Problem 229 - A Complete Resource Book in Chemistry for JEE Main

Question

If the \(\left[\mathrm{H}^{+}\right]\)is increased by 10 times, its \(\mathrm{pH}\) will change by ______ units.

Step-by-Step Solution

Answer

The pH will decrease by 1 unit.

Step 1: Understand the pH formula

The pH of a solution is calculated using the formula \( \text{pH} = -\log_{10} [\mathrm{H}^+] \). This equation relates the hydrogen ion concentration \([\mathrm{H}^+]\) to the pH.

Step 2: Calculate new function with increased concentration

If the concentration of \([\mathrm{H}^+]\) is increased by 10 times, the new concentration will be denoted as \( [\mathrm{H}^+]_{new} = 10 \times [\mathrm{H}^+] \). We can express the related pH as \( \text{pH}_{new} = -\log_{10} (10 \times [\mathrm{H}^+]) \).

Step 3: Simplify the logarithmic equation

Using the log property \( \log_{10} (a \times b) = \log_{10} a + \log_{10} b \), we have \( \text{pH}_{new} = - \log_{10} (10) - \log_{10} ([\mathrm{H}^+]) \). It follows that \( \text{pH}_{new} = -1 - \log_{10} [\mathrm{H}^+] \).

Step 4: Calculate change in pH

The original pH is \( \text{pH} = - \log_{10} [\mathrm{H}^+] \). The new pH is \( \text{pH}_{new} = -1 - \log_{10} [\mathrm{H}^+] = \text{pH} - 1 \). Therefore, the change in pH is 1 unit.

Key Concepts

pH Formula

The pH formula is a key concept in chemistry that helps us understand the acidity or basicity of a solution. It's like a translator between the concentration of hydrogen ions in a solution and a number we can easily interpret. The formula used is:

• pH = -\( \log_{10} \left[ \text{H}^+ \right] \)

This simple yet powerful expression means that the pH is the negative logarithm (base 10) of the hydrogen ion concentration. The negative sign indicates that as the hydrogen ion concentration increases, the pH value actually decreases. Likewise, if the concentration decreases, the pH value increases.By using this formula, you can understand that a low pH value indicates a high concentration of hydrogen ions, suggesting an acidic solution. Conversely, a high pH value suggests a low concentration of hydrogen ions, indicating a basic (or alkaline) environment.

Hydrogen Ion Concentration

Hydrogen ion concentration, often written as

• \( \left[ \text{H}^+ \right] \)

refers to the number of hydrogen ions present in a solution. It is a crucial factor in determining the pH level of that solution. Hydrogen ions are simply protons, and their concentration in a solution tells us how acidic that solution is. The relationship between pH and hydrogen ion concentration is inversely logarithmic. For instance, if the concentration of hydrogen ions increases by a factor of 10, the new concentration can be represented as

• \( [\text{H}^+]_{new} = 10 \times [\text{H}^+] \)

This results in the pH decreasing by 1 unit. Understanding hydrogen ion concentration is essential as it directly influences the chemical nature and behavior of the solution.

Logarithmic Properties

In order to fully grasp how pH and hydrogen ion concentration are related, it's important to understand logarithmic properties. Logarithms are mathematical tools that can simplify the multiplication and division of numbers into addition and subtraction, thanks to properties like

• \( \log_{10}(a \times b) = \log_{10} a + \log_{10} b \)

Through this property, we can simplify complex calculations, like finding the pH of a solution with increased hydrogen ion concentration. For the exercise, we applied the property:

• \( \text{pH}_{new} = -\log_{10}(10 \times [\text{H}^+]) = -\log_{10} 10 - \log_{10} [\text{H}^+] \)

This translates to

• \( \text{pH}_{new} = -1 - \log_{10} [\text{H}^+] \)

Understanding these properties not only helps simplify the complex relationships in chemistry but also deepens our mathematical comprehension of logarithmic functions.