Problem 249 - A Complete Resource Book in Chemistry for JEE Main

Question

\(\mathrm{pOH}\) of an aqueous solution is \(5.0 .\) If the conc. of \(\left[\mathrm{H}^{+}\right]\) in that solution is \(10^{-x} \mathrm{M}\), the value of \(\mathrm{x}\) is

Step-by-Step Solution

Answer

The value of \( x \) is 9.0.

Step 1: Understanding the pOH Definition

The pOH of a solution is defined as the negative logarithm (base 10) of the hydroxide ion concentration: \( \text{pOH} = -\log [\text{OH}^-] \). In this problem, we are given \( \text{pOH} = 5.0 \).

Step 2: Calculate [OH⁻]

To find the hydroxide ion concentration \([\text{OH}^-]\), use the definition of pOH: \[ [\text{OH}^-] = 10^{-\text{pOH}} = 10^{-5.0} \]. This means \([\text{OH}^-] = 10^{-5} \text{ M}\).

Step 3: Relationship Between pH and pOH

For aqueous solutions at 25°C, the sum of pH and pOH is always 14: \( \text{pH} + \text{pOH} = 14 \). Given \( \text{pOH} = 5.0 \), calculate \( \text{pH} \) as follows: \( \text{pH} = 14 - 5.0 = 9.0 \).

Step 4: Calculate [H⁺] from pH

The hydrogen ion concentration \([\text{H}^+]\) can be calculated from the pH: \[ [\text{H}^+] = 10^{-\text{pH}} = 10^{-9.0} \].

Step 5: Relate [H⁺] to Given Expression

We are given that \([\text{H}^+] = 10^{-x} \text{ M}\). Since from our calculation \([\text{H}^+] = 10^{-9.0} \text{ M}\), this implies that \( x = 9.0 \).

Key Concepts

Understanding Hydroxide Ion Concentration

The hydroxide ion concentration is a crucial part of understanding the behavior of alkaline or basic solutions. Hydroxide ions, denoted as \([ ext{OH}^-]\), contribute to a solution's basicity.
In terms of the pOH scale, which measures how basic a solution is, it is defined as the negative log of this concentration. In mathematical form, pOH is expressed as:

• \( \text{pOH} = -\log [\text{OH}^-] \)

Given the pOH of 5.0 in our exercise, we find the hydroxide ion concentration by reversing the logarithmic process. We do this by calculating:

• \([\text{OH}^-] = 10^{\text{-pOH}} = 10^{-5.0} = 10^{-5} \text{ M}\)

This helps determine how basic the aqueous solution actually is.

Exploring the Relationship Between pH and pOH

The relationship between pH and pOH is foundational for understanding aqueous solutions, especially at a temperature of 25°C. The sum of pH and pOH for such solutions is always 14. This relationship is described by the equation:

• \( \text{pH} + \text{pOH} = 14 \)

This makes it easy to convert between pOH and pH. Given our previous value of \( \text{pOH} = 5.0 \), we can calculate pH by simply subtracting from 14, giving:

• \( \text{pH} = 14 - 5.0 = 9.0 \)

Understanding this equation allows one to switch between measuring acidity and basicity.

Determining Hydrogen Ion Concentration

Hydrogen ion concentration is a core concept when discussing the acidity of a solution. It is denoted as \([\text{H}^+]\). The pH, which tells us how acidic a solution is, is calculated with the formula:

• \( \text{pH} = -\log [\text{H}^+] \)

Therefore, to find \([\text{H}^+]\) given the pH, we reverse the logarithm:

• \([\text{H}^+] = 10^{-\text{pH}} \)

From our determined pH of 9.0, the calculation becomes:

• \([\text{H}^+] = 10^{-9.0} \text{ M}\)

This value is critical for understanding the solution's acidic nature and, in this case, confirms the calculation, concluding \( x = 9.0 \) if \([\text{H}^+] = 10^{-x} \text{ M}\).