Question

Novocaine, used as a local anesthetic by dentists, is a weak base \(\left(K_{\mathrm{b}}=8.91 \times 10^{-6}\right) .\) What is the ratio of the concentration of the base to that of its acid in the blood plasma \((\mathrm{pH}=7.40)\) of a patient? (As an approximation, use the \(K_{\mathrm{a}}\) values at \(25^{\circ} \mathrm{C}\).)

Short answer

The ratio of the concentration of the base to the acid is approximately 0.028.

Step-by-step solution

Understanding the Relationship Between pH and pOH

The pH of the blood plasma is given as 7.40. For a complete understanding, we need to determine the pOH in order to relate it to the concentration of hydroxide ions (OH-). This can be done using the formula: \( \text{pH} + \text{pOH} = 14 \). Substituting 7.40 for the pH, we have: \( \text{pOH} = 14 - 7.40 = 6.60 \).

Calculate Hydroxide Ion Concentration

The pOH is related to the concentration of hydroxide ions, \([\text{OH}^{-}]\), by the equation \( \text{pOH} = -\log [\text{OH}^{-}] \). Thus, \([\text{OH}^{-}] = 10^{-6.60} \approx 2.51 \times 10^{-7} \text{ M} \).

Relate Hydroxide Ion Concentration to the Base

For the weak base, according to the equation \( B + H_2O \rightleftharpoons BH^+ + OH^- \), the concentration of the base \([B]\) is related to \([\text{OH}^{-}]\) by the base dissociation constant equation: \( K_b = \frac{[BH^+][\text{OH}^{-}]}{[B]} \). Rearranging, \([B] = \frac{[BH^+][\text{OH}^{-}]}{K_b}\).

Determine the Concentration Ratio Using the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation for bases is \( \text{pH} = \text{pK}_a + \log \frac{[\text{base}]}{[\text{acid}]} \). First, find \( pK_a \) based on \( K_a \) which can be derived from \( K_w / K_b \) since \( K_w = 1.0 \times 10^{-14} \) at 25°C. Calculate \( K_a = \frac{1.0 \times 10^{-14}}{8.91 \times 10^{-6}} \approx 1.12 \times 10^{-9} \). Then \( pK_a = -\log (1.12 \times 10^{-9}) \approx 8.95 \).

Calculate the Ratio of the Base to Acid

Using the Henderson-Hasselbalch equation: \( 7.40 = 8.95 + \log \frac{[\text{base}]}{[\text{acid}]} \). Rearrange to find: \( \log \frac{[\text{base}]}{[\text{acid}]} = 7.40 - 8.95 = -1.55 \). Therefore, \( \frac{[\text{base}]}{[\text{acid}]} = 10^{-1.55} \approx 0.028 \).

Key concepts

pH and pOH relationship

pH and pOH are essential concepts in chemistry for understanding the acidity and basicity of solutions. The pH scale measures how acidic or basic a solution is on a scale of 0 to 14, with 7 being neutral.
To find the pOH of a solution, we use the simple relationship:

• pH + pOH = 14

This equation indicates that if you know the pH, you can easily calculate the pOH by subtracting the pH from 14.
In our exercise, blood plasma has a pH of 7.40. By using the above relationship, we find:

• pOH = 14 - 7.40 = 6.60

This pOH value helps us to determine concentrations of ions which contribute to the overall chemical equilibrium in the solution.

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a straightforward way to relate the pH of a solution to the concentration ratio of a weak acid and its conjugate base, or vice versa. For bases, the equation is:

• \[\text{pH} = \text{pK}_a + \log \left( \frac{[\text{base}]}{[\text{acid}]} \right)\]

This equation helps in calculating the ratio of base to acid, which is crucial for buffer solutions in chemical equilibrium.
In this exercise, we start by converting the base dissociation constant \(K_b\) to \(K_a\) using

• \[K_a = \frac{K_w}{K_b}\]

, where \(K_w\) is the ion-product constant of water at 25°C.
After finding \(K_a\), we calculate the pK_a and then utilize the Henderson-Hasselbalch equation to find the base-to-acid ratio.
This provides insight into how Novocaine behaves in blood plasma to provide effective local anesthesia.

Weak base dissociation constant

A weak base, unlike a strong base, partly dissociates in water, which makes its reaction state stable but incomplete. The extent of this dissociation is quantified by the base dissociation constant, \(K_b\).

• \[K_b = \frac{[BH^+][\text{OH}^-]}{[B]}\]

Here, \([BH^+]\) and \([\text{OH}^-]\) are the concentrations of the conjugate acid and hydroxide ions, respectively, and \([B]\) is the concentration of the base.
The value of \(K_b\) helps us understand the strength of the base in accepting protons.
In this exercise, the given \(K_b\) for Novocaine was used to find \(K_a\), which is further used in the Henderson-Hasselbalch equation.
Knowing the \(K_b\) allows us to predict the reaction's extent, necessary for calculating ion concentrations, especially when dealing with substances that act as weak bases in physiological environments like the human body.