Problem 45 - Chemistry An Atoms First Approach

Question

Carbonate buffers are important in regulating the pH of blood at \(7.40 .\) If the carbonic acid concentration in a sample of blood is 0.0012 \(M,\) determine the bicarbonate ion concentration required to buffer the \(\mathrm{pH}\) of blood at \(\mathrm{pH}=7.40\). \(\mathrm{H}_{2} \mathrm{CQ}_{3}(a g) \rightleftharpoons \mathrm{HCQ}_{3}^{-}(a g)+\mathrm{H}^{+}(a g) \quad \) \(K_{\mathrm{a}}=4.3 \times 10^{-7}\)

Step-by-Step Solution

Answer

To maintain the pH of blood at 7.40, the bicarbonate ion concentration (\(HCO_3^-\)) must be approximately 0.00234 M when the carbonic acid concentration is 0.0012 M.

Step 1: Write the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation relates the pH, pKa, acid concentration, and conjugate base concentration in a solution. It has the form: \[pH = pKa + \log\left(\frac{[A^-]}{[HA]}\right)\] where: - \(pH\) is the pH of the solution - \(pKa\) is the negative logarithm of the acid dissociation constant, \(K_a\) - \([A^-]\) is the concentration of the conjugate base (bicarbonate ion, \(HCO_3^-\)) - \([HA]\) is the concentration of the weak acid (carbonic acid, \(H_2CO_3\)) We will use this equation to solve for the bicarbonate ion concentration (\([HCO_3^-]\)).

Step 2: Calculate the pKa

In order to use the Henderson-Hasselbalch equation, we first need to find the pKa. The pKa is the negative logarithm of Ka, which is given as \(4.3 \times 10^{-7}\). \[pKa = -\log(K_a)\] Plug in the value of Ka: \[pKa = -\log(4.3 \times 10^{-7})\] Calculate the pKa value: \[pKa \approx 6.366\]

Step 3: Plug known values into the Henderson-Hasselbalch equation

We can now update the Henderson-Hasselbalch equation with the known values of pH, pKa, and the carbonic acid concentration (\([H_2CO_3]\)): \[7.40 = 6.366 + \log\left(\frac{[HCO_3^-]}{0.0012}\right)\]

Step 4: Solve for bicarbonate ion concentration ([HCO_3^-])

To find the bicarbonate ion concentration, first subtract the pKa value from the pH. \[7.40 - 6.366 = \log\left(\frac{[HCO_3^-]}{0.0012}\right)\] Next, remove the logarithm using the exponentiation with base 10. \[\frac{[HCO_3^-]}{0.0012} = 10^{(7.40 - 6.366)}\] Now, multiply both sides by 0.0012 to find the bicarbonate ion concentration. \[[HCO_3^-] = 0.0012 \times 10^{(7.40 - 6.366)}\] Calculate the value for [HCO_3^-]: \[[HCO_3^-] \approx 0.00234\,M\]

Step 5: Interpret the result

To maintain the pH of blood at 7.40, the bicarbonate ion concentration (\(HCO_3^-\)) must be approximately 0.00234 M when the carbonic acid concentration is 0.0012 M.

Key Concepts

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a fundamental tool in chemistry that provides insight into how the pH of a solution is related to the concentration of an acid and its conjugate base. For buffer solutions, the equation is expressed as:\[ pH = pK_a + \log\left(\frac{[A^-]}{[HA]}\right) \]Here,

• pH represents the acidity or basicity level of the solution.
• pK_a is the negative logarithm of the acid dissociation constant, providing a measure of the strength of the acid.
• [A^-] is the concentration of the conjugate base, and
• [HA] is the concentration of the weak acid.

The equation helps in understanding how buffers maintain pH against the addition of small amounts of strong acids or bases. Buffer solutions resist drastic changes in pH by balancing the concentrations of these key components. This balance is crucial when working with biological systems, where pH must be carefully regulated to maintain proper physiological function.
When using the equation, we often first convert the K_a to pK_a, followed by substituting known values to solve for unknown concentrations, such as finding how much [A^-] is required to achieve a desired pH.

pH Regulation

Regulating pH is essential in many biological processes and industrial applications. pH regulation refers to the process of keeping the pH within a narrow range to prevent harmful effects. In biological systems, enzymes, proteins, and cellular functions operate optimally only within certain pH limits. Buffers play a vital role in maintaining this balance in solutions where a very specific pH range must be upheld, such as in the human body, which maintains blood pH around 7.40. They work by neutralizing added acids or bases. For instance, in blood, the carbonic acid-bicarbonate buffer system is key. When excess hydrogen ions are present, bicarbonate ions ( HCO_3^−) can absorb these to form carbonic acid, while a deficiency can push the reaction in the opposite direction to supply more hydrogen ions, thus maintaining pH. These systems are crucial because any significant deviation in pH can lead to metabolic disturbances or enzymatic inactivity and severe physiological effects. pH regulation is a complex, yet vital, process for homeostasis in living organisms.

Acid-base Equilibrium

Acid-base equilibrium is the state of balance between acids and bases in a solution, ensuring a stable pH level. In chemistry, acids donate protons (H^+), while bases accept them. This equilibrium can be described using the reaction:\[ HA \rightleftharpoons H^+ + A^- \]In this reversible reaction, HA represents the acid, H^+ the hydronium ion, and A^− the conjugate base. The position of equilibrium shifts depending on the concentrations of the acid and base, temperature, and pressure.Buffers assist in stabilizing the acid-base equilibrium by providing a source/reservoir of acid and base. When an external source of acid or base is introduced, the equilibrium can shift, but the buffer system compensates by adjusting the concentrations of [H^+] and [A^-], thus keeping the pH relatively stable. This mechanism is especially important in systems such as blood, where specific pH levels (like 7.40) are essential for proper physiological functioning.
Understanding acid-base equilibrium in buffer solutions allows scientists and medical professionals to predict and control the behavior of pH-sensitive environments effectively.