Question

When \(1.0 \mathrm{~mL}\) of \(1.0 \mathrm{M} \mathrm{HCl}\) is added to \(50 . \mathrm{mL}\) of \(1.0 M \mathrm{NaCl}\), the \(\left[\mathrm{H}^{+}\right.\)] changes from \(1 \times 10^{-7} M\) to \(2.0 \times 10^{-2} M\). Calculate the initial \(\mathrm{pH}\) and the \(\mathrm{pH}\) change in the solution.

Short answer

The initial pH was 7, and the change in pH was -5.3.

Step-by-step solution

Calculate initial pH

To find the initial pH of the NaCl solution, use the concentration of H+ from the water's autoionization, which is given as \(1 \times 10^{-7} M\). The pH is calculated using the formula pH = -log[H+]. So, pH = -log(1 \times 10^{-7}) = 7.

Calculate final pH

After adding the HCl, the final concentration [H+] is given as \(2.0 \times 10^{-2} M\). Calculate the final pH using the formula pH = -log[H+]. Therefore, pH = -log(2.0 \times 10^{-2}) = 1.7.

Calculate pH change

Subtract the initial pH from the final pH to find the pH change: Initial pH = 7, final pH = 1.7. The pH change is 7 - 1.7 = 5.3.

Key concepts

Autoionization of Water

Water is not just a solvent; it also undergoes a remarkable self-transformation known as autoionization. Autoionization is the process where water molecules spontaneously split into hydrogen ions (\text{H}^+), also known as protons, and hydroxide ions (\text{OH}^-). This is described by the equation: \
\[2\text{H}_2\text{O} \rightleftharpoons \text{H}_3\text{O}^+ + \text{OH}^-\
\] The concentration of hydrogen ions produced by autoionization is very low, typically around \(1 \times 10^{-7} M\), at 25°C which is considered neutral pH. Understanding this concept is crucial when discussing the pH of solutions because it sets the foundation for the pH scale itself.

Consider a scenario where pure water is the starting point. Here, we say the water is 'neutral' because the concentrations of hydrogen ions and hydroxide ions are equal, maintaining a delicate balance which influences the pH value. In the exercise, the presence of NaCl does not change the autoionization of water, and thus the initial [H+] remains at \(1 \times 10^{-7} M\).

Hydrogen Ion Concentration

The term 'hydrogen ion concentration' refers to the amount of hydrogen ions present in a solution. It is crucial because the [H+] indicates the acidity or basicity of a solution. A high concentration of hydrogen ions (represented by a high [H+] value) means the solution is acidic, while a low concentration signifies a basic or alkaline solution.

In the given exercise, we examine the change in hydrogen ion concentration when a strong acid, HCl, is added to a neutral salt solution of NaCl. The significant increase in [H+] from \(1 \times 10^{-7} M\) to \(2.0 \times 10^{-2} M\) indicates a shift towards a more acidic environment. This shift is crucial as it directly affects the calculation of pH and the understanding of the solution's properties.

pH Scale

The pH scale is a measure of acidity or basicity of a water-based solution. It ranges typically from 0 to 14, with 7 being neutral (the pH of pure water at 25°C). Values below 7 indicate an acidic solution and values above 7 indicate a basic or alkaline solution. The scale is logarithmic, meaning each whole pH value below 7 is ten times more acidic than the next higher value, and each whole pH value above 7 is ten times less acidic (or more basic) than the one below it.

Applying this to our example, the initial pH of the solution was 7, which is neutral. After adding HCl, the solution's pH dropped to 1.7, demonstrating a substantial increase in acidity. Recognizing this shift on the pH scale helps students visualize the magnitude of the change in acidity.

Logarithmic Calculations

Logarithmic calculations are pivotal when working with pH because the pH scale itself is logarithmic. The pH is the negative base 10 logarithm of the hydrogen ion concentration: \[ \text{pH} = -\log[\text{H}^+] \]

In the provided solution, logarithmic calculations were used to determine the initial and final pH values. Initially, the pH was calculated as 7 (\( -\log(1 \times 10^{-7}) \)), and the final pH was found to be 1.7 (\( -\log(2.0 \times 10^{-2}) \)). It's important to grasp the concept of logs to understand the dramatic change in hydrogen ion concentration that a relatively small numerical change in pH represents. In essence, each unit change in pH corresponds to a tenfold change in [H+], which is significant in chemical terms and has practical implications in many scientific fields.