A 15 -cm-diameter aluminum ball is to be heated from \(80^{\circ} \mathrm{C}\) to an average temperature of \(200^{\circ} \mathrm{C}\). Taking the average density and specific heat of aluminum in this temperature range to be \(\rho=2700 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c_{p}=0.90 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\), respectively, determine the amount of energy that needs to be transferred to the aluminum ball. Answer: \(515 \mathrm{~kJ}\)

Specific heat, symbolized as c, is a material property that stands for the amount of heat needed to raise the temperature of one kilogram of a substance by one degree Celsius (or Kelvin). In other words, it is a measure of how much thermal energy a substance can soak up before its temperature changes significantly.

The specific heat of a substance is a critical aspect in heat transfer calculations because it reflects how substances absorb and store heat. For instance, in the given exercise, the specific heat of aluminum was given as 0.90 kJ/kg\(\bullet\)K. This relatively low specific heat means that aluminum heats up quickly compared to substances with a higher specific heat. The ability of a substance to resist temperature changes also has practical applications, such as in constructing buildings or manufacturing cookware.

Thermal energy transfer involves moving heat from one object or substance to another. The method of transfer can occur in three primary ways: conduction, convection, and radiation. Conduction happens through direct contact, convection is through fluid movement (like air or water), and radiation transfers energy through electromagnetic waves.

In the textbook exercise, the quantity of thermal energy that must be transferred to the aluminum ball to heat it is what we are solving for, denoted as Q. The formula used in the exercise, Q = mc\(\Delta\)T, succinctly expresses thermal energy transfer in cases of conduction or convection where no phase change occurs. It states that the amount of energy transferred (Q) is the product of the mass (m), specific heat (c), and change in temperature (\(\Delta\)T).

Temperature change is a concept that measures the difference in temperature as a substance absorbs or releases heat. It is denoted as \(\Delta\)T in heat transfer equations, calculating the shift in temperature from an initial state T₁ to a final state T₂, described as \(\Delta\)T = T₂ - T₁.

This variable is essential in understanding how much energy is required to achieve a specific temperature increase, as seen in the exercise where the aluminum ball’s temperature rises from 80°C to 200°C. The broader the temperature change, the more energy is typically needed, provided the mass and specific heat remain constant. This relationship is crucial in various applications, from climate control in buildings to managing thermal conditions in industrial processes.