Multiple Choice
If a block of aluminum has a volume of exactly at , what is its volume at ? The coefficient of linear expansion for aluminum is.
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20. Heat and Temperature
Volume Thermal Expansion
3:44 minutesProblem 15
Textbook Question
An aluminum sphere is 8.75 cm in diameter. What will be its % change in volume if it is heated from 30°C to 140°C?
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Determine the formula for the volume of a sphere: \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the sphere. The diameter is given as 8.75 cm, so calculate the radius as \( r = \frac{8.75}{2} \) cm.
The volume change due to thermal expansion can be calculated using the formula \( \Delta V = \beta V_0 \Delta T \), where \( \beta \) is the coefficient of volumetric expansion for aluminum, \( V_0 \) is the initial volume, and \( \Delta T \) is the temperature change.
Find the initial volume \( V_0 \) by substituting the radius into the sphere volume formula: \( V_0 = \frac{4}{3} \pi r^3 \).
Calculate the temperature change \( \Delta T = T_{\text{final}} - T_{\text{initial}} = 140^{\circ}C - 30^{\circ}C \). Use the given temperature values.
Substitute \( \beta \), \( V_0 \), and \( \Delta T \) into the formula \( \Delta V = \beta V_0 \Delta T \) to find the change in volume. Then, calculate the percentage change in volume using \( \%\,\Delta V = \frac{\Delta V}{V_0} \times 100 \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Thermal Expansion
Thermal expansion refers to the increase in volume of a material as its temperature rises. For solids, liquids, and gases, this phenomenon occurs due to the increased kinetic energy of particles, causing them to move apart. The degree of expansion can be quantified using the coefficient of volumetric expansion, which varies by material.
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Volume of a Sphere
The volume of a sphere is calculated using the formula V = (4/3)πr³, where r is the radius. In this case, the radius can be determined from the diameter provided. Understanding how to calculate the volume is essential for determining the change in volume due to thermal expansion.
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Percentage Change in Volume
Percentage change in volume is calculated by taking the difference between the final and initial volumes, dividing by the initial volume, and then multiplying by 100. This metric allows for a clear understanding of how much the volume has increased relative to its original size, which is crucial for interpreting the effects of temperature changes.
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