To find the pH of a [tex]$HCl$[/tex] solution with a given concentration of hydrogen ions ([tex]\(\left[ H ^{+}\right]\)[/tex]), we can utilize the formula for pH, which is:
[tex]\[ \text{pH} = -\log \left[ H_3O^+ \right] \][/tex]
For this problem, the concentration of [tex]\(\left[ H^+ \right]\)[/tex] is given as [tex]\(0.01 \, M\)[/tex].
Step-by-step solution:
1. Identify the concentration of hydrogen ions:
[tex]\[ \left[ H^+ \right] = 0.01 \, M \][/tex]
2. Use the pH formula:
[tex]\[ \text{pH} = -\log \left( 0.01 \right) \][/tex]
3. Recall that [tex]\(\log_{10} 0.01\)[/tex] is the power to which 10 must be raised to give 0.01. Note that:
[tex]\[ 0.01 = 10^{-2} \][/tex]
4. Thus,
[tex]\[ \log_{10} 0.01 = -2 \][/tex]
5. Substituting [tex]\(-2\)[/tex] into the pH formula:
[tex]\[ \text{pH} = -(-2) = 2 \][/tex]
Therefore, the pH of the solution is:
[tex]\[ \boxed{2} \][/tex]
