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 49.73 g, is pressed into a leaf of 2.700 𝜇m thickness, what is the area of the leaf? (Put m^2 in your answer)
(b) If, instead, the gold is drawn out into a cylindrical fiber of radius 2.400 𝜇m, what is the length of the fiber? (Put km in your answer)
To answer these questions, we'll need to use the formula for density:
Density = Mass / Volume
Since we know the density of gold and its mass, we can rearrange the formula to find the volume of the gold sample.
Volume = Mass / Density
Once we know the volume, we can answer parts (a) and (b) of the question by calculating the area and length respectively.
(a) To find the area of the leaf:
Area = Volume / Thickness
(b) To find the length of the fiber:
Length = Volume / (π * Radius^2)
Let's calculate the answers step by step.
(a) Area of the leaf:
Given:
Density of gold = 19.32 g/cm^3
Mass of the gold sample = 49.73 g
Thickness of the leaf = 2.700 μm (convert to cm by dividing by 10^4)
Step 1: Calculate the volume of the gold sample:
Volume = Mass / Density
Volume = 49.73 g / 19.32 g/cm^3
Volume = 2.57 cm^3
Step 2: Calculate the area of the leaf:
Area = Volume / Thickness
Area = 2.57 cm^3 / (2.700 μm / 10^4 cm/μm)
Area = 2.57 cm^3 / (2.700 x 10^-4 cm)
Area = 9.52 x 10^6 cm^2
Area = 9.52 m^2 (since 1 m^2 = 10^4 cm^2)
Therefore, the area of the leaf is 9.52 m^2.
(b) Length of the fiber:
Given:
Density of gold = 19.32 g/cm^3
Mass of the gold sample = 49.73 g
Radius of the fiber = 2.400 μm (convert to cm by dividing by 10^4)
Step 1: Calculate the volume of the gold sample (same as in part (a)):
Volume = Mass / Density
Volume = 49.73 g / 19.32 g/cm^3
Volume = 2.57 cm^3
Step 2: Calculate the length of the fiber:
Length = Volume / (π * Radius^2)
Length = 2.57 cm^3 / (π * (2.400 μm / 10^4 cm/μm)^2)
Length = 2.57 cm^3 / (π * (2.400 x 10^-4 cm)^2)
Length = 2.57 cm^3 / (π * 5.76 x 10^-8 cm^2)
Length = 1.42 x 10^14 cm
Length = 1.42 x 10^11 m (since 1 km = 10^5 cm)
Therefore, the length of the fiber is 1.42 x 10^11 km.
To solve these problems, we need to use the formula for density:
Density = mass / volume
For part (a), we are given the mass of the gold and the thickness of the leaf. We need to find the area of the leaf.
We can start by rearranging the formula for density:
Mass = Density * Volume
Since the density is given in g/cm^3, we need to convert the thickness from micrometers (𝜇m) to centimeters (cm). There are 10,000 micrometers in one centimeter.
Now, let's calculate the area:
1. Convert the thickness to centimeters:
Thickness = 2.700 𝜇m * (1 cm / 10000 𝜇m) = 0.00027 cm
2. Calculate the volume:
Volume = thickness * area
We want to find the area, so rearrange the formula:
Area = Volume / thickness
Substituting in the values:
Area = (Mass / Density) / Thickness
= (49.73 g) / (19.32 g/cm^3) / (0.00027 cm)
= 9.596 cm^2
Therefore, the area of the leaf is 9.596 m^2.
Now, let's move on to part (b).
For part (b), we are given the radius of the cylindrical fiber. We need to find the length of the fiber.
The volume of a cylinder is given by the formula:
Volume = π * radius^2 * length
We can rearrange the formula to solve for the length:
Length = Volume / (π * radius^2)
To calculate the volume, we can multiply the cross-sectional area (which we already calculated in part (a)) by the length of the fiber:
Volume = Area * length
Rearranging the formula:
Length = Volume / Area
Substituting in the values:
Length = (Mass / Density) / Area
= (49.73 g) / (19.32 g/cm^3) / (9.596 cm^2)
= 0.265 km
Therefore, the length of the fiber is 0.265 km.
since volume is mass/density, the volume of gold is 2.574 cm^3
since volume = area * thickness,
2.4*10^-4 a cm^3 = 2.574 cm^3
a = 10725 cm^2
volume is πr^2L, so
π(2.4*10^-4 cm)^2 * L = 2.574 cm^3
L = 1.422*10^7 cm = 142.2 km