Understanding the Base Protonation Constant

Oct 13, 2025 by pynkocean.com 0 views

Introduction

In chemistry, the base protonation constant, denoted as $K_b$ , is a crucial parameter that describes the strength of a base. It is defined as the equilibrium constant for the reaction of a base with water to form its conjugate acid and hydroxide ions. In this article, we will focus on calculating the pH of a 1.5 M solution of 1-H-imidazole, a weak base with a known base protonation constant $K_b$ of $9.0 \times 10^{-8}$ at $25^{\circ} C$ .

Understanding the Base Protonation Constant

The base protonation constant $K_b$ is related to the acid dissociation constant $K_a$ by the following equation:

K_b = \frac{K_w}{K_a}

where $K_w$ is the water dissociation constant, which is equal to $1.0 \times 10^{-14}$ at $25^{\circ} C$ . The acid dissociation constant $K_a$ is a measure of the strength of an acid, with higher values indicating stronger acids.

Calculating the pH of 1-H-Imidazole

To calculate the pH of a 1.5 M solution of 1-H-imidazole, we can use the following equation:

pH = \frac{1}{2} \left( pK_b + \log \left[ \frac{C}{C - x} \right] \right)

where $pK_b$ is the negative logarithm of the base protonation constant $K_b$ , $C$ is the initial concentration of the base, and $x$ is the concentration of the base that has reacted with water to form its conjugate acid.

Step 1: Calculate the pK_b

First, we need to calculate the pK_b of 1-H-imidazole. We can do this by taking the negative logarithm of the base protonation constant $K_b$ :

pK_b = -\log \left( K_b \right)

pK_b = -\log \left( 9.0 \times 10^{-8} \right)

pK_b = 7.045

Step 2: Calculate the Concentration of the Base that has Reacted with Water

Next, we need to calculate the concentration of the base that has reacted with water to form its conjugate acid. We can do this by using the following equation:

x = \frac{K_b C}{K_b + \frac{K_w}{C}}

where $x$ is the concentration of the base that has reacted with water, $K_b$ is the base protonation constant, $C$ is the initial concentration of the base, and $K_w$ is the water dissociation constant.

Plugging in the values, we get:

x = \frac{9.0 \times 10^{-8} \times 1.5}{9.0 \times 10^{-8} + \frac{1.0 \times 10^{-14}}{1.5}}

x = 1.33 \times 10^{-7}

Step 3: Calculate the pH

Finally, we can calculate the pH of the solution using the following equation:

pH = \frac{1}{2} \left( pK_b + \log \left[ \frac{C}{C - x} \right] \right)

where $pK_b$ is the negative logarithm of the base protonation constant $K_b$ , $C$ is the initial concentration of the base, and $x$ is the concentration of the base that has reacted with water.

Plugging in the values, we get:

pH = \frac{1}{2} \left( 7.045 + \log \left[ \frac{1.5}{1.5 - 1.33 \times 10^{-7}} \right] \right)

pH = 7.045

Conclusion

In this article, we calculated the pH of a 1.5 M solution of 1-H-imidazole using the base protonation constant $K_b$ of $9.0 \times 10^{-8}$ at $25^{\circ} C$ . We found that the pH of the solution is approximately 7.045. This result is consistent with the expected pH of a weak base solution.

References

[1] Atkins, P. W., & De Paula, J. (2010). Physical chemistry. Oxford University Press.
[2] Levine, I. N. (2008). Physical chemistry. McGraw-Hill.

Note

The pH calculation was performed using the following assumptions:

The base protonation constant $K_b$ is equal to $9.0 \times 10^{-8}$ at $25^{\circ} C$ .
The initial concentration of the base is 1.5 M.
The water dissociation constant $K_w$ is equal to $1.0 \times 10^{-14}$ at $25^{\circ} C$ .

Q: What is the base protonation constant $K_b$ and how is it related to the acid dissociation constant $K_a$ ?

A: The base protonation constant $K_b$ is a measure of the strength of a base, and it is related to the acid dissociation constant $K_a$ by the following equation:

K_b = \frac{K_w}{K_a}

where $K_w$ is the water dissociation constant, which is equal to $1.0 \times 10^{-14}$ at $25^{\circ} C$ .

Q: How is the pH of a solution of 1-H-imidazole calculated?

A: The pH of a solution of 1-H-imidazole can be calculated using the following equation:

pH = \frac{1}{2} \left( pK_b + \log \left[ \frac{C}{C - x} \right] \right)

Q: What is the pK_b of 1-H-imidazole and how is it calculated?

A: The pK_b of 1-H-imidazole is calculated by taking the negative logarithm of the base protonation constant $K_b$ :

pK_b = -\log \left( K_b \right)

pK_b = -\log \left( 9.0 \times 10^{-8} \right)

pK_b = 7.045

Q: How is the concentration of the base that has reacted with water calculated?

A: The concentration of the base that has reacted with water can be calculated using the following equation:

x = \frac{K_b C}{K_b + \frac{K_w}{C}}

Q: What is the pH of a 1.5 M solution of 1-H-imidazole and how is it calculated?

A: The pH of a 1.5 M solution of 1-H-imidazole can be calculated using the following equation:

pH = \frac{1}{2} \left( pK_b + \log \left[ \frac{C}{C - x} \right] \right)

where $pK_b$ is the negative logarithm of the base protonation constant $K_b$ , $C$ is the initial concentration of the base, and $x$ is the concentration of the base that has reacted with water.

Plugging in the values, we get:

pH = \frac{1}{2} \left( 7.045 + \log \left[ \frac{1.5}{1.5 - 1.33 \times 10^{-7}} \right] \right)

pH = 7.045

Q: What are the assumptions made in the pH calculation?

A: The following assumptions are made in the pH calculation:

The base protonation constant $K_b$ is equal to $9.0 \times 10^{-8}$ at $25^{\circ} C$ .
The initial concentration of the base is 1.5 M.
The water dissociation constant $K_w$ is equal to $1.0 \times 10^{-14}$ at $25^{\circ} C$ .

These assumptions are consistent with the given data and are commonly used in chemical calculations.

Q: What is the significance of the pH calculation?

A: The pH calculation is significant because it provides a quantitative measure of the acidity or basicity of a solution. In this case, the pH calculation shows that a 1.5 M solution of 1-H-imidazole has a pH of approximately 7.045, which is consistent with the expected pH of a weak base solution.

Q: What are the limitations of the pH calculation?

A: The pH calculation has several limitations, including:

The assumption that the base protonation constant $K_b$ is constant over the entire concentration range.
The assumption that the water dissociation constant $K_w$ is constant over the entire concentration range.
The assumption that the initial concentration of the base is known accurately.

These limitations should be taken into account when interpreting the results of the pH calculation.