Sure, let's solve this problem step-by-step.
1. Understand the Saponification Value:
- The saponification value is the amount of potassium hydroxide (KOH), in milligrams, needed to saponify 1 gram of fat or oil.
- For olive oil, this value is given as 190 mg KOH per gram of oil.
2. Determine the Ratio of Molar Masses:
- The molar mass of KOH (potassium hydroxide) is approximately 56 g/mol.
- The molar mass of NaOH (sodium hydroxide) is approximately 40 g/mol.
- The ratio of the molar mass of NaOH to the molar mass of KOH is:
[tex]\[
\text{Ratio} = \frac{\text{Molar mass of NaOH}}{\text{Molar mass of KOH}} = \frac{40 \, \text{g/mol}}{56 \, \text{g/mol}}
\][/tex]
Simplifying this ratio:
[tex]\[
\text{Ratio} = \frac{40}{56} = \frac{5}{7}
\][/tex]
3. Calculate the Saponification Value with NaOH:
- Since the saponification value with KOH is given as 190 mg KOH per gram of oil, we need to convert this to the equivalent value using NaOH.
- Let's denote the saponification value with NaOH as [tex]\(SV_{NaOH}\)[/tex].
To find [tex]\(SV_{NaOH}\)[/tex], we use the ratio we determined above:
[tex]\[
SV_{NaOH} = 190 \, \text{mg KOH/g oil} \times \frac{5}{7}
\][/tex]
Carry out the multiplication:
[tex]\[
SV_{NaOH} = 190 \times \frac{5}{7} = 190 \times 0.7142857
\][/tex]
[tex]\[
SV_{NaOH} \approx 135.71 \, \text{mg NaOH/g oil}
\][/tex]
Thus, the saponification value of olive oil when using NaOH (sodium hydroxide) to make soap is approximately 135.71 mg NaOH per gram of oil.
