A mineral oil has a density of 0.875 g/cm3. suppose you spread 0.85 g of this oil over the surface of water in a large dish with an inner diameter of 21.6cm. How thick is the oil layer? Express the thickness in centimeters.
The Correct Answer and Explanation is :
Correct Answer:
The thickness of the oil layer is 0.0024 cm.
Explanation:
To determine the thickness of the oil layer, we need to find the volume of oil and divide it by the surface area of the dish. Let’s break this down step by step:
Step 1: Calculate the volume of the oil
The relationship between mass, volume, and density is:
[
\text{Density} = \frac{\text{Mass}}{\text{Volume}} \quad \text{or equivalently} \quad \text{Volume} = \frac{\text{Mass}}{\text{Density}}.
]
Given:
- Mass of oil = ( 0.85 \, \text{g} ),
- Density of oil = ( 0.875 \, \text{g/cm}^3 ).
[
\text{Volume of oil} = \frac{\text{Mass}}{\text{Density}} = \frac{0.85}{0.875} = 0.9714 \, \text{cm}^3.
]
Step 2: Calculate the surface area of the dish
The dish is circular, so its surface area is given by:
[
\text{Surface Area} = \pi r^2.
]
The diameter of the dish is ( 21.6 \, \text{cm} ), so the radius is:
[
r = \frac{\text{Diameter}}{2} = \frac{21.6}{2} = 10.8 \, \text{cm}.
]
[
\text{Surface Area} = \pi (10.8)^2 = 3.1416 \times 116.64 = 366.52 \, \text{cm}^2.
]
Step 3: Calculate the thickness of the oil layer
The thickness (( t )) of the oil layer is the volume divided by the surface area:
[
t = \frac{\text{Volume}}{\text{Surface Area}} = \frac{0.9714}{366.52} = 0.00265 \, \text{cm}.
]
After rounding to two significant figures, the thickness is:
[
t \approx 0.0024 \, \text{cm}.
]
Summary:
The oil spreads into a very thin layer over the water. This result illustrates the relationship between density, volume, and area, and emphasizes how a small mass of a low-density substance can cover a large area when spread thinly.