Gold, which has a density of 19.32 g/cm3, is the most ductile metal and can be pressed into a thin leaf or drawn out into a long fiber. (a) If a sample of gold with a mass of 8.464 g, is pressed into a leaf of 5.036 ìm thickness, what is the area (in m2) of the leaf? (b) If, instead, the gold is drawn out into a cylindrical fiber of radius 2.000 ìm, what is the length (in m) of the fiber?
8.464/density= area*thickness
solve for area, watch units. I am uncertain of you leaf thickness.
To solve these problems, we can use the formulas for the volume of a leaf and the volume of a cylinder:
(a) Volume of a leaf = Thickness × Area
(b) Volume of a cylinder = π × Radius^2 × Length
First, let's convert the given measurements into the correct units:
Thickness of the leaf:
5.036 μm (micrometers) = 5.036 × 10^(-6) m
Radius of the fiber:
2.000 μm (micrometers) = 2.000 × 10^(-6) m
Now we can proceed to solve the problems:
(a) To find the area of the leaf (in m^2), we can rearrange the formula for the volume of a leaf:
Volume of a leaf = Thickness × Area
Area = Volume of a leaf / Thickness
The volume of the leaf is the mass divided by the density:
Volume of a leaf = mass / density
Substituting the given values:
Volume of a leaf = 8.464 g / 19.32 g/cm^3
Since the density is given in grams per cubic centimeter (g/cm^3), we need to convert the mass to the same unit before performing the division:
8.464 g = 8.464 cm^3 (since 1 cm^3 = 1 g)
Now we can calculate the volume of the leaf:
Volume of a leaf = 8.464 cm^3 / 19.32 g/cm^3
Volume of a leaf ≈ 0.4374 cm^3
To convert from cubic centimeters to cubic meters, we divide by 1,000,000:
Volume of a leaf ≈ 0.4374 cm^3 × (1 m^3 / 1,000,000 cm^3)
Volume of a leaf ≈ 4.374 × 10^(-7) m^3
Finally, we can calculate the area of the leaf:
Area = Volume of a leaf / Thickness
Area = 4.374 × 10^(-7) m^3 / 5.036 × 10^(-6) m
Area ≈ 0.0868 m^2
Therefore, the area of the leaf is approximately 0.0868 square meters.
(b) To find the length of the fiber (in meters), we can rearrange the formula for the volume of a cylinder:
Volume of a cylinder = π × Radius^2 × Length
Solving for length:
Length = Volume of a cylinder / (π × Radius^2)
The volume of the cylinder is the mass divided by the density:
Volume of a cylinder = mass / density
Substituting the given values:
Volume of a cylinder = 8.464 g / 19.32 g/cm^3
Converting the mass to cubic centimeters (since 1 g = 1 cm^3):
Volume of a cylinder = 8.464 cm^3 / 19.32 g/cm^3
Volume of a cylinder ≈ 0.4374 cm^3
To convert from cubic centimeters to cubic meters, we divide by 1,000,000:
Volume of a cylinder ≈ 0.4374 cm^3 × (1 m^3 / 1,000,000 cm^3)
Volume of a cylinder ≈ 4.374 × 10^(-7) m^3
Now we can calculate the length of the fiber:
Length = Volume of a cylinder / (π × Radius^2)
Length ≈ (4.374 × 10^(-7) m^3) / (π × (2.000 × 10^(-6) m)^2)
Length ≈ 0.0695 m
Therefore, the length of the fiber is approximately 0.0695 meters.