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The concentration of \( \mathrm{H}^{+} \) ions in water is an essential concept when studying water's autoionization. At 25掳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掳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 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鈥檚 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.

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鈥檚 number.When you know the concentration of \( \mathrm{H}^{+} \) ions in a solution (like in pure water at 25掳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.