To solve this problem, we need to determine the amount of hydrogen bromide (HBr) required to achieve the desired pH of the solution. Let's break it down step by step.
### Step 1: Understand the Relationship Between pH and [H⁺]
The pH of a solution is given by the formula:
[tex]\[ \text{pH} = -\log[\text{H}^+] \][/tex]
Where:
- [tex]\([\text{H}^+]\)[/tex] is the concentration of hydrogen ions in the solution in moles per liter (M).
### Step 2: Calculate the Hydrogen Ion Concentration
We are given that the pH of the solution should be 2.00. Using the pH formula, we can determine the concentration of hydrogen ions [tex]\([\text{H}^+]\)[/tex].
[tex]\[
\text{pH} = 2.00 \implies -\log[\text{H}^+] = 2.00
\][/tex]
To find [tex]\([\text{H}^+]\)[/tex], we need to rearrange the formula:
[tex]\[
[\text{H}^+] = 10^{-\text{pH}}
\][/tex]
Plugging in the given pH:
[tex]\[
[\text{H}^+] = 10^{-2.00}
\][/tex]
Calculating this:
[tex]\[
[\text{H}^+] = 0.01 \, \text{M}
\][/tex]
### Step 3: Relate [H⁺] to the Amount of HBr
Hydrogen bromide (HBr) is a strong acid, which means it completely dissociates in water. This means every mole of HBr added to the water will produce one mole of [tex]\(\text{H}^+\)[/tex] ions. Therefore, the concentration of HBr needed is equal to the concentration of [tex]\(\text{H}^+\)[/tex] ions required.
So, [tex]\([\text{HBr}] = [\text{H}^+] = 0.01 \, \text{M}\)[/tex].
### Step 4: Calculate the Amount of HBr in Moles
The concentration calculated above is in moles per liter (M). Since we are preparing 1.00 L of solution, the amount in moles of HBr required will be equivalent to its concentration.
[tex]\[
\text{Amount of HBr (in moles)} = \text{Volume of solution (in L)} \times \text{Concentration (in M)}
\][/tex]
Given that the volume of the solution is 1.00 L:
[tex]\[
\text{Amount of HBr} = 1.00 \, \text{L} \times 0.01 \, \text{M} = 0.01 \, \text{moles}
\][/tex]
### Final Answer:
The amount of HBr that must be dissolved in 1.00 L of water to prepare a solution whose pH is 2.00 is [tex]\(0.01\)[/tex] moles.
