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Calculate the temperature change of a 1 cm³ object if its volume expands by 0.0018 cm³ given a linear expansivity of 2.0 x 10⁻⁵ K⁻¹.

Physics

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Step-by-step solution

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To find the temperature change Θ\Theta, we use the formula for volume expansion:

ΔV=γV0Θ\Delta V = \gamma V_0 \Theta

where:

  • ΔV\Delta V is the change in volume
  • γ\gamma is the coefficient of volume expansivity
  • V0V_0 is the initial volume
  • Θ\Theta is the change in temperature

We are given:

  • Initial volume (V0V_0) = 1 cm31 \, \text{cm}^3
  • Change in volume (ΔV\Delta V) = 0.0018 cm30.0018 \, \text{cm}^3
  • Linear expansivity (α\alpha) = 2.0×10−5 K−12.0 \times 10^{-5} \, \text{K}^{-1}
  1. Calculate the coefficient of volume expansivity (γ\gamma).

    For an isotropic material, the volume expansivity is approximately three times the linear expansivity:

    γ=3α\gamma = 3\alpha γ=3×(2.0×10−5 K−1)\gamma = 3 \times (2.0 \times 10^{-5} \, \text{K}^{-1}) γ=6.0×10−5 K−1\gamma = 6.0 \times 10^{-5} \, \text{K}^{-1}
  2. Rearrange the volume expansion formula to solve for Θ\Theta.

    Θ=ΔVγV0\Theta = \frac{\Delta V}{\gamma V_0}
  3. Substitute the values into the formula.

    Θ=0.0018 cm3(6.0×10−5 K−1)(1 cm3)\Theta = \frac{0.0018 \, \text{cm}^3}{(6.0 \times 10^{-5} \, \text{K}^{-1})(1 \, \text{cm}^3)} Θ=0.00186.0×10−5 K\Theta = \frac{0.0018}{6.0 \times 10^{-5}} \, \text{K} Θ=1.8×10−36.0×10−5 K\Theta = \frac{1.8 \times 10^{-3}}{6.0 \times 10^{-5}} \, \text{K} Θ=0.3×10(−3−(−5)) K\Theta = 0.3 \times 10^{(-3 - (-5))} \, \text{K} Θ=0.3×102 K\Theta = 0.3 \times 10^2 \, \text{K} Θ=30 K\Theta = 30 \, \text{K}

    Since a change in temperature of 1 K1 \, \text{K} is equivalent to a change of 1 ∘C1 \, ^\circ\text{C}, the temperature change is 30 ∘C30 \, ^\circ\text{C}.

    Comparing this to the given options:
    (A) 25∘C25^\circ\text{C}
    (B) 20∘C20^\circ\text{C}
    (C) 30∘C30^\circ\text{C}
    (D) 50∘C50^\circ\text{C}

    The calculated value matches option (C).

    The final answer is \boxed{\text{(C) 30^\circ C}}

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