Question

Some solar-heated homes use large beds of rocks to store heat. (a) How much heat is absorbed by \(100.0 \mathrm{~kg}\) of rocks if their temperature increases by \(12^{\circ} \mathrm{C} ?\) (Assume that \(c=\) \(\left.0.82 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C} .\right)\) (b) Assume that the rock pile has total surface area \(2 \mathrm{~m}^{2}\). At maximum intensity near the earth's surface, solar power is about 170 watts \(/ \mathrm{m}^{2}\). (1 watt \(=1 \mathrm{~J} / \mathrm{s} .\) ) How many minutes will it take for solar power to produce the \(12^{\circ} \mathrm{C}\) increase in part \((\mathrm{a}) ?\)

Short answer

Answer: It will take approximately 48.24 minutes for solar power to increase the temperature of the rock pile by 12°C.

Step-by-step solution

Part (a): Calculate the heat absorbed by the rocks

We have the mass (m) of rocks = 100 kg, specific heat capacity (c) = 0.82 J/(g·C), and the change in temperature (ΔT) = 12°C. To find the heat absorbed (Q), we use the formula: Q = m × c × ΔT First, let's convert the mass of rocks from kg to g, as the specific heat capacity is given in J/(g·C). m = 100 kg × 1000 g/kg = 100000 g Now, plug in the values: Q = 100000 g × 0.82 J/(g·C) × 12°C Q = 984000 J Thus, the heat absorbed by the 100 kg of rocks is 984,000 J.

Part (b): Determine the time required for solar power to produce the temperature increase

We are given the rock pile's total surface area (A) = 2 m², and the solar power intensity (P_intensity) = 170 W/m². To find the total solar power (P_total) received by the rock pile, we multiply the intensity by the surface area: P_total = P_intensity × A P_total = 170 W/m² × 2 m² P_total = 340 W Note: We know that 1 W = 1 J/s. So, we have P_total = 340 J/s. Now, let's find out how many seconds (t) it will take for solar power to produce the required heat (Q) using the formula: t = Q / P_total t = 984000 J / 340 J/s t = 2894.12 s To convert this time in seconds to minutes, simply divide by 60: t_minutes = t / 60 t_minutes ≈ 48.24 minutes Therefore, it will take approximately 48.24 minutes for solar power to produce the 12°C increase in temperature.