Question

Given the molar concentration of hydrogen ion, calculate the concentration of hydroxide ion: (a) \(\left[\mathrm{H}^{+}\right]=0.025\) (b) \(\left[\mathrm{H}^{+}\right]=0.000017\)

Short answer

(a) [\mathrm{OH}^{-}] = 4.0 \times 10^{-13}; (b) [\mathrm{OH}^{-}] = 5.88 \times 10^{-10}

Step-by-step solution

Understand the Relationship Between Ion Concentrations

According to the principle of water's ion product at 25°C, the product of the molar concentrations of hydrogen ions \([\mathrm{H}^{+}]\) and hydroxide ions \([\mathrm{OH}^{-}]\) is constant and equal to \[ K_w = [\mathrm{H}^{+}][\mathrm{OH}^{-}] = 1.0 \times 10^{-14} \]

Calculate Hydroxide Ion Concentration for (a)

Given \([\mathrm{H}^{+}] = 0.025\), use the formula \[ [\mathrm{OH}^{-}] = \frac{K_w}{[\mathrm{H}^{+}]} \]Plug in the given value \([\mathrm{H}^{+}] = 0.025\):\[ [\mathrm{OH}^{-}] = \frac{1.0 \times 10^{-14}}{0.025} \]Calculate the result:\[ [\mathrm{OH}^{-}] = 4.0 \times 10^{-13} \]

Calculate Hydroxide Ion Concentration for (b)

For \([\mathrm{H}^{+}] = 0.000017\), use the same formula:\[ [\mathrm{OH}^{-}] = \frac{1.0 \times 10^{-14}}{0.000017} \]Perform the calculation:\[ [\mathrm{OH}^{-}] = 5.88 \times 10^{-10} \]

Key concepts

Hydrogen Ion Concentration

In chemistry, particularly when dealing with acids and bases, the concentration of hydrogen ions \([\text{H}^+]\) plays a critical role in understanding the nature and strength of solutions. The hydrogen ion concentration is a measure of the number of hydrogen ions present in a solution. In other words, it determines the acidity of the solution. The more hydrogen ions present, the more acidic the solution is, and the lower the pH value. You can relate the hydrogen ion concentration directly to pH through the equation:

• pH = -\(\log_{10}([\text{H}^+])\)

Knowing the hydrogen ion concentration allows one to calculate not only the pH but also the hydroxide ion concentration when dealing with water or aqueous solutions. The calculations demonstrate the inverse relationship between hydrogen ion concentration and pH. This basic understanding is fundamental, as it sets the foundation for exploring how changes in ion concentrations affect equilibria, especially in buffer solutions and during titrations.

Ion Product of Water

Water, a vital component of life, also participates in a delicate balance known as its ion product. At 25°C, the product of the concentrations of hydrogen ions \([\text{H}^+]\) and hydroxide ions \([\text{OH}^-]\) in pure water is constant and equal to \([ K_w ] \). This relationship is expressed by:\[ K_w = [\text{H}^+][\text{OH}^-] = 1.0 \times 10^{-14} \]This equation embodies the idea of the ion product of water, ensuring that a change in one ion's concentration is countered by a change in the other, maintaining the value of \([ K_w ] \) constant. Key points to note:

• The constant \([ K_w ]\) value changes with temperature but is commonly considered at 25°C for most calculations.
• This balance allows chemists to predict concentrations of one ion if the other ion's concentration is known.
• Any aqueous solution will have a \([\text{H}^+]\) and a \([\text{OH}^-]\) that satisfies this ion product relationship.

Understanding this concept is crucial in fields like environmental science, medicine, and any domain where pH control is vital.

Hydroxide Ion Concentration

Hydroxide ion concentration plays a complementary role to hydrogen ion concentration when it comes to determining a solution's basicity. By using known values of \([\text{H}^+]\) and applying the ion product of water, one can easily uncover the \([\text{OH}^- ]\) values. When you know the hydrogen ion concentration and want to find the hydroxide concentration, use the formula:

• \( [\text{OH}^-] = \frac{K_w}{[\text{H}^+]} \)

Through this calculation, you understand the inverse relationship wherein a rise in the \([\text{H}^+]\) leads to a decrease in \([\text{OH}^-]\), thus affecting pH and shift in equilibrium.By using these relationships, chemists and students alike can dive deeper into various pH-based applications, adjust pH levels in different settings, and further comprehend reactions in both acidic and basic environments. Remember, the goal is to bring clarity to the way ions interact in solutions, a topic central to many Chemistry applications and problem-solving exercises.