Certainly! Let's carefully analyze the scenario with the given cabbage pH indicator color key and the provided options for hydrogen ion concentration.
We are given a colored indicator key for cabbage that corresponds to different pH ranges. The solution turns blue, which indicates a pH of approximately 8-9. Our goal is to determine the correct hydrogen ion concentration ([H⁺]) for this pH range from the given options.
The pH of a solution is defined as the negative logarithm (base 10) of the hydrogen ion concentration:
[tex]\[ \text{pH} = -\log[H^+] \][/tex]
To match the pH range of 8-9, we need to find the concentration of hydrogen ions that produces these pH values.
1. For pH of 8:
[tex]\[ 8 = -\log[H^+] \][/tex]
[tex]\[ [H^+] = 10^{-8} \text{ M} \][/tex]
2. For pH of 9:
[tex]\[ 9 = -\log[H^+] \][/tex]
[tex]\[ [H^+] = 10^{-9} \text{ M} \][/tex]
Next, we examine the provided concentration options to see which one corresponds to a pH in the 8-9 range:
- [tex]\(1 \times 10^{-2} \text{ M}\)[/tex]:
[tex]\[ \text{pH} = -\log(1 \times 10^{-2}) = 2 \][/tex]
This pH is too acidic.
- [tex]\(5 \times 10^{-2} \text{ M}\)[/tex]:
[tex]\[ \text{pH} = -\log(5 \times 10^{-2}) \approx 1.3 \][/tex]
This pH is even more acidic.
- [tex]\(5 \times 10^{-4} \text{ M}\)[/tex]:
[tex]\[ \text{pH} = -\log(5 \times 10^{-4}) \approx 3.3 \][/tex]
This pH is still too acidic.
- [tex]\(1 \times 10^{-8} \text{ M}\)[/tex]:
[tex]\[ \text{pH} = -\log(1 \times 10^{-8}) = 8 \][/tex]
This pH falls within the desired range of 8-9.
Therefore, the correct hydrogen ion concentration for a solution that turns blue with a cabbage indicator is:
[tex]\[ \boxed{1 \times 10^{-8} \text{ M}} \][/tex]
