In an experiment to estimate the size of a molecule, an oil droplet of mass 0.9 mg and density 918 kg/m3 spreads out into a circle of radius 41.8 cm on water surface. Calculate the diameter of the oil molecule.
To calculate the diameter of the oil molecule, we need to use the formula for the spread of the oil droplet on the water surface.
The formula to calculate the radius of the oil droplet is given by:
r = √(9ηvl/2gρd)
where:
- r is the radius of the oil droplet (41.8 cm in this case)
- η is the viscosity of water (assumed to be 0.001 kg/m/s)
- v is the velocity of the oil droplet (which can be calculated using the mass and density)
- l is the length traveled by the oil droplet
- g is the acceleration due to gravity (9.8 m/s^2)
- ρ is the density of water (1000 kg/m^3)
- d is the diameter of the oil molecule (what we want to calculate)
First, let's calculate the velocity of the oil droplet:
v = m / (πr^2ρ)
where:
- m is the mass of the oil droplet (0.9 mg)
- r is the radius of the oil droplet (41.8 cm)
Converting the mass from mg to kg:
m = 0.9 mg * (10^-6 kg/mg) = 9 * 10^-7 kg
Converting the radius from cm to m:
r = 41.8 cm * (1 m/100 cm) = 0.418 m
Converting the density of water from kg/m^3 to g/cm^3:
ρ = 1000 kg/m^3 * (10^-3 g/kg) / (1 cm/10^-2 m)^3 = 1 g/cm^3
Substituting the values into the formula to calculate the velocity:
v = (9 * 10^-7 kg) / (π * (0.418 m)^2 * 1 g/cm^3) ≈ 5.055 * 10^-4 m/s
Now, let's substitute the known values into the formula for the radius r:
41.8 cm = √(9 * (0.001 kg/m/s) * (5.055 * 10^-4 m/s) * l/(2 * 9.8 m/s^2) * 918 kg/m^3 * d)
Simplifying:
(41.8 cm)^2 = 0.000457 * l * d
Rearranging the equation to solve for the diameter d:
d = (41.8 cm)^2 / (0.000457 * l)
Since the length traveled by the oil droplet is not provided in the question, we are unable to calculate the exact diameter of the oil molecule.