At the freezing point of water \(\left(0^{\circ} \mathrm{C}\right), K_{w}=1.2 \times 10^{-15}\) Calculate \(\left[\mathrm{H}^{+}\right]\) and \(\left[\mathrm{OH}^{-}\right]\) for a neutral solution at this temperature.

Short Answer

Expert verified
In a neutral solution at 0°C with \(K_w = 1.2 \times 10^{-15}\), both the hydrogen ion concentration and the hydroxide ion concentration are approximately \(3.5 \times 10^{-8} M\).

Step by step solution

01

Write the Ion Product of Water Expression

For a neutral solution at 0°C, the ion product of water, \(K_w\), is given as \(1.2 \times 10^{-15}\). We can write the expression for the ion product of water as follows: \[K_w = \left[\mathrm{H}^{+}\right]^2\] Here, \(K_w\) is the ion product constant for water. The concentrations of hydrogen ions and hydroxide ions are equal in a neutral solution, since the number of hydrogen ions and the number of hydroxide ions are equal.
02

Solve for the Hydrogen Ion Concentration

Plug in the given value for \(K_w\) and solve for \(\left[\mathrm{H}^{+}\right]\): \(1.2\times 10^{-15} = \left[\mathrm{H}^{+}\right]^2\) To find the hydrogen ion concentration, take the square root of both sides: \(\left[\mathrm{H}^{+}\right] = \sqrt{1.2\times 10^{-15}}\)
03

Calculate the Hydrogen Ion Concentration

Evaluate the square root: \(\left[\mathrm{H}^{+}\right] \approx 3.5 \times 10^{-8} M\)
04

Find the Hydroxide Ion Concentration

Since the solution is neutral, the hydroxide ion concentration equals the hydrogen ion concentration: \(\left[\mathrm{OH}^{-}\right] = \left[\mathrm{H}^{+}\right]\) Thus, \(\left[\mathrm{OH}^{-}\right] \approx 3.5 \times 10^{-8} M\). In conclusion, for a neutral solution at 0°C, both the hydrogen ion concentration and the hydroxide ion concentration are approximately \(3.5 \times 10^{-8} M\).

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