Question

A \(15-\mathrm{cm}\)-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.

Short answer

Given: density of aluminum = 2700 kg/m³, specific heat of aluminum = 0.90 kJ/kg.K. Answer: The amount of energy required to heat the aluminum ball from 80°C to 200°C is 623.88 kJ.

Step-by-step solution

Calculate the volume of the aluminum ball

To calculate the volume of the aluminum ball, we can use the formula for the volume of a sphere: \(V = \dfrac{4}{3} \pi r^3\) where \(V\) is the volume, and \(r\) is the radius of the sphere. First, we need to find the radius (half the diameter) in meters, as given dimensions are in centimeters. \(r = \dfrac{15}{2} cm = 0.075 m\) Now, we can plug in the value of the radius into the volume formula: \(V = \dfrac{4}{3} \pi (0.075)^3 = 0.00177 m^3\)

Calculate the mass of the aluminum ball

Now that we have the volume, we can calculate the mass of the aluminum ball using the formula: mass = density × volume We are given the density of aluminum as \(\rho = 2700 \mathrm{~kg} / \mathrm{m}^{3}\). Plugging the values into the formula: mass = \((2700 \mathrm{~kg}/\mathrm{m}^{3}) \times 0.00177\mathrm{m}^{3} = 4.59\mathrm{kg}\)

Calculate the change in temperature

The change in temperature can be calculated by finding the difference between the final and initial temperatures: change in temperature = final temperature - initial temperature change in temperature = \(200^\circ C - 80^\circ C = 120^\circ C\) Note that the temperature change is given in Celsius and can be used directly with the given specific heat in \(kJ/kg.K\).

Calculate the required energy

Now that we have the mass, specific heat, and change in temperature, we can use the energy equation to find the required energy to heat the aluminum ball: Energy = mass × specific heat × change in temperature Energy = \((4.59\mathrm{kg}) \times (0.90\mathrm{~kJ}/\mathrm{kg} \cdot \mathrm{K}) \times (120\mathrm{K}) = 623.88 \mathrm{kJ}\) So, the amount of energy required to heat the aluminum ball from \(80^{\circ}\mathrm{C}\) to an average temperature of \(200^{\circ}\mathrm{C}\) is 623.88 kJ.