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Difficulty: HardHeat Capacity and Specific Heat Capacity

If two solid spheres, AA and BB, constructed from the same uniform metallic material, have radii in the ratio 2:12:1 respectively, then supplying equal quantities of heat energy to both spheres will cause sphere BB to experience a temperature rise eight times that of sphere AA.

Answer: Answer

Answer

The statement is true because the mass and heat capacity of a uniform solid sphere scale with the cube of its radius (r3r^3), giving sphere AA eight times the heat capacity of sphere BB. For an equal input of thermal energy, sphere BB undergoes eight times the temperature rise of sphere AA.
The statement is correct because volume scales as r3r^3, giving sphere AA eight times the mass and heat capacity of sphere BB. As a result, sphere BB undergoes eight times the temperature increase of sphere AA when absorbing equal heat energy.

Step-by-Step Solution

1
Relate the masses of the two spheres using their radii ratio.
mA=ρVA=ρ⋅43π(2rB)3=8(ρ⋅43πrB3)=8mBm_A = \rho V_A = \rho \cdot \frac{4}{3}\pi (2r_B)^3 = 8 \left(\rho \cdot \frac{4}{3}\pi r_B^3\right) = 8 m_B.
Mass is proportional to volume for uniform density, and volume scales as r3r^3.
2
Express the heat capacities of both spheres.
CA=mAc=8mBc=8CBC_A = m_A c = 8 m_B c = 8 C_B.
Heat capacity C=mcC = mc is an extensive property proportional to mass, while specific heat capacity cc is constant for a given material.
3
Compare the temperature rises for equal heat energy QQ.
ΔTB=QCB=Q18CA=8(QCA)=8ΔTA\Delta T_B = \frac{Q}{C_B} = \frac{Q}{\frac{1}{8}C_A} = 8 \left(\frac{Q}{C_A}\right) = 8 \Delta T_A.
Temperature rise ΔT=QC\Delta T = \frac{Q}{C} is inversely proportional to heat capacity when heat input QQ is identical.

Key Concept

Extensive nature of heat capacity and its scaling with volume (r3r^3) versus intensive specific heat capacity.
Estimated Time:1m 30s
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Topics

Heat Capacity and Specific Heat Capacity

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