Question

Calculate the number of \(\mathrm{H}^{+}(a q)\) ions in \(1.0 \mathrm{~mL}\) of pure water at \(25^{\circ} \mathrm{C}\)

Short answer

There are approximately \(6.022 \times 10^{13}\) H⁺ ions in 1.0 mL of pure water at 25°C.

Step-by-step solution

Find the concentration of H₃O₊ ions in water at 25°C

At 25°C, the autoionization constant of water (Kw) is given by: \[K_\mathrm{w} = [\mathrm{H}^{+}][\mathrm{OH}^{-}] = 1.0 \times 10^{-14} \mathrm{mol^2/L^2}\] Since the concentrations of \(\mathrm{H}^{+}\) ions and \(\mathrm{OH}^{-}\) ions are equal in pure water, we have: \[[\mathrm{H}^{+}] = [\mathrm{OH}^{-}]\] So, to find the concentration of \(\mathrm{H}^{+}\) ions, we can write: \[[\mathrm{H}^{+}]^2 = 1.0 \times 10^{-14}\]

Solve for the concentration of H⁺ ions

To find the concentration of \(\mathrm{H}^{+}\) ions, take the square root of both sides of the equation: \[[\mathrm{H}^{+}] = \sqrt{1.0 \times 10^{-14}}\] \[[\mathrm{H}^{+}] = 1 .0 \times 10^{-7} \mathrm{M}\] The concentration of H⁺ ions in water at 25°C is \(1 .0 \times 10^{-7} \mathrm{M}\).

Convert the volume of water to liters

We are given the volume of water as 1.0 mL. Convert this to liters: \[1.0 \mathrm{mL} \times \frac{1 \mathrm{L}}{1000 \mathrm{mL}} = 0.001 \mathrm{L}\]

Calculate the number of moles of H⁺ ions in the given volume of water

To find the number of moles of H⁺ ions present in 1.0 mL of water, multiply the concentration of H⁺ ions by the volume of water in liters: \[\mathrm{moles\:of\:H^+} = (1.0 \times 10^{-7} \mathrm{M}) \times (0.001 \mathrm{L}) = 1.0 \times 10^{-10} \mathrm{mol}\]

Calculate the number of H⁺ ions using Avogadro's number

Finally, to find the number of \(\mathrm{H}^{+}\) ions in 1.0 mL of water, multiply the number of moles by Avogadro's number (\(6.022 \times 10^{23} \mathrm{ions/mol}\)): \[\mathrm{number\:of\:H^+} = (1.0 \times 10^{-10}\mathrm{mol}) \times (6.022 \times 10^{23} \mathrm{ions/mol}) = 6.022 \times 10^{13} \mathrm{ions}\] So there are approximately \(6.022 \times 10^{13}\) H⁺ ions in 1.0 mL of pure water at 25°C.

Key concepts

Concentration of H⁺ Ions

The concentration of \( \mathrm{H}^{+} \) ions in water is an essential concept when studying water's autoionization. At 25°C, pure water undergoes a slight dissociation into \( \mathrm{H}^{+} \) and \( \mathrm{OH}^{-} \) ions, resulting in an equilibrium constant known as the water autoionization constant, denoted by \( K_w \). This constant is expressed as: \[ K_w = [\mathrm{H}^{+}][\mathrm{OH}^{-}] = 1.0 \times 10^{-14} \mathrm{mol^2/L^2} \] In pure water, the concentrations of \( \mathrm{H}^{+} \) ions and \( \mathrm{OH}^{-} \) ions are equal. Thus, we can say: \[ [\mathrm{H}^{+}] = \sqrt{1.0 \times 10^{-14}} = 1.0 \times 10^{-7} \mathrm{M} \] This concentration tells us that in each liter of pure water at 25°C, there are \( 1.0 \times 10^{-7} \) moles of \( \mathrm{H}^{+} \) ions. By knowing this value, we can progress to determining quantities such as the number of moles or even the exact count of ions with further calculations.

Avogadro's Number

Avogadro's number is a key concept in chemistry that defines the number of particles, typically atoms or ions, in one mole of a substance. This constant is approximately \( 6.022 \times 10^{23} \) particles/mol.This means when you have one mole of \( \mathrm{H}^{+} \) ions, you're essentially holding \( 6.022 \times 10^{23} \) ions. Avogadro's number is crucial for converting between moles of a substance and the actual number of discrete particles.Using our exercise as an example, if we've calculated the moles of \( \mathrm{H}^{+} \) ions in a certain volume of water, we can use Avogadro’s number to find out exactly how many \( \mathrm{H}^{+} \) ions are present. So even though the notion of a mole connects the atomic scale to the human scale, Avogadro's number gives us a practical way to count out these billions and billions of tiny entities that make up matter.

Moles of H⁺ Ions

The concept of moles is a cornerstone of chemistry, especially when dealing with tiny particles like \( \mathrm{H}^{+} \) ions. A mole allows chemists to count particles at the atomic scale by using a large number, Avogadro’s number.When you know the concentration of \( \mathrm{H}^{+} \) ions in a solution (like in pure water at 25°C, where it's \( 1.0 \times 10^{-7} \mathrm{M} \)), and you also know the volume of the solution, you can calculate the moles of \( \mathrm{H}^{+} \) ions present using the formula: \[ \mathrm{moles\ of\ H^+} = [\mathrm{H}^{+}] \times \text{Volume in liters} \]In the given problem, we converted the 1.0 mL water volume to liters (0.001 L) and multiplied it by the concentration, resulting in \( 1.0 \times 10^{-10} \) moles of \( \mathrm{H}^{+} \) ions. This calculation is simple yet powerful, as it bridges concentration with real, countable units that can be further analyzed or used in additional computations.