Question

Phosphate buffers are important in regulating the \(\mathrm{pH}\) of intra- cellular fluids at pH values generally between 7.1 and \(7.2 .\) a. What is the concentration ratio of \(\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\) to \(\mathrm{HPO}_{4}^{2-}\) inintracellular 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 pH of intracellular fluid? $$ \mathrm{H}_{3} \mathrm{PO}_{4}(a q) \rightleftharpoons \mathrm{H}_{2} \mathrm{PO}_{4}^{-}(a q)+\mathrm{H}^{+}(a q) \quad K_{2}=7.5 \times 10^{-3} $$

Short answer

The concentration ratio of \(H_2PO_4^-\) to \(HPO_4^{2-}\) in intracellular fluid at pH 7.15 is approximately \(0.861:1\). A buffer composed of \(H_3PO_4\) and \(H_2PO_4^-\) is ineffective in buffering the pH of intracellular fluid because its pKa value is 2.12, which is significantly lower than the desired pH range of 7.1 and 7.2. The efficiency of a buffer is optimal when the pKa is near the pH of the system being buffered, and in this case, the pKa is too far from the desired pH range of intracellular fluid.

Step-by-step solution

Use the Henderson-Hasselbalch equation with given pH value and Ka value

To find the concentration ratio, we need to use the Henderson-Hasselbalch equation. The equation is as follows: \[ pH = pK_a + \log{\left(\frac{[A^-]}{[HA]}\right)} \] Where: - pH is the pH of the solution, - pKa is the negative logarithm (base 10) of the acid dissociation constant (Ka) of the acid, - [A^-] is the concentration of the conjugate base, and - [HA] is the concentration of the weak acid. The given pH is 7.15 and the Ka value is \(6.2 \times 10^{-8}\). Therefore, the pK_a value is defined as: \[ pK_a = \log{\left(\frac{1}{Ka}\right)} = -\log{(6.2 \times 10^{-8})} \] Calculate the pKa value:

Calculate the pKa value

Using the given Ka value, we can calculate pKa as follows: \[ pK_a = -\log{(6.2 \times 10^{-8})} \approx 7.21 \] Now we can use the pH value of 7.15 and pKa value of 7.21 to find the concentration ratio of \(H_2PO_4^{-}\) to \(HPO_4^{2-}\) by substituting these values into the Henderson-Hasselbalch equation.

Determine the concentration ratio

Now we can use the pH, 7.15, and the pKa, 7.21, in the Henderson-Hasselbalch equation to find the concentration ratio of \(H_2PO_4^-\) to \(HPO_4^{2-}\): \[ 7.15 = 7.21 + \log{\left(\frac{[HPO_4^{2-}]}{[H_2PO_4^{-}]}\right)} \] Rearrange the equation to determine the concentration ratio: \[ \log{\left(\frac{[HPO_4^{2-}]}{[H_2PO_4^{-}]}\right)} = 7.15 - 7.21 = -0.06 \] \[ \frac{[HPO_4^{2-}]}{[H_2PO_4^{-}]} = 10^{-0.06} \approx 0.861 \] Thus, the concentration ratio of \(H_2PO_4^-\) to \(HPO_4^{2-}\) in intracellular fluid at pH 7.15 is approximately \(0.861:1\).

Explain the inefficiency of \(H_3PO_4\) and \(H_2PO_4^-\) as a buffer in intracellular fluid

In the second part of the exercise, we are given that: \[ H_3PO_4 \leftrightharpoons H_2PO_4^- + H^+ \quad K_2 = 7.5 \times 10^{-3} \] Now we need to calculate the pK_a2 value to determine whether it falls within the desired pH range for intracellular fluid buffering: \[ pK_a2 = -\log{(7.5 \times 10^{-3})} \] Then: \[ pK_a2 = -\log{(7.5 \times 10^{-3})} \approx 2.12 \] Now let's compare the pKa values of both reactions: For the phosphate buffer: \[ pK_a = -\log{(6.2 \times 10^{-8})} = 7.21 \] For the \(H_3PO_4\) buffer: \[ pK_a2 = -\log{(7.5 \times 10^{-3})} = 2.12 \] The pKa of the \(H_3PO_4\) buffer reaction is 2.12, which is significantly lower than the desired pH range of intracellular fluid (7.1 and 7.2). Thus, a buffer composed of \(H_3PO_4\) and \(H_2PO_4^-\) is ineffective in buffering the pH of intracellular fluid as the buffering capacity of the \(H_3PO_4\) buffer reacts more strongly with added acidic or basic compounds, leading to a larger change in pH. The efficiency of a buffer is optimal when the pKa is near the pH of the system being buffered.

Key concepts

pH of intracellular fluid

Intracellular fluid, the fluid inside our cells, plays a critical role in maintaining the right pH, which is crucial for various cellular functions. Typically, this pH is slightly basic, ranging from 7.1 to 7.2. This narrow range is maintained by various buffering systems that prevent drastic changes in pH even when acids or bases are introduced into or produced by the cells.
Within this environment, phosphate buffers are particularly significant. They help stabilize the pH, making sure that cellular processes, including enzyme activities and biochemical reactions, proceed optimally. The phosphate buffer system is favored because it can effectively maintain the pH near 7 due to its specific action at the pH levels observed in intracellular fluids.
This ability to buffer is pivotal for cell function because cells rely on a remarkably consistent internal pH. Deviations may lead to dysfunction or, worst-case, damage or death of the cell. Thus, maintaining a stable intracellular pH is one of the essential features of cell homeostasis.

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a fundamental formula in understanding how buffer solutions work. It helps in calculating the pH of a buffer solution or in finding the concentration ratio of an acid to its conjugate base within the solution. The equation is represented as:
\[ pH = pK_a + \log{\left(\frac{[A^-]}{[HA]}\right)} \]
Where:

• \( pH \) is the measure of acidity or basicity of the solution.
• \( pK_a \) is the negative logarithm of the acid dissociation constant (\( K_a \)), reflecting the strength of the acid.
• \( [A^-] \) is the concentration of the conjugate base.
• \( [HA] \) is the concentration of the weak acid.

This equation elucidates the relationship between pH and pKa and highlights how buffers resist changes in pH by adjusting the ratio of acid to conjugate base. In the context of intracellular buffers like phosphates, this equation allows us to calculate how different concentrations of phosphate ions help maintain the desired pH of 7.15 by maintaining precise ratios that counteract pH fluctuations. Understanding the Henderson-Hasselbalch equation is vital for anyone studying acid-base balance in biological systems.

acid dissociation constant

The acid dissociation constant, \( K_a \), is a critical concept for understanding acids in chemistry, particularly in the realm of buffers. It provides insight into the strength of an acid by measuring its ability to donate protons to a base in a solution. Mathematically, it is expressed as:
\[ K_a = \frac{[H^+][A^-]}{[HA]} \]
In this equation:

• \([H^+]\) is the concentration of hydrogen ions.
• \([A^-]\) is the concentration of the conjugate base.
• \([HA]\) is the concentration of the undissociated acid.

A larger \( K_a \) value indicates a stronger acid, which dissociates more in solution. Conversely, a smaller \( K_a \) means a weaker acid. The related measure, \( pK_a \), which is \(-\log{K_a}\), is often used to provide a more manageable scale for comparing acid strengths.
In the context of phosphate buffers in intracellular fluids, such as \({H_2PO_4}^-\) and \(HPO_4^{2-}\), the dissociation constants help determine the appropriate acid-base ratio to maintain the pH close to the desired value. Ensuring the proper \( K_a \) and hence \( pK_a \) values allows these compounds to act effectively as buffers, resisting drastic changes in pH despite acid or base additions, thus preserving cellular integrity.