Question

A) Determine the \(\mathrm{pH}\) of a solution with a hydrogen ion concentration of \(3.5 \times 10^{-4}\) B) If a solution has a \(\mathrm{pH}\) of \(4.25\), what is the hydrogen ion concentration?

Short answer

A) The pH of the solution with a hydrogen ion concentration of \(3.5 \times 10^{-4}\) is 3.46. B) The hydrogen ion concentration of the solution with a pH of 4.25 is \(5.62 \times 10^{-5}\) moles per liter.

Step-by-step solution

Understand the formula for pH

The formula for pH is: \[ \mathrm{pH} = -\log [\mathrm{H^+}] \] where \([\mathrm{H^+}]\) is the concentration of hydrogen ions in solution. The square brackets denote concentration.

Calculate the pH (part A)

Given that the hydrogen ion concentration is \(3.5 \times 10^{-4}\), substitute this value into the formula to find the pH: \[ \mathrm{pH} = -\log (3.5 \times 10^{-4}) = 3.46 \]

Understand how to find the hydrogen ion concentration from pH

To find the hydrogen ion concentration from the pH, rearrange the pH formula as follows: \[ [\mathrm{H^+}] = 10^{-\mathrm{pH}} \]

Calculate the hydrogen ion concentration (part B)

Since the pH is given as 4.25, substitute this value into the rearranged formula: \[ [\mathrm{H^+}] = 10^{-4.25} = 5.62 \times 10^{-5} \] So the hydrogen ion concentration of the solution is \(5.62 \times 10^{-5}\) moles per liter.

Key concepts

Hydrogen Ion Concentration

Understanding hydrogen ion concentration is crucial when studying acidity and basicity in chemistry. This term specifically refers to the number of hydrogen ions, denoted as \( \text{H}^+ \), present in a solution. It is measured in moles per liter (M), indicating the molarity of \( \text{H}^+ \).
When we look at a solution and say it is 'acidic', what we're really saying is that it has a higher concentration of hydrogen ions. If a solution has a hydrogen ion concentration of \( 3.5 \times 10^{-4} \) M, we interpret that solution as being acidic. Conversely, a smaller concentration of hydrogen ions implies a solution is more basic (or alkaline).
In scientific practice, accurately measuring these tiny concentrations requires sensitive instruments. However, in educational settings and for practical calculations, we use the given molarity to find the pH level, which tells us about the acidity of the solution in a more digestible way.

Logarithmic Scale

A logarithmic scale is a non-linear scale used for a vast range of scientific measurements. It is especially useful when dealing with quantities that can vary greatly. The pH scale is one such utility, expressing the acidity or basicity of a solution on a logarithmic scale.
Here's how it works: on the pH scale, each number is 10 times more or less acidic than the next one. For example, a solution with a pH of 3 is ten times more acidic than a solution with a pH of 4. This concept makes sense if you consider the underlying formula, \( \text{pH} = -\log [\text{H}^+] \), where the concentration of hydrogen ions is inversely related to the pH value.
The reason we use a logarithmic scale for pH is due to the wide range of \( \text{H}^+ \) concentrations we might encounter – from very acidic solutions (\textgreater1 M) to very dilute (\textless\textless1 M). Without a logarithmic scale, comparing such varied concentrations would be cumbersome and less intuitive.

Concentration of H+

When discussing acidity, the concentration of \( \text{H}^+ \) ions is paramount. To properly grasp this, you need to understand that the concentrations are typically very small because the solutions are often dilute. In the exercise, a concentration of \( 5.62 \times 10^{-5} \) M was found when the pH was 4.25, showcasing this exceptionally low molarity.
To arrive at the concentration from the pH, the calculation was made using the inverse of the logarithmic function, as the formula \( [\text{H}^+] = 10^{-\text{pH}} \) illustrates. This mathematical operation, the antilog, helps us step back from the pH value to the actual molarity of \( \text{H}^+ \). Whenever you're handling pH calculations, remember that you're working with a scale: a decrease in pH means an increase in the \( \text{H}^+ \) ion concentration, and vice versa, moving us along a continuum from highly acidic to highly basic solutions.