Question

Calculate the hydrogen ion concentration, in moles per liter, for solutions with each of the following \(\mathrm{pH}\) or \(\mathrm{pOH}\) values. a. \(\mathrm{pOH}=0.90\) b. \(\mathrm{pH}=0.90\) c. \(\mathrm{pOH}=10.3\) d. \(\mathrm{pH}=5.33\)

Short answer

a. The hydrogen ion concentration for a solution with \(\mathrm{pOH}=0.90\) is \(7.94 \times 10^{-14} \,\text{moles per liter}\). b. The hydrogen ion concentration for a solution with \(\mathrm{pH}=0.90\) is \(1.26 \times 10^{-1} \,\text{moles per liter}\). c. The hydrogen ion concentration for a solution with \(\mathrm{pOH}=10.3\) is \(1.99 \times 10^{-4} \,\text{moles per liter}\). d. The hydrogen ion concentration for a solution with \(\mathrm{pH}=5.33\) is \(4.67 \times 10^{-6} \,\text{moles per liter}\).

Step-by-step solution

Convert the given \(\mathrm{pOH}\) to \(\mathrm{pH}\)

Using the relationship: \(\mathrm{pH}+\mathrm{pOH}=14\), we can find the \(\mathrm{pH}\) value: $$\mathrm{pH} = 14 - \mathrm{pOH} = 14 - 0.90 = 13.1$$

Calculate the hydrogen ion concentration[\(\mathrm{H}^+\)] using \(\mathrm{pH}\) value

Using the relationship between \(\mathrm{pH}\) and hydrogen ion concentration: $$\mathrm{pH}=-\log_{10}[\mathrm{H}^+]$$ we can find the hydrogen ion concentration: $$13.1 = -\log_{10}[\mathrm{H}^+]$$ To find the hydrogen ion concentration, we can rewrite the equation as: $$[\mathrm{H}^+] = 10^{-13.1}$$ Calculating the concentration, we obtain: $$[\mathrm{H}^+] = 7.94 \times 10^{-14} \,\text{moles per liter}$$ b. \(\mathrm{pH}=0.90\)

Calculate the hydrogen ion concentration[\(\mathrm{H}^+\)] using \(\mathrm{pH}\) value

Using the relationship between \(\mathrm{pH}\) and hydrogen ion concentration: $$0.90 = -\log_{10}[\mathrm{H}^+]$$ $$[\mathrm{H}^+] = 10^{-0.90}$$ Calculating the concentration, we obtain: $$[\mathrm{H}^+] = 1.26 \times 10^{-1} \,\text{moles per liter}$$ c. \(\mathrm{pOH}=10.3\)

Convert the given \(\mathrm{pOH}\) to \(\mathrm{pH}\)

Using the relationship: \(\mathrm{pH}+\mathrm{pOH}=14\), we can find the \(\mathrm{pH}\) value: $$\mathrm{pH} = 14 - \mathrm{pOH} = 14 - 10.3 = 3.7$$

Calculate the hydrogen ion concentration[\(\mathrm{H}^+\)] using \(\mathrm{pH}\) value

Using the relationship between \(\mathrm{pH}\) and hydrogen ion concentration: $$3.7 = -\log_{10}[\mathrm{H}^+]$$ $$[\mathrm{H}^+] = 10^{-3.7}$$ Calculating the concentration, we obtain: $$[\mathrm{H}^+] = 1.99 \times 10^{-4} \,\text{moles per liter}$$ d. \(\mathrm{pH}=5.33\)

Calculate the hydrogen ion concentration[\(\mathrm{H}^+\)] using \(\mathrm{pH}\) value

Using the relationship between \(\mathrm{pH}\) and hydrogen ion concentration: $$5.33 = -\log_{10}[\mathrm{H}^+]$$ $$[\mathrm{H}^+] = 10^{-5.33}$$ Calculating the concentration, we obtain: $$[\mathrm{H}^+] = 4.67 \times 10^{-6} \,\text{moles per liter}$$

Key concepts

pH calculations

Understanding how to calculate pH is a crucial part of acid-base chemistry. The pH of a solution tells us how acidic or basic it is: lower pH values (< 7) mean the solution is acidic, while higher pH values (> 7) signify a basic or alkaline solution. The neutral point is pH 7. To determine pH, we use the formula:\[ \mathrm{pH} = -\log_{10} [\text{H}^+] \]where \([\text{H}^+]\) is the concentration of hydrogen ions in moles per liter. Calculating pH from hydrogen ion concentration is straightforward by simply taking the negative logarithm (base 10) of the concentration. This formula allows us to move between hydrogen ion concentration and the pH scale, helping us understand the acidity or basicity of a solution.
By mastering pH calculations, you can determine the acidity level of any given solution, a fundamental skill in chemistry.

pOH and pH relationship

The pH and pOH scales are interrelated in a simple and elegant way. Together, they explain the full nature of a solution's acidity and basicity. This relationship is given by: \[ \mathrm{pH} + \mathrm{pOH} = 14 \]This equation is derived from the ion-product constant for water at 25°C, where the concentrations of hydrogen ions \([\text{H}^+]\) and hydroxide ions \([\text{OH}^-]\) are multiplied to always equal \(1 \times 10^{-14}\).
This fundamental relationship enables chemists to calculate pH when pOH is known and vice versa, by simply subtracting one from 14.
This is particularly useful in cases where pH is not directly measured.
When dealing with basic (alkaline) solutions, pOH provides a more straightforward measurement, as it focuses on hydroxide ions rather than hydrogen ions.

• pH
  • A measure of acidity.
  • Lower values mean more acidic.
• pOH
  • A measure of basicity.
  • Lower values mean more basic.

Understanding the interplay between pH and pOH is critical for grasping the full picture of solution chemistry.

hydrogen ion concentration

Hydrogen ion concentration \([\text{H}^+]\) is an expression of the number of hydrogen ions present in a solution, and it is often expressed in terms of moles per liter. This factor is crucial for determining the acidity of a solution.
Highly acidic solutions have high \([\text{H}^+]\) while basic solutions have low \([\text{H}^+]\). The pH is inversely related to \([\text{H}^+]\):\[ \mathrm{pH} = -\log_{10} [\text{H}^+] \]This is a guiding principle for predicting the chemical behavior and reactivity of solutions.
To find \([\text{H}^+]\) from pH: take the antilogarithm (raising 10 to the power of negative pH):\[ [\text{H}^+] = 10^{-\mathrm{pH}} \]This transformation helps us connect abstract pH values to a more tangible count of ions in solution.
Understanding \([\text{H}^+]\) is like understanding the heartbeat of a solution due to its direct role in signaling solution's acidity.

• The higher the \([\text{H}^+]\), the more acidic the solution.
• The lower the \([\text{H}^+]\), the more basic the solution.

Mastery of this concept lets you determine the acidity level and predict chemical interactions in various environments.

logarithmic functions in chemistry

Logarithmic functions play a critical role in chemistry, particularly in calculations involving pH, pOH, and concentrations of hydrogen and hydroxide ions. The logarithmic relationship simplifies the large range of concentration values into a manageable scale, aiding in easier comprehension and calculation.
In essence, a logarithm answers the question: 'To what power must a specific base be raised, to obtain a given number?' For pH calculations, we use base 10 logarithms:\[ \mathrm{pH} = -\log_{10}[\text{H}^+] \]This equation tells us the pH value is the negative base 10 logarithm of the hydrogen ion concentration. Similarly, the relationship for hydroxide concentration and pOH is given by:\[ \mathrm{pOH} = -\log_{10}[\text{OH}^-] \]Logarithmic functions simplify multiplicative relationships to additive ones, which are much easier to manipulate mathematically.
This simplification is indispensable in various fields of science and engineering, as it allows professionals to work with data that spans many orders of magnitude.

Grasping the concept of logarithmic functions in chemistry is vital for interpreting and predicting how different solutions will behave under different conditions.