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

A thin flat metal plate has an initial surface area of 0.5 m20.5\text{ m}^2 at 10∘C10^\circ\text{C}. When the plate is heated to a final temperature of 110∘C110^\circ\text{C}, its surface area increases by 1.8×10−3 m21.8 \times 10^{-3}\text{ m}^2. What is the coefficient of cubical expansivity of the metal?

  1. A
    1.8×10−5 K−11.8 \times 10^{-5}\text{ K}^{-1}
  2. B
    3.6×10−5 K−13.6 \times 10^{-5}\text{ K}^{-1}
  3. 5.4×10−5 K−15.4 \times 10^{-5}\text{ K}^{-1}Answer
  4. D
    1.08×10−4 K−11.08 \times 10^{-4}\text{ K}^{-1}

Answer

5.4×10−5 K−15.4 \times 10^{-5}\text{ K}^{-1}
The correct answer is derived by first finding the area expansivity β=ΔAA0ΔT=3.6×10−5 K−1\beta = \frac{\Delta A}{A_0 \Delta T} = 3.6 \times 10^{-5}\text{ K}^{-1}. Since β=2α\beta = 2\alpha, the linear expansivity α=1.8×10−5 K−1\alpha = 1.8 \times 10^{-5}\text{ K}^{-1}. The cubical expansivity is γ=3α=5.4×10−5 K−1\gamma = 3\alpha = 5.4 \times 10^{-5}\text{ K}^{-1}.

Step-by-Step Solution

1
Calculate the temperature change ΔT\Delta T
ΔT=110∘C−10∘C=100 K\Delta T = 110^\circ\text{C} - 10^\circ\text{C} = 100\text{ K}
Expansion calculations depend on the temperature increase.
2
Determine the coefficient of area expansivity β\beta
β=ΔAA0ΔT=1.8×10−30.5×100=3.6×10−5 K−1\beta = \frac{\Delta A}{A_0 \Delta T} = \frac{1.8 \times 10^{-3}}{0.5 \times 100} = 3.6 \times 10^{-5}\text{ K}^{-1}
The area expansion formula is ΔA=A0βΔT\Delta A = A_0 \beta \Delta T.
3
Calculate the coefficient of linear expansivity α\alpha
α=β2=3.6×10−52=1.8×10−5 K−1\alpha = \frac{\beta}{2} = \frac{3.6 \times 10^{-5}}{2} = 1.8 \times 10^{-5}\text{ K}^{-1}
Area expansivity is twice the linear expansivity (β=2α\beta = 2\alpha).
4
Calculate the coefficient of cubical expansivity γ\gamma
γ=3α=3×(1.8×10−5)=5.4×10−5 K−1\gamma = 3\alpha = 3 \times (1.8 \times 10^{-5}) = 5.4 \times 10^{-5}\text{ K}^{-1}
Cubical expansivity is three times the linear expansivity (γ=3α\gamma = 3\alpha).

Key Concept

Relationship between linear (α\alpha), area (β\beta), and volume (γ\gamma) expansivities: β=2α\beta = 2\alpha and γ=3α\gamma = 3\alpha.
Estimated Time:1m 30s
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Topics

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

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