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### Question 1: Calculation of [tex]\( \text{OH}^- \)[/tex] ion concentration
A solution with a [tex]\( pH = 13 \)[/tex] has approximately how many moles of [tex]\( \text{OH}^- \)[/tex] ions per liter?
1. Definition of pH and pOH: pH and pOH are related by the equation:
[tex]\[ \text{pH} + \text{pOH} = 14 \][/tex]
2. Calculate pOH:
[tex]\[ \text{pOH} = 14 - \text{pH} \][/tex]
Substituting [tex]\( \text{pH} = 13 \)[/tex] into the equation:
[tex]\[ \text{pOH} = 14 - 13 = 1 \][/tex]
3. Determine the concentration of [tex]\( \text{OH}^- \)[/tex] ions: The concentration of [tex]\( \text{OH}^- \)[/tex] ions is given by:
[tex]\[ [\text{OH}^-] = 10^{-\text{pOH}} \][/tex]
Since [tex]\( \text{pOH} = 1 \)[/tex]:
[tex]\[ [\text{OH}^-] = 10^{-1} = 0.1 \ \text{moles per liter} \][/tex]
So, the concentration of [tex]\( \text{OH}^- \)[/tex] ions is approximately [tex]\( 0.1 \)[/tex] moles per liter.
### Question 2: Calculation of [tex]\( \text{H}^+ \)[/tex] ion concentration
How many moles of [tex]\( \text{H}^+ \)[/tex] would this same solution have per liter?
1. Definition of pH: The concentration of [tex]\( \text{H}^+ \)[/tex] ions is given by:
[tex]\[ [\text{H}^+] = 10^{-\text{pH}} \][/tex]
2. Using [tex]\( \text{pH} = 13 \)[/tex]:
[tex]\[ [\text{H}^+] = 10^{-13} = 1 \times 10^{-13} \ \text{moles per liter} \][/tex]
So, the concentration of [tex]\( \text{H}^+ \)[/tex] ions is [tex]\( 1 \times 10^{-13} \)[/tex] moles per liter.
### Question 3: Calculation of pH from [tex]\( \text{H}^+ \)[/tex] concentration
A different solution with a [tex]\( \text{H}^+ \)[/tex] concentration of [tex]\( 1.0 \times 10^{-4} \)[/tex] would have a pH of:
1. Definition of pH:
[tex]\[ \text{pH} = -\log_{10} [\text{H}^+] \][/tex]
2. Using [tex]\( [\text{H}^+] = 1.0 \times 10^{-4} \)[/tex]:
[tex]\[ \text{pH} = -\log_{10} (1.0 \times 10^{-4}) \][/tex]
3. Simplifying, the [tex]\( \log_{10} \)[/tex] of [tex]\( 1.0 \times 10^{-4} \)[/tex] is [tex]\(-4\)[/tex]:
[tex]\[ \text{pH} = -(-4) = 4 \][/tex]
So, the pH of the solution with an [tex]\( \text{H}^+ \)[/tex] concentration of [tex]\( 1.0 \times 10^{-4} \)[/tex] is 4.
### Final Answers
1. [tex]\( \text{OH}^- \)[/tex] concentration: [tex]\( 0.1 \)[/tex] moles per liter
2. [tex]\( \text{H}^+ \)[/tex] concentration: [tex]\( 1 \times 10^{-13} \)[/tex] moles per liter
3. pH of the different solution: [tex]\( 4 \)[/tex]
### Question 1: Calculation of [tex]\( \text{OH}^- \)[/tex] ion concentration
A solution with a [tex]\( pH = 13 \)[/tex] has approximately how many moles of [tex]\( \text{OH}^- \)[/tex] ions per liter?
1. Definition of pH and pOH: pH and pOH are related by the equation:
[tex]\[ \text{pH} + \text{pOH} = 14 \][/tex]
2. Calculate pOH:
[tex]\[ \text{pOH} = 14 - \text{pH} \][/tex]
Substituting [tex]\( \text{pH} = 13 \)[/tex] into the equation:
[tex]\[ \text{pOH} = 14 - 13 = 1 \][/tex]
3. Determine the concentration of [tex]\( \text{OH}^- \)[/tex] ions: The concentration of [tex]\( \text{OH}^- \)[/tex] ions is given by:
[tex]\[ [\text{OH}^-] = 10^{-\text{pOH}} \][/tex]
Since [tex]\( \text{pOH} = 1 \)[/tex]:
[tex]\[ [\text{OH}^-] = 10^{-1} = 0.1 \ \text{moles per liter} \][/tex]
So, the concentration of [tex]\( \text{OH}^- \)[/tex] ions is approximately [tex]\( 0.1 \)[/tex] moles per liter.
### Question 2: Calculation of [tex]\( \text{H}^+ \)[/tex] ion concentration
How many moles of [tex]\( \text{H}^+ \)[/tex] would this same solution have per liter?
1. Definition of pH: The concentration of [tex]\( \text{H}^+ \)[/tex] ions is given by:
[tex]\[ [\text{H}^+] = 10^{-\text{pH}} \][/tex]
2. Using [tex]\( \text{pH} = 13 \)[/tex]:
[tex]\[ [\text{H}^+] = 10^{-13} = 1 \times 10^{-13} \ \text{moles per liter} \][/tex]
So, the concentration of [tex]\( \text{H}^+ \)[/tex] ions is [tex]\( 1 \times 10^{-13} \)[/tex] moles per liter.
### Question 3: Calculation of pH from [tex]\( \text{H}^+ \)[/tex] concentration
A different solution with a [tex]\( \text{H}^+ \)[/tex] concentration of [tex]\( 1.0 \times 10^{-4} \)[/tex] would have a pH of:
1. Definition of pH:
[tex]\[ \text{pH} = -\log_{10} [\text{H}^+] \][/tex]
2. Using [tex]\( [\text{H}^+] = 1.0 \times 10^{-4} \)[/tex]:
[tex]\[ \text{pH} = -\log_{10} (1.0 \times 10^{-4}) \][/tex]
3. Simplifying, the [tex]\( \log_{10} \)[/tex] of [tex]\( 1.0 \times 10^{-4} \)[/tex] is [tex]\(-4\)[/tex]:
[tex]\[ \text{pH} = -(-4) = 4 \][/tex]
So, the pH of the solution with an [tex]\( \text{H}^+ \)[/tex] concentration of [tex]\( 1.0 \times 10^{-4} \)[/tex] is 4.
### Final Answers
1. [tex]\( \text{OH}^- \)[/tex] concentration: [tex]\( 0.1 \)[/tex] moles per liter
2. [tex]\( \text{H}^+ \)[/tex] concentration: [tex]\( 1 \times 10^{-13} \)[/tex] moles per liter
3. pH of the different solution: [tex]\( 4 \)[/tex]
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