To solve the problem step by step, we will follow the instructions provided in the video transcript and break it down into clear steps.
### Step 1: Identify Given Data
- Molar mass of palmitic acid = 256 g/mol
- Mass of palmitic acid in solution = 5.12 g/L
- Area to be covered = 500 cm²
- Area covered by one molecule = 0.2 nm²
### Step 2: Convert Area Covered by One Molecule
Convert the area covered by one molecule from nm² to cm²:
\[
0.2 \, \text{nm}^2 = 0.2 \times 10^{-14} \, \text{cm}^2
\]
### Step 3: Calculate Number of Molecules Required
To find the number of molecules needed to cover the area of 500 cm², we use the area covered by one molecule:
\[
\text{Number of molecules} = \frac{\text{Total area}}{\text{Area per molecule}} = \frac{500 \, \text{cm}^2}{0.2 \times 10^{-14} \, \text{cm}^2}
\]
Calculating this gives:
\[
\text{Number of molecules} = \frac{500}{0.2 \times 10^{-14}} = 2.5 \times 10^{16}
\]
### Step 4: Relate Number of Molecules to Mass
Using Avogadro's number (approximately \(6.023 \times 10^{23}\) molecules/mol), we can find the number of moles:
\[
\text{Number of moles} = \frac{\text{Number of molecules}}{6.023 \times 10^{23}} = \frac{2.5 \times 10^{16}}{6.023 \times 10^{23}} \approx 4.15 \times 10^{-8} \, \text{mol}
\]
### Step 5: Calculate Mass of Palmitic Acid Required
Using the molar mass of palmitic acid:
\[
\text{Mass} = \text{Number of moles} \times \text{Molar mass} = 4.15 \times 10^{-8} \, \text{mol} \times 256 \, \text{g/mol} \approx 1.06 \times 10^{-5} \, \text{g}
\]
### Step 6: Relate Mass to Volume of Solution
We know that the concentration of the solution is 5.12 g/L. To find the volume of the solution required to obtain the calculated mass:
\[
\text{Volume} = \frac{\text{Mass}}{\text{Concentration}} = \frac{1.06 \times 10^{-5} \, \text{g}}{5.12 \, \text{g/L}} \approx 2.07 \times 10^{-6} \, \text{L}
\]
### Step 7: Convert Volume to cm³
Since \(1 \, \text{L} = 1000 \, \text{cm}^3\):
\[
\text{Volume in cm}^3 = 2.07 \times 10^{-6} \, \text{L} \times 1000 \approx 2.07 \times 10^{-3} \, \text{cm}^3
\]
### Step 8: Calculate X
Now, we need to find \(X\) where \(X = \frac{V}{100}\):
\[
X = \frac{2.07 \times 10^{-3} \, \text{cm}^3}{100} \approx 2.07 \times 10^{-5}
\]
### Final Answer
Thus, the value of \(X\) is approximately \(2.07 \times 10^{-5}\).
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