You can obtain a rough estimate of the size of a molecule by the following simple experiment. Let a droplet of oil spread out on a smooth water surface. The resulting oil slick will be approximately one molecule thick. Given an oil droplet of mass 10.00 multiplied by 10-7 kg and density 1174 kg/m3 that spreads out into a circle of radius 41.8 cm on the water surface, what is the diameter of an oil molecule?
Density=mass/volume
Volume=mass/density
(10.00*10^-7 kg)/(1174 kg/m^3)= 8.518*10^-10 m^3
Volume=(height)(cross-sectional area)
Volume=h*pi*(.418 m)^2
Volume=h(.549 m^2)
h(.549 m^2)=8.518*10^-10 m^3
h=(8.518*10^-10 m^3)/(.549 m^2)
h~1.55*10^-9
To find the diameter of an oil molecule, we can use the formula for the volume of a cylinder:
V = πr^2h
Where:
V = Volume of the oil droplet (approximately equal to the volume of the oil slick)
r = Radius of the oil slick (41.8 cm)
h = Height of the droplet (thickness of one molecule)
Since the oil slick is approximately one molecule thick, we can assume that h is equal to the diameter of an oil molecule (d).
Also, the mass of the oil droplet is given as 10.00 × 10^-7 kg, and the density of the oil is given as 1174 kg/m^3. The volume (V) of the droplet can be calculated as:
V = mass / density
Substituting the given values, we get:
V = (10.00 × 10^-7 kg) / (1174 kg/m^3)
V ≈ 8.529 × 10^-11 m^3
Now we can calculate the diameter of an oil molecule (d):
V = πr^2h
8.529 × 10^-11 m^3 = π(41.8 cm)^2 × d
Converting radius to meters:
41.8 cm = 0.418 m
Simplifying the equation:
8.529 × 10^-11 m^3 = π(0.418 m)^2 × d
Solving for d:
d ≈ (8.529 × 10^-11 m^3) / (π(0.418 m)^2)
Calculating:
d ≈ 0.000645 nm
Therefore, the diameter of an oil molecule is approximately 0.000645 nanometers.
To determine the diameter of an oil molecule using the given information, we can follow these steps:
1. Calculate the area of the oil slick:
- The area of a circle can be calculated using the formula A = πr^2, where A is the area and r is the radius.
- In this case, the radius is given as 41.8 cm, so we need to convert it to meters by dividing by 100 (since 1 meter = 100 cm).
- Calculate the area using the formula: A = π(41.8/100)^2.
2. Estimate the number of oil molecules in the oil slick:
- Since the slick is approximately one molecule thick, the number of molecules can be approximated by dividing the mass of the droplet by the mass of a single molecule.
- The density of the oil is given as 1174 kg/m^3, which gives us the mass/volume ratio.
- So, the volume of the droplet can be calculated as V = M / ρ, where M is the mass of the droplet and ρ is the density of the oil.
- Calculate the volume of the droplet using the given values.
- Based on the assumption that the droplet has formed a slick that is one molecule thick, the volume of the droplet can be considered to be equal to the volume of the slick.
- Divide the volume of the slick by the volume of a single oil molecule to estimate the number of oil molecules in the slick.
3. Calculate the diameter of the oil molecule:
- The diameter of a sphere can be calculated using the formula D = 2√(V/π), where D represents the diameter and V is the volume of the molecule.
- Divide the volume of a single oil molecule by the number of oil molecules calculated in the previous step.
- Calculate the square root of this value.
- Multiply the square root by 2 to obtain the diameter of the oil molecule.
By following these steps and plugging in the given values, you can calculate the diameter of an oil molecule.