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Difficulty: MediumThermal Expansion of Solids (Linear, Area, and Volume Expansivity)

A solid brass cube with an edge length of 10 cm10\text{ cm} at 15∘C15^\circ\text{C} is heated to a temperature of 115∘C115^\circ\text{C}. If the linear expansivity of brass is 2.0×10−5 K−12.0 \times 10^{-5}\text{ K}^{-1}, what is the increase in the volume of the cube in cm3\text{cm}^3?

Answer: 6 cm^3

Answer

The increase in the volume of the brass cube is 6.0 cm36.0\text{ cm}^3.
The volume expansion of a solid is given by ΔV=V1γΔT\Delta V = V_1 \gamma \Delta T. The initial volume of the cube is V1=(10 cm)3=1000 cm3V_1 = (10\text{ cm})^3 = 1000\text{ cm}^3 and the temperature change is ΔT=115∘C−15∘C=100 K\Delta T = 115^\circ\text{C} - 15^\circ\text{C} = 100\text{ K}. Because the expansion occurs in three dimensions, the volume expansivity is γ=3α=3×2.0×10−5=6.0×10−5 K−1\gamma = 3\alpha = 3 \times 2.0 \times 10^{-5} = 6.0 \times 10^{-5}\text{ K}^{-1}. Substituting these values yields ΔV=1000×6.0×10−5×100=6.0 cm3\Delta V = 1000 \times 6.0 \times 10^{-5} \times 100 = 6.0\text{ cm}^3.

Step-by-Step Solution

1
Calculate the initial volume of the cube
V1=(10 cm)3=1000 cm3V_1 = (10\text{ cm})^3 = 1000\text{ cm}^3
The volume of a cube is calculated using V=L3V = L^3, where LL is the edge length.
2
Determine the change in temperature
ΔT=115∘C−15∘C=100 K\Delta T = 115^\circ\text{C} - 15^\circ\text{C} = 100\text{ K}
The change in temperature is the difference between the final and initial temperatures.
3
Calculate the volume expansivity (cubic expansivity)
γ=3α=3×(2.0×10−5 K−1)=6.0×10−5 K−1\gamma = 3\alpha = 3 \times (2.0 \times 10^{-5}\text{ K}^{-1}) = 6.0 \times 10^{-5}\text{ K}^{-1}
Volume expansivity γ\gamma is three times the linear expansivity α\alpha for isotropic solids.
4
Compute the increase in volume
ΔV=V1γΔT=1000×(6.0×10−5)×100=6.0 cm3\Delta V = V_1 \gamma \Delta T = 1000 \times (6.0 \times 10^{-5}) \times 100 = 6.0\text{ cm}^3
The formula for volume expansion is ΔV=V1γΔT\Delta V = V_1 \gamma \Delta T.

Key Concept

Thermal expansion of solids: volume expansion (ΔV=V1γΔT\Delta V = V_1 \gamma \Delta T) and the relationship between linear and volume expansivity (γ=3α\gamma = 3\alpha).
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Topics

Thermal Expansion of Solids (Linear, Area, and Volume Expansivity)

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