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

Question

The number of \(\mathrm{H}^{+}\)ions present in \(\mathrm{lcm}^{3}\) 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 in \( 1 \text{ lcm}^3 \) of the solution is \( 6.02 \times 10^{10} \) (option c).

Step 1: Understand pH Definition

The pH of a solution is defined as the negative logarithm (base 10) of the hydrogen ion concentration: \( \text{pH} = -\log_{10} \left[ \mathrm{H}^+ \right] \). We need to find \( \left[ \mathrm{H}^+ \right] \) given that the pH is 10.

Step 2: Calculate Hydrogen Ion Concentration

Since \( \text{pH} = -\log_{10} \left[ \mathrm{H}^+ \right] \), if \( \text{pH} = 10 \), then \( \left[ \mathrm{H}^+ \right] = 10^{-10} \text{ M} \). This is the hydrogen ion concentration in moles per liter.

Step 3: Convert Volume from lcm³ to Liters

1 \( \text{lcm}^3 \) is actually \( 1 \text{ cm}^3 \) or \( 1 \text{ mL} \), which is \( 10^{-3} \text{ L} \) (since there are 1000 mL in a L).

Step 4: Calculate Number of Hydrogen Ions in 1 lcm³

Using the concentration \( \left[ \mathrm{H}^+ \right] = 10^{-10} \text{ M} \), and knowing that 1 \( \text{lcm}^3 \) is \( 10^{-3} \text{ L} \), the number of moles of \( \mathrm{H}^+ \) ions in \( 1 \text{ mL} \) is \( 10^{-10} \times 10^{-3} = 10^{-13} \text{ moles} \).

Step 5: Use Avogadro's Number to Find Total Ions

Avogadro's number tells us there are \( 6.02 \times 10^{23} \) molecules (or ions) per mole. Therefore, the number of \( \mathrm{H}^+ \) ions in \( 10^{-13} \) moles is \( 10^{-13} \times 6.02 \times 10^{23} = 6.02 \times 10^{10} \).

Key Concepts

Hydrogen Ion Concentration

To understand hydrogen ion concentration, think of it as how many hydrogen ions (H^+) are present in a specific volume of a solution. The concept is closely tied to the pH scale. This scale runs from 0 to 14 and tells you how acidic or basic a solution is.
When you have the pH value, like a pH of 10, you can calculate the (H^+) ion concentration using the formula: \( [H^+] = 10^{- ext{pH}}\).
In simple terms, a pH of 10 means the hydrogen ion concentration is \( 10^{-10} ext{ M} \) (M stands for molarity, or moles per liter).
Understanding this concept is crucial when dealing with chemical reactions, as it affects how molecules interact in solutions.

Avogadro's Number

Avogadro's number is a fundamental constant in chemistry that is used to connect moles to the number of particles. This number, \( 6.02 \times 10^{23} \), tells us how many molecules, atoms, or ions are in one mole of a substance.
For hydrogen ions, using Avogadro's number helps convert moles of (H^+) ions into the actual number of (H^+) ions.
For instance, if you find that you have \(10^{-13}\) moles of (H^+) ions in a solution, then to find the actual count of these ions, you multiply the moles by Avogadro’s number: \(10^{-13} \times 6.02 \times 10^{23}\), giving \(6.02 \times 10^{10}\) ions.
This step is important in scientific calculations for moving from theoretical values to real-world applications.

Molarity

Molarity is a way to express how concentrated a solution is by focusing on how many moles of solute are present in one liter of solution.
For example, if a solution has a molarity of \(10^{-10} ext{ M}\), it means there are \(10^{-10}\) moles of hydrogen ions in every liter of that solution.
When we are asked to calculate pH or solve for hydrogen ion concentration, understanding molarity is key. Such concentration measurements help in predicting how acids and bases will behave in reactions.
Always remember that molarity connects to many other chemical concepts, making it a fundamental idea to grasp in chemistry.