What is the [tex]$\left[ H _3 O ^{+}\right]$[/tex] for a solution with a [tex]$pH$[/tex] of 3.40?

A. [tex]$2.51 \times 10^{-4} M$[/tex]
B. [tex]$3.98 \times 10^{-4} M$[/tex]
C. [tex]$2.51 \times 10^3 M$[/tex]
D. [tex]$3.98 \times 10^4 M$[/tex]



Answer :

To determine the hydronium ion concentration [tex]\(\left[ \text{H}_3\text{O}^+ \right]\)[/tex] given the pH of a solution, we use the definition of pH, which is:

[tex]\[ \text{pH} = -\log \left( \left[ \text{H}_3\text{O}^+ \right] \right) \][/tex]

Given this equation, we can solve for [tex]\(\left[ \text{H}_3\text{O}^+ \right]\)[/tex] by rearranging it as follows:

[tex]\[ \left[ \text{H}_3\text{O}^+ \right] = 10^{-\text{pH}} \][/tex]

In this specific problem, the pH is given as 3.40. Plugging in the value of pH into the formula, we get:

[tex]\[ \left[ \text{H}_3\text{O}^+ \right] = 10^{-3.40} \][/tex]

Evaluating this expression yields:

[tex]\[ \left[ \text{H}_3\text{O}^+ \right] = 0.00039810717055349735\; \text{M} \][/tex]

To express this in scientific notation for easier comparison with the given options, we can rewrite it as:

[tex]\[ \left[ \text{H}_3\text{O}^+ \right] = 3.98 \times 10^{-4}\; \text{M} \][/tex]

Thus, the correct answer is:

[tex]\[ 3.98 \times 10^{-4}\; \text{M} \][/tex]

Therefore, the [tex]\(\left[ \text{H}_3\text{O}^+ \right]\)[/tex] for the solution is:

[tex]\[ \boxed{3.98 \times 10^{-4}\; \text{M}} \][/tex]