Question

What is the \(\mathrm{pH}\) of a solution when \(\left[\mathrm{H}^{+}\right]\) is \(3.44 \times 10^{-4} \mathrm{M}\) ?

Short answer

The pH of the solution is approximately 3.46.

Step-by-step solution

Understanding pH Definition

The \mathrm{pH} of a solution is defined as the negative logarithm (base 10) of the hydrogen ion concentration \( [\mathrm{H}^+] \). The formula to calculate \mathrm{pH} is: \\[ \pH = -\log_{10}( [\mathrm{H}^{+}]) \\] \This formula tells us how we can relate the hydrogen ion concentration in a solution to its \mathrm{pH}.

Applying the Formula

Given \( [\mathrm{H}^+] = 3.44 \times 10^{-4} \mathrm{M} \). We use the equation from Step 1 to find the \mathrm{pH}: \\[ \pH = -\log_{10}(3.44 \times 10^{-4}) \\] \This step involves substituting the given hydrogen ion concentration into the \mathrm{pH} formula.

Calculating the Logarithm

Compute the logarithm: \\[ \\log_{10}(3.44) + \log_{10}(10^{-4}) = \\log_{10}(3.44) - 4 \\] \First, calculate \log_{10}(3.44) \(\approx 0.5377\), and then subtract 4: \\[ \pH = -(0.5377 - 4) = 3.4623 \\] \The logarithm is determined by using a calculator or log table.

Finalizing the pH Value

From Step 3, we concluded that \(pH \approx 3.4623\). Now, we confirm this is the \mathrm{pH} of the solution, since the value is reasonable for an acidic solution with a given hydrogen ion concentration.

Key concepts

Hydrogen Ion Concentration

When we talk about the concentration of hydrogen ions, we are referring to the abundance of \( \mathrm{H}^{+}\) ions in a solution. This is often represented as \([\mathrm{H}^{+}]\),and its measurement is crucial as it helps us determine the acidity of a solution.

• Concentration is usually given in moles per liter (Molarity, \( \mathrm{M} \)).
• It directly impacts the calculation of \( \mathrm{pH}\), which is a scale used to specify how acidic or basic a water-based solution is.
• The smaller the \([\mathrm{H}^{+}]\),the less acidic the solution, and vice versa.

Understanding hydrogen ion concentration sets the foundation for calculating \( \mathrm{pH} \).By knowing the concentration, we can move on to the calculation step involving logarithms.

Logarithm Base 10

The concept of logarithms, particularly with base 10, is a mathematical tool we use to simplify the process of pH calculation. A logarithm, in simple terms, indicates the power to which a number must be raised to obtain another number.

• For example, in our case, \(\log_{10}(100) = 2 \),because \(10^2 = 100\).
• In \(\mathrm{pH}\) calculations, we use base 10 logarithms to handle the typically tiny values of \([\mathrm{H}^{+}]\).
• The formula \( \mathrm{pH} = -\log_{10}([\mathrm{H}^{+}]) \)transforms the concentration into a manageable scale.

The negative sign in the formula inverts the scale, ensuring that higher concentrations (more acidic solutions) have lower \(\mathrm{pH}\) values. This use of logarithms helps simplify and standardize how we express acidity.

Acidic Solution

An acidic solution is a solution with a higher concentration of hydrogen ions, \([\mathrm{H}^{+}]\), compared to pure water. This results in a \(\mathrm{pH}\)value less than 7.

• Solutions are considered acidic when their \(\mathrm{pH}\) falls below 7.
• The more hydrogen ions present, the more acidic the solution.
• Common examples include vinegar, lemon juice, and battery acid.

In the given exercise, we calculated a \( \mathrm{pH} \)value of approximately 3.4623, indicating that the solution is indeed acidic. This is consistent with the concentration of \([\mathrm{H}^{+}]\)provided, confirming it as an acidic solution on the \( \mathrm{pH} \) scale. Understanding these concepts assists in predicting chemical behavior and potential reactivity of the solution.