Problem 107 - A Complete Resource Book in Chemistry for JEE Main

Question

The number of \(\mathrm{H}^{+}\)ions present in \(1 \mathrm{~cm}^{3}\) of a solution whose \(\mathrm{pH}\) is 10 is (a) \(10^{-10}\) (b) \(10^{-13}\) (c) \(6.02 \times 10^{10}\) (d) \(6.02 \times 10^{13}\)

Step-by-Step Solution

Answer

The number of \(\mathrm{H}^{+}\) ions is \(6.02 \times 10^{10}\).

Step 1: Understanding pH

The pH of a solution is the negative logarithm (base 10) of the hydrogen ion concentration, which can be expressed as: \[ \text{pH} = -\log[\text{H}^+] \]Given: \( \text{pH} = 10 \).

Step 2: Convert pH to Concentration

To find the concentration of hydrogen ions, we use the formula: \[[\text{H}^+] = 10^{-\text{pH}} \]Substituting the given pH:\[[\text{H}^+] = 10^{-10} \] mol/L

Step 3: Calculate Volume in Liters

Given the volume is \(1 \text{ cm}^3\). Convert this to liters since 1 liter = 1000 \text{ cm}^3, the volume in liters is:\[ \frac{1}{1000} = 10^{-3} \text{ L} \]

Step 4: Calculate Number of Ions

Use the concentration of \(\mathrm{H}^{+}\) and the volume to find the total number of \(\mathrm{H}^{+}\) ions. Number of mols of ions is:\[ 10^{-10} \text{ mol/L} \times 10^{-3} \text{ L} = 10^{-13} \text{ mols} \]The number of ions is:\[ 10^{-13} \times 6.022 \times 10^{23} = 6.022 \times 10^{10} \] ions.

Key Concepts

Understanding pH Value

The pH value is a crucial concept when dealing with solutions, especially in chemistry and biology. It measures the acidity or basicity of a solution. The pH scale ranges from 0 to 14:

• 0 to 6 indicates an acidic solution
• 7 is neutral, like pure water
• 8 to 14 indicates a basic or alkaline solution

The pH is calculated as the negative logarithm (base 10) of the hydrogen ion concentration of the solution, expressed with the formula:\[ \text{pH} = -\log[\mathrm{H}^+] \]This means that if you know the pH value, you can deduce the concentration of hydrogen ions. In the exercise, the given pH is 10. This tells us that the solution is slightly basic. With this value, you can calculate the concentration of hydrogen ions, giving insight into the solution's properties.

Determining Solution Concentration

Concentration tells us how much of a certain substance is present in a specified volume of solution. When we talk about hydrogen ion concentration, it informs us about the number of \(\mathrm{H}^{+}\) ions per liter of the solution.
To find this concentration from a given pH, use the formula:\[ [\mathrm{H}^+] = 10^{-\text{pH}} \]In the exercise, with a pH of 10, the concentration becomes:\[ [\mathrm{H}^+] = 10^{-10} \text{ mol/L} \]This tells us there are very few hydrogen ions in each liter of this solution, confirming it is quite basic. Understanding concentration will allow you to further calculate how many ions are present in any given volume of the solution by considering how the number of moles relates to the Avogadro's number.

Calculating the Number of Ions

To find the number of \(\mathrm{H}^{+}\) ions in a specific volume, you first need to calculate the number of moles of \(\mathrm{H}^{+}\) ions, then convert this to the actual number of ions. Here’s how it’s done:
1. Start with the concentration: for a pH of 10, you have \( 10^{-10} \text{ mol/L} \).2. Convert the volume from \(\text{cm}^3\) to liters as calculations are typically done in liters. Given \(1 \text{ cm}^3\), that's:\[ \text{Volume in liters} = \frac{1}{1000} = 10^{-3} \text{ L} \]3. Calculate the number of moles of ions:\[ \text{Moles of ions} = 10^{-10} \text{ mol/L} \times 10^{-3} \text{ L} = 10^{-13} \text{ mols} \]4. Use Avogadro's number \( (6.022 \times 10^{23}) \) to find the number of ions:\[ \text{Number of ions} = 10^{-13} \times 6.022 \times 10^{23} = 6.022 \times 10^{10} \] ions.This calculation shows you the step-by-step process of not only understanding the concentration but also how it translates into the actual quantity of ions in a specific volume, which is fundamental in many scientific and practical applications.