{conjugate base}}]}}{{[\text{{weak acid}}]}} \right) \]

Step 2: Solving for the Ratio of H2PO4- to HPO42- using the Henderson-Hasselbalch Equation

We're given pH = 7.15 and Ka = 6.2 x 10^(-8). First, let's substitute Ka into our equation to get the pKa value. pKa is simply -log(Ka) so, pKa = -log(6.2 x 10^-8). This brings us to, \( pH = pKa + \log \left( \frac{{[\text{{conjugate base}}]}}{{[\text{{weak acid}}]}} \right) \), Since our interest is in the ratio of the concentrations, let's rearrange the above equation: \( \frac{{[\text{{conjugate base}}]}}{{[\text{{weak acid}}]}} = 10^{(pH - pKa)} \) Substituting the given values into the equation: \( \frac{{[\mathrm{HPO}_{4}^{2-}]}}{{[\mathrm{H}_{2} \mathrm{PO}_{4}^{-}]}} = 10^{(7.15 - (-log(6.2 x 10^-8)))} \) This calculation will yield the desired concentration ratio.

Step 3: Explaining the Ineffectiveness of the H3PO4/H2PO4- Buffer

Buffers are most effective when the pH of the solution is within 1 pH unit of the pKa of the buffering acid-base system. This is derived from the Henderson-Hasselbalch equation. If pH is more than 1 unit away from the pKa, the buffer system loses its effectiveness. In this case, for the H3PO4/H2PO4- buffer system, we can work out the pKa from the provided Ka value: pKa = -log(7.5 x 10^-3). Comparing this pKa value to the pH range of interest (7.1-7.2), we can clearly see that the pKa is not within 1 pH unit of this range, hence, the buffer system will not be effective. This relatively higher pKa value means the buffer system is not optimized for controlling pH in the intracellular environment; it will not be able to absorb or donate protons effectively to resist changes to the pH value.

Key Concepts

pH regulation

The regulation of pH is crucial for maintaining the stability and proper function of biological systems. The pH scale, ranging from 0 to 14, is a measure of the concentration of hydrogen ions \( H^+ \) in a solution. On this scale, lower values indicate more acidic environments, and higher values indicate more basic or alkaline environments. Maintaining the pH within a specific range is essential because even slight changes could significantly affect biochemical reactions.
In living cells and fluids, the pH is often tightly regulated by buffering systems, which can accept or donate protons to counteract pH fluctuations. The Henderson-Hasselbalch equation is an indispensable tool in understanding these systems, as it allows us to calculate the pH of a solution based on the concentration ratio of a weak acid and its conjugate base. This equation helps predict how well a buffer can maintain pH when acids or bases are added.
By understanding pH regulation, one can appreciate how different biological processes are optimized under specific pH conditions, ensuring that enzymes and other catalytic proteins function correctly.

buffer systems

Buffer systems are mixtures that minimize changes in the concentration of hydrogen ions \( H^+ \) when acids or bases are added to a solution. These systems are vital in biological contexts, where precise pH levels are necessary for chemical reactions to proceed efficiently. A buffer typically consists of a weak acid and its conjugate base, or a weak base and its conjugate acid.
The phosphate buffer, for example, is crucial in cellular fluids, where the phosphate ion \( ext{H}_2 ext{PO}_4^- \) acts as a weak acid donating protons, and \( ext{HPO}_4^{2-} \) acts as a conjugate base accepting protons. The effectiveness of a buffer depends on its range, which is typically within 1 pH unit of the pKa of the acid in the buffer.
Among several buffer systems, the phosphate buffer stands out due to its optimal pKa, close to the 7.2 pH level ideal for most intracellular processes. By absorbing excess protons or providing them, buffer systems maintain stability, which is fundamental for life.

acid-base equilibrium

Acid-base equilibrium refers to the balance in the concentration of acids and bases in a solution, crucial for maintaining a stable pH. It involves the reversible reactions between an acid and its conjugate base or a base and its conjugate acid. In an equilibrium, the rate of the forward reaction equals the rate of the reverse reaction, leading to a stable concentration ratio of reactants and products.
In the case of acid-base chemistry, this equilibrium is expressed in terms of the acid dissociation constant, \( K_a \), which provides insight into the strength of an acid. A higher \( K_a \) value indicates a stronger acid that dissociates more in solution, while a lower value suggests a weaker acid. Understanding this equilibrium is vital because it underpins the action of buffer systems which stabilize pH.
The Henderson-Hasselbalch equation utilizes this equilibrium concept to relate the concentration of weak acids and their conjugate bases to pH, offering a comprehensive understanding of how buffer systems operate. Maintaining equilibrium ensures a consistent environment, essential for sustaining various biological reactions and cellular signals.