Question

When the \(\mathrm{pH}\) of a solution is 1 , the concentration of hydronium ions is \(10^{-1} M=0.1 M\). Assume that the volume of this solution is \(500 \mathrm{~mL}\) and that the solution is not buffered. What is the pH after \(500 \mathrm{~mL}\) of pure water is added? You will need a calculator with a logarithm function to answer this question.

Short answer

The new pH after dilution is approximately 1.3.

Step-by-step solution

Understand the Problem

We need to find the new pH after diluting a 0.1 M hydronium ion solution by adding 500 mL of pure water to the existing 500 mL solution.

Determine Initial Moles of Hydronium Ions

Calculate the initial moles of hydronium ions. Moles = concentration \(\times\) volume (L). Initially, \(0.1\ M\ \times\ 0.5\ L = 0.05\ moles\).

Calculate New Concentration After Dilution

The total volume after adding 500 mL of water is 1000 mL or 1 L. New concentration = \(\frac{0.05\ moles}{1\ L} = 0.05\ M\).

Calculate the New pH

Use the equation \(\mathrm{pH} = -\log[\text{H}^{+}]\). Substitute 0.05 M: \(\mathrm{pH} = -\log(0.05)\). Use a calculator to find that \(\mathrm{pH} \approx 1.3\).

Key concepts

Hydronium Ion Concentration

Hydronium ion concentration is a key factor in determining the acidity of a solution. It is commonly expressed in molarity (M), which measures moles of hydronium ions per liter of solution. A higher hydronium ion concentration corresponds to a lower pH value, indicating a more acidic solution. Conversely, a lower concentration results in higher pH values, representing a more basic or neutral solution.
Understanding how these ions interact in a solution is essential for calculating the pH. The relationship is encapsulated in the formula: \[\text{pH} = -\log[\text{H}_3\text{O}^+]\]This equation utilizes logarithms to represent the hydronium ion concentration on a more manageable scale. When dealing with real-world chemical solutions, having precise calculations for these concentrations helps us understand the chemical environment more intuitively.

Solution Dilution

Dilution is a process that involves decreasing the concentration of solute in a solution by adding more solvent. It doesn't change the number of solute particles; instead, it increases the volume, thereby spreading the solute particles over a larger amount of solvent. This is an essential concept in chemistry, especially when working with solutions of varying concentrations.
To calculate how dilution affects concentration, use the simple equation:\[\text{C}_1 \times \text{V}_1 = \text{C}_2 \times \text{V}_2\]

• \(\text{C}_1\) is the initial concentration.
• \(\text{V}_1\) is the initial volume.
• \(\text{C}_2\) is the final concentration after dilution.
• \(\text{V}_2\) is the final volume.

By diluting the solution, you effectively change the concentration which in turn affects the pH value. For example, if you start with 500 ml of a 0.1M solution and add another 500 ml of water, the concentration halves.

Logarithms in Chemistry

Logarithms play a critical role in chemistry, particularly when measuring pH levels. Since the range of hydrogen ion concentrations is vast, from about 1M to 10^-14M, logarithms allow chemists to work with more manageable numbers.
The logarithmic scale used in pH calculations helps simplify these large differences in concentration because it is a base-10 scale. This means each unit change represents a tenfold change in acidity or basicity. Therefore, when you calculate \[\text{pH} = -\log[\text{H}_3\text{O}^+]\]you are mapping out a very broad range of hydrogen ion concentrations into a more concise form. This simplicity allows for easier comparison and more intuitive understanding of a solution's acidity.

Acid-Base Chemistry

Acid-base chemistry is a central part of understanding many natural and industrial processes. Acids are substances that donate protons (or accept electrons), while bases accept protons (or donate electrons). When an acid dissolves in water, it increases the concentration of hydronium ions, which decreases the pH.
Conversely, bases increase the hydroxide ions which decreases the hydronium ions' concentration, thus increasing the pH. The pH scale, ranging from 0 to 14, helps categorize solutions:

• Acidic solutions: \(\text{pH}\) below 7
• Neutral solutions: \(\text{pH} = 7\)
• Basic solutions: \(\text{pH}\) above 7

Understanding these properties is key in predicting how substances will interact in various conditions. When diluting an acidic solution, like in the given exercise, you decrease its acidity by reducing the concentration of hydronium ions, thus causing the pH to rise.