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

A metal rod has a linear expansivity of 1.5×10−5 K−11.5 \times 10^{-5}\text{ K}^{-1}. What is the volume expansivity of a sphere made from the same metal?

  1. A
    1.5×10−5 K−11.5 \times 10^{-5}\text{ K}^{-1}
  2. B
    3.0×10−5 K−13.0 \times 10^{-5}\text{ K}^{-1}
  3. 4.5×10−5 K−14.5 \times 10^{-5}\text{ K}^{-1}Answer
  4. D
    6.0×10−5 K−16.0 \times 10^{-5}\text{ K}^{-1}

Answer

The volume expansivity of the sphere is 4.5×10−5 K−14.5 \times 10^{-5}\text{ K}^{-1}.
The volume expansivity γ\gamma of a uniform solid object is related to its linear expansivity α\alpha by γ=3α\gamma = 3\alpha. Multiplying 1.5×10−5 K−11.5 \times 10^{-5}\text{ K}^{-1} by 33 gives 4.5×10−5 K−14.5 \times 10^{-5}\text{ K}^{-1}.

Step-by-Step Solution

1
Identify the mathematical relationship between linear expansivity (α\alpha) and volume (cubical) expansivity (γ\gamma).
γ=3α\gamma = 3\alpha
For an isotropic solid, volume expansion occurs equally in three dimensions, making the volume coefficient three times the linear coefficient.
2
Substitute the given value of linear expansivity into the formula and calculate.
γ=3×(1.5×10−5 K−1)=4.5×10−5 K−1\gamma = 3 \times (1.5 \times 10^{-5}\text{ K}^{-1}) = 4.5 \times 10^{-5}\text{ K}^{-1}
Obtain the numeric value of volume expansivity.

Key Concept

Relationship between linear and volume expansivity of solids
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Topics

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

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