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 8.556 g, is pressed into a leaf of 5.931 μm thickness, what is the area of the leaf?
The Correct Answer and Explanation is :
To find the area of the leaf formed from the gold, we can use the relationship between mass, density, volume, and thickness.
Step 1: Understand the variables involved
- Density of gold (ρ): 19.32 g/cm³
- Mass of gold (m): 8.556 g
- Thickness of the leaf (t): 5.931 μm = 5.931 × 10⁻⁶ m
Step 2: Convert the mass of gold to volume
Density (ρ) is defined as mass (m) per unit volume (V): ρ=mV\rho = \frac{m}{V}
Rearranging this formula, we can solve for the volume of the gold: V=mρV = \frac{m}{\rho}
Now, plug in the known values. First, we need to convert the density from g/cm³ to g/m³ (since the thickness is given in meters). We know that 1 cm = 10⁻² m, so: 1 g/cm3=106 g/m31 \, \text{g/cm}^3 = 10⁶ \, \text{g/m}^3
Thus: ρ=19.32 g/cm3=19.32×106 g/m3\rho = 19.32 \, \text{g/cm}^3 = 19.32 \times 10⁶ \, \text{g/m}^3
Now, calculate the volume: V=8.556 g19.32×106 g/m3V = \frac{8.556 \, \text{g}}{19.32 \times 10⁶ \, \text{g/m}^3}
Step 3: Calculate the area of the leaf
The volume (V) of the gold is related to the area (A) and the thickness (t) of the leaf: V=A×tV = A \times t
Rearrange this to solve for the area (A): A=VtA = \frac{V}{t}
Now we can substitute the values to calculate the area.
Final Answer:
Let’s carry out the calculations: V=8.55619.32×106=4.429×10−7 m3V = \frac{8.556}{19.32 \times 10^6} = 4.429 \times 10^{-7} \, \text{m}^3
Now, calculate the area: A=4.429×10−7 m35.931×10−6 m=0.0747 m2A = \frac{4.429 \times 10^{-7} \, \text{m}^3}{5.931 \times 10^{-6} \, \text{m}} = 0.0747 \, \text{m}^2
Thus, the area of the gold leaf is approximately 0.0747 m².
Explanation:
The process starts by determining the volume of gold using its mass and density. After that, we use the relationship between volume, area, and thickness to find the area of the leaf. Since gold is pressed into a thin leaf, the thickness is crucial, and knowing the density of gold helps calculate the volume, which is ultimately linked to the area of the leaf.