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

A rectangular metal sheet has an initial area of 2.0 m22.0\text{ m}^2 at 20∘C20^\circ\text{C}. If the linear expansivity of the metal is 2.5×10−5 K−12.5 \times 10^{-5}\text{ K}^{-1}, what is the final temperature required for its area to increase by 0.005 m20.005\text{ m}^2?

  1. 70∘C70^\circ\text{C}Answer
  2. B
    120∘C120^\circ\text{C}
  3. C
    50∘C50^\circ\text{C}
  4. D
    53.3∘C53.3^\circ\text{C}

Answer

The final temperature required is 70∘C70^\circ\text{C}.
Area expansion is governed by ΔA=A0βΔT\Delta A = A_0 \beta \Delta T, where the area expansivity β=2α\beta = 2\alpha. Substituting A0=2.0 m2A_0 = 2.0\text{ m}^2, ΔA=0.005 m2\Delta A = 0.005\text{ m}^2, and β=5.0×10−5 K−1\beta = 5.0 \times 10^{-5}\text{ K}^{-1} gives a temperature rise ΔT=50∘C\Delta T = 50^\circ\text{C}. Adding the initial temperature of 20∘C20^\circ\text{C} yields a final temperature of 70∘C70^\circ\text{C}.

Step-by-Step Solution

1
Determine the area expansivity (superficial expansivity) β\beta from the linear expansivity α\alpha.
β=2α=2×2.5×10−5 K−1=5.0×10−5 K−1\beta = 2\alpha = 2 \times 2.5 \times 10^{-5}\text{ K}^{-1} = 5.0 \times 10^{-5}\text{ K}^{-1}
Area expansion depends on area expansivity, which is twice the linear expansivity.
2
Calculate the temperature change ΔT\Delta T using the formula ΔA=A0βΔT\Delta A = A_0 \beta \Delta T.
\(\Delta T = \frac{\Delta A}{A_0 \beta} = \frac{0.005\text{ m}^2}{2.0\text{ m}^2 \times 5.0 \times 10^{-5}\text{ K}^{-1}} = \frac{5 \times 10^{-3}}{1.0 \times 10^{-4}} = 50\text{ K}\)
Rearranging the expansion formula isolates the temperature change variable.
3
Calculate the final temperature T2T_2 by adding ΔT\Delta T to the initial temperature T1T_1.
T2=T1+ΔT=20∘C+50∘C=70∘CT_2 = T_1 + \Delta T = 20^\circ\text{C} + 50^\circ\text{C} = 70^\circ\text{C}
The final temperature is the sum of the initial temperature and the rise in temperature.

Key Concept

Thermal Expansion of Solids (Area Expansivity)
Estimated Time:1m 30s
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Topics

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

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