To determine how the [tex]\( H^{+} \)[/tex] concentration of a solution with a pH of 4 compares with that of a solution with a pH of 2, we use the relationship given by the pH scale:
[tex]\[
pH = -\log \left[ H^{+} \right]
\][/tex]
Let's begin by finding the hydrogen ion concentration for each pH value.
1. Hydrogen ion concentration for pH 4:
[tex]\[
pH = 4 \implies 4 = -\log \left[ H^{+} \right]
\][/tex]
Solving for [tex]\( H^{+} \)[/tex]:
[tex]\[
\left[ H^{+} \right] = 10^{-4}
\][/tex]
2. Hydrogen ion concentration for pH 2:
[tex]\[
pH = 2 \implies 2 = -\log \left[ H^{+} \right]
\][/tex]
Solving for [tex]\( H^{+} \)[/tex]:
[tex]\[
\left[ H^{+} \right] = 10^{-2}
\][/tex]
Next, we find the ratio of the hydrogen ion concentrations:
3. Ratio of hydrogen ion concentrations:
[tex]\[
\text{Ratio} = \frac{\left[ H^{+} \right]_{\text{pH 4}}}{\left[ H^{+} \right]_{\text{pH 2}}} = \frac{10^{-4}}{10^{-2}} = 10^{-4 - (-2)} = 10^{-4 + 2} = 10^{-2} = 0.01
\][/tex]
Thus, the [tex]\( H^{+} \)[/tex] concentration in a solution with a pH of 4 is [tex]\( 0.01 \)[/tex] times that of a solution with a pH of 2.
This corresponds to option C:
C. The [tex]\( H^{+} \)[/tex] concentration in a solution with a pH of 4 is 0.01 times that of a solution with a pH of 2.
