Question

Calculate the \(\mathrm{pH}\) of each of the following solutions from the information given. a. \(\left[\mathrm{H}^{+}\right]=4.78 \times 10^{-2} M\) b. \(\mathrm{pOH}=4.56\) c. \(\left[\mathrm{OH}^{-}\right]=9.74 \times 10^{-3} M\) d. \(\left[\mathrm{H}^{+}\right]=1.24 \times 10^{-8} M\)

Short answer

a. \(\mathrm{pH = -log[4.78 \times 10^{-2}] = 1.32}\) b. \(\mathrm{pH = 14 - 4.56 = 9.44}\) c. \(\mathrm{pOH = -log[9.74 \times 10^{-3}] = 2.01; pH = 14 - 2.01 = 11.99}\) d. \(\mathrm{pH = -log[1.24 \times 10^{-8}] = 7.91}\)

Step-by-step solution

Calculate pH from known [H+]

We can directly obtain the pH from the given [H+]. Using the formula: \[\mathrm{pH = -log[H^+]}\] \[\mathrm{pH = -log[4.78 \times 10^{-2}]}\] In case b:

Calculate pH from known pOH

We can obtain the pH by using the relationship: \[\mathrm{pH + pOH = 14}\] Knowing the pOH value, we can calculate the pH, so \[\mathrm{pH = 14 - pOH}\] \[\mathrm{pH = 14 - 4.56}\] In case c:

Calculate pH from known [OH-]

We can obtain the pOH from [OH-], first, using the formula: \[\mathrm{pOH = -log[OH^{-}]}\] \[\mathrm{pOH = -log[9.74 \times 10^{-3}]}\] Next, we can calculate the pH, using the relation between pH and pOH: \[\mathrm{pH + pOH = 14}\] \[\mathrm{pH = 14 - pOH}\] In case d:

Calculate pH from known [H+]

We can directly obtain the pH from the given [H+]. Using the formula: \[\mathrm{pH = -log[H^+]}\] \[\mathrm{pH = -log[1.24 \times 10^{-8}]}\]

Key concepts

pH formula

The pH formula is a straightforward way to determine the acidity or basicity of an aqueous solution. The pH of a solution is calculated using the formula: \[\text{pH} = -\log_{10}[H^+]\]where

• [H⁺] is the concentration of hydrogen ions in moles per liter (M).
• "log" refers to the logarithm to the base 10.

The negative sign in front of the log means that as the hydrogen ion concentration increases, the pH value decreases, indicating a more acidic solution.
This relationship is crucial because it measures how acidic or basic a solution is on a scale typically ranging from 0 to 14.
Lower pH values (< 7) signify acidic solutions, while higher pH values (> 7) represent basic solutions.
This understanding allows scientists and students alike to easily gauge the nature of a solution based on hydrogen ion concentration.

pOH relationship

To understand the pOH relationship, we need to delve into how pH and pOH signify the acidity and basicity of a solution. pOH is a measure of hydroxide ion concentration [OH^-].
The relationship between pH and pOH is governed by the equation:\[\text{pH} + \text{pOH} = 14\]This equation highlights the balance between hydrogen and hydroxide ions in water, which is always maintained to produce the neutral pH of 7 in pure water.
This implies that if you know the pH of a solution, you can easily determine its pOH, and vice versa.
For instance, a pOH of 4.56 can be used to find the pH by subtracting from 14, thus yielding a pH of 9.44.
This balance ensures that any change in the concentration of H⁺ ions is mirrored by an inverse change in OH^- ions, maintaining the water's equilibrium.

acidic and basic solutions

Solutions' acidity or basicity is determined by comparing their pH value to the scale established for water. Pure water at 25°C has a pH of 7, marking it as neutral.

• Acidic Solutions: These have a pH less than 7. This means there is a higher concentration of hydrogen ions [H⁺] compared to hydroxide ions [OH^-]. Examples include lemon juice and vinegar.
• Basic (or Alkaline) Solutions: These have a pH greater than 7. They contain more hydroxide ions compared to hydrogen ions. Common examples are soap and baking soda solutions.

Understanding whether a solution is acidic or basic is essential in various applications, from cooking to chemical manufacturing.
It also plays a critical role in biological systems, where cell function often relies on maintaining a specific pH level to proceed effectively.
Knowing the nature of a solution helps us predict its reactivity and its effect on the environment or biological systems.

hydrogen ion concentration

The concentration of hydrogen ions [H⁺] in a solution is a key factor in determining its pH. The more hydrogen ions present, the more acidic the solution is.
This is expressed in scientific notation due to the small quantities typically encountered. For example, an [H⁺] concentration of 4.78 × 10^-2 M would signify a highly acidic solution.
Conversely, a small concentration of hydrogen ions, such as 1.24 × 10^-8 M, would result in a more basic solution, indicating fewer hydrogen ions compared to hydroxide ions.
By calculating the negative logarithm (base 10) of the hydrogen ion concentration, we determine the pH, thus allowing us to gauge the solution’s acidic or basic nature.
This calculation is pivotal in fields ranging from environmental science to medicine, where pH control is crucial in processes like metabolism and drug formulation.