Question

Question:(II) An aluminum sphere is 8.75 cm in diameter. What will be its %change in volume if it is heated from 30°C to 160°C?

Short answer

The percentage change in the volume of the sphere is \(0.97\% \).

Step-by-step solution

Identification of given data

• The coefficient of volumetric expansion of the aluminum is\(\beta = 75 \times {10^{ - 6}}\;{\rm{/^\circ C}}\).
• The final temperature of the aluminum sphere is\({T_f} = 160^\circ {\rm{C}}\).
• The initial temperature of the aluminum sphere is\({T_i} = 30^\circ {\rm{C}}\).
• The diameter of the aluminum sphere is \({d_i} = 8.75{\rm{ cm}}\).

Understanding the percentage change in the volume of the aluminum sphere

The percentage change in the volume of the sphere is the ratio of the change in its volume to its original volume.

The change in the volume of the sphere depends on its initial volume and the volumetric expansion coefficient.

Determination of the percentage change in the volume of the sphere

The percentage change in the volume of a sphere can be expressed as shown below:

\(\begin{aligned}{c}\Delta V\% &= \frac{{\Delta V}}{{{V_i}}} \times 100\% \\ &= \frac{{{V_i}\beta \Delta T}}{{{V_i}}} \times 100\% \\ &= \beta \left( {{T_f} - {T_i}} \right) \times 100\% \end{aligned}\)

Substitute the values in the above equation.

\(\begin{aligned}{c}\Delta V\% &= 75 \times {10^{ - 6}}\;{\rm{/^\circ C}} \times \left( {160^\circ {\rm{C}} - 30^\circ {\rm{C}}} \right) \times 100\% \\ &= 0.97\% \end{aligned}\)

Thus, the percentage change in the volume of the sphere is \(0.97\% \).