Examkin

Question

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

A solid metal sphere has an initial volume of 1000 cm31000\text{ cm}^3 at 20∘C20^\circ\text{C}. If it is heated to a final temperature of 70∘C70^\circ\text{C} and the linear expansivity of the metal is 2.0×10−5 K−12.0 \times 10^{-5}\text{ K}^{-1}, what is the increase in the volume of the sphere?

  1. 3.0 cm33.0\text{ cm}^3Answer
  2. B
    1.0 cm31.0\text{ cm}^3
  3. C
    2.0 cm32.0\text{ cm}^3
  4. D
    4.2 cm34.2\text{ cm}^3

Answer

The increase in the volume of the sphere is 3.0 cm33.0\text{ cm}^3.
The value of 3.0 cm33.0\text{ cm}^3 is correct because the volume expansion requires the cubical expansivity γ=3α=6.0×10−5 K−1\gamma = 3\alpha = 6.0 \times 10^{-5}\text{ K}^{-1}. Multiplying this coefficient by the initial volume (1000 cm31000\text{ cm}^3) and the temperature change (50 K50\text{ K}) yields ΔV=1000×6.0×10−5×50=3.0 cm3\Delta V = 1000 \times 6.0 \times 10^{-5} \times 50 = 3.0\text{ cm}^3.

Step-by-Step Solution

1
Calculate the temperature change (ΔT\Delta T).
ΔT=70∘C−20∘C=50∘C=50 K\Delta T = 70^\circ\text{C} - 20^\circ\text{C} = 50^\circ\text{C} = 50\text{ K}
Thermal expansion depends on the change in temperature rather than the initial or final temperature alone.
2
Determine the cubical (volume) expansivity (γ\gamma) from linear expansivity (α\alpha).
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}
For an isotropic solid, volume expands in three orthogonal dimensions, making cubical expansivity equal to three times linear expansivity.
3
Calculate the increase in volume (ΔV\Delta V).
ΔV=V1γΔT=1000 cm3×(6.0×10−5 K−1)×50 K=3.0 cm3\Delta V = V_1 \gamma \Delta T = 1000\text{ cm}^3 \times (6.0 \times 10^{-5}\text{ K}^{-1}) \times 50\text{ K} = 3.0\text{ cm}^3
Substitute initial volume, volume expansivity, and temperature change into the volume expansion formula.

Key Concept

Relationship between linear and volume expansivity (γ=3α\gamma = 3\alpha) and application of the volume expansion formula.
Estimated Time:1m 30s
Rate this question

Topics

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

More questions from this topic