A drop of olive oil of radius 0.25 mm spreads into a circular film of diameter 20cm on the water surface. Estimate the size of the oil molecule.

Explanation

To estimate the size of an olive oil molecule based on the information provided, we will follow these steps:

### Step 1: Calculate the volume of the olive oil drop
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

Given the radius of the olive oil drop is \( 0.25 \, \text{mm} \), we need to convert this into meters:
\[ r = 0.25 \, \text{mm} = 0.25 \times 10^{-3} \, \text{m} = 2.5 \times 10^{-4} \, \text{m} \]

Now, substituting the radius into the volume formula:
\[ V = \frac{4}{3} \pi (2.5 \times 10^{-4})^3 \]

Calculating this:
\[ V = \frac{4}{3} \pi (1.5625 \times 10^{-11}) \approx \frac{4}{3} \times 3.14159 \times 1.5625 \times 10^{-11} \approx 6.576 \times 10^{-11} \, \text{m}^3 \]

### Step 2: Calculate the area of the circular film
The area \( A \) of a circle is given by the formula:
\[ A = \pi R^2 \]
where \( R \) is the radius of the circle.

Given the diameter of the circular film is \( 20 \, \text{cm} \), we convert this to meters:
\[ R = \frac{20 \, \text{cm}}{2} = 10 \, \text{cm} = 0.1 \, \text{m} \]

Now, substituting the radius into the area formula:
\[ A = \pi (0.1)^2 = \pi (0.01) \approx 3.14159 \times 0.01 \approx 0.0314 \, \text{m}^2 \]

### Step 3: Calculate the thickness of the oil layer
The volume of the oil drop will be equal to the volume of the oil layer spread over the area:
\[ V = A \times t \]
where \( t \) is the thickness of the oil layer.

Rearranging the formula to find \( t \):
\[ t = \frac{V}{A} \]

Substituting the values we calculated:
\[ t = \frac{6.576 \times 10^{-11}}{0.0314} \approx 2.095 \times 10^{-9} \, \text{m} \]

### Step 4: Estimate the size of the oil molecule
To estimate the size of an oil molecule, we can assume that the thickness of the oil layer corresponds to a single layer of molecules. If we assume that the oil molecules are roughly spherical, we can estimate the size \( d \) of an oil molecule as follows:
\[ d \approx t \]

Thus, the estimated size of an olive oil molecule is approximately:
\[ d \approx 2.095 \, \text{nm} \, (\text{nanometers}) \]

### Final Answer
The estimated size of an olive oil molecule is approximately \( 2.1 \, \text{nm} \).

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