one liter (1000cm3) of oil is spiled onto a smooth lake. if the oil spreads out uniformly until it makes an oil slick justone molecure thick, with adjcent molecules just touching estimate the diameter of the oil slick. Assume the oil molecules have a diameter of 2X10-10 m.
Assuming the oil molecules fill the entire space, then the thickness of the film is the diameter of the molecules: 2*10^-10 m = 2*10^-8 cm
1000cm^3 / 2*10^-8cm = 5*10^10 cm^2
That's the area of the oil slick.
a = pi/4 d^2, so the diameter d of the slick is
d = √(4*5*10^10/pi)
= 2*10^5/pi √5
= 252313 cm
= 2.52 km
wow. Seems kind of large. Better check my math.
To estimate the diameter of the oil slick, we can first calculate the number of oil molecules in one liter of oil. Then we can use this number to determine the size of the oil slick.
Step 1: Calculate the number of oil molecules in one liter of oil.
Given:
Volume of oil = 1 liter = 1000 cm³
Diameter of oil molecule = 2 × 10^(-10) m
First, convert the volume from cm³ to m³:
1 liter = 1000 cm³ = 1000 × 10^(-6) m³
Now we need to find the number of oil molecules using Avogadro's number (6.022 × 10^23 molecules per mole):
Number of oil molecules = (Volume of oil / Volume of one oil molecule) × Avogadro's number
To find the volume of one oil molecule, we can consider it as a sphere:
Volume of one oil molecule = (4/3) × π × (diameter/2)³
Plugging in the values:
Volume of one oil molecule = (4/3) × π × (2 × 10^(-10) m / 2)³
Volume of one oil molecule ≈ (4/3) × 3.1416 × (10^(-10) m)³
Now we can calculate the number of oil molecules:
Number of oil molecules = (1000 × 10^(-6) m³) / [(4/3) × 3.1416 × (10^(-10) m)³]
Number of oil molecules ≈ 8.47 × 10^9 molecules
Step 2: Estimate the diameter of the oil slick.
We can assume that the oil slick forms a single layer with adjacent molecules just touching. In this case, the area of the oil slick represents the cross-sectional area of all the oil molecules lying flat.
Area of oil slick = Number of oil molecules × Cross-sectional area of one oil molecule
The cross-sectional area of one oil molecule is π × (diameter/2)²:
Area of oil slick = (8.47 × 10^9 molecules) × [π × (2 × 10^(-10) m / 2)²]
Area of oil slick ≈ (8.47 × 10^9 molecules) × π × (10^(-10) m)²
Assuming the oil slick is circular, we can use the formula for the area of a circle:
Area of oil slick = π × (radius)²
Now we can calculate the radius of the oil slick:
π × (radius)² ≈ (8.47 × 10^9 molecules) × π × (10^(-10) m)²
Simplifying the equation:
(radius)² ≈ (8.47 × 10^9 molecules) × (10^(-10) m)²
(radius)² ≈ (8.47 × 10^(-1)) m²
radius ≈ √(8.47 × 10^(-1)) m
Finally, we can estimate the diameter of the oil slick:
Diameter of oil slick ≈ 2 × radius
Calculating:
Diameter of oil slick ≈ 2 × √(8.47 × 10^(-1)) m
After evaluating this expression, the diameter of the oil slick will be obtained.
Note: Since we made assumptions and approximations throughout the calculation, the estimated diameter may not be perfectly accurate, but it should provide a reasonable estimate based on the given information.