The earth receives \(1.8 \times 10^{14} \mathrm{kJ} / \mathrm{s}\) of solar energy. What mass of solar material is converted to energy over a 24 -h period to provide the daily amount of solar energy to the earth? What mass of coal would have to be burned to provide the same amount of energy? (Coal releases \(32 \mathrm{kJ}\) of energy per gram when burned.)

To find the total energy received, we'll multiply the energy received per second by the number of seconds in a day. Total solar energy = Energy received per second × Number of seconds in a day Total solar energy = \(1.8 \times 10^{14} \mathrm{kJ/s} \times 24 \times 60 \times 60 \mathrm{s}\) Total solar energy = \(1.8 \times 10^{14} \mathrm{kJ/s} \times 86,400 \mathrm{s}\) Total solar energy = \(1.5552 \times 10^{19} \mathrm{kJ}\)

Einstein's mass-energy equivalence formula is given by: \(E = mc^2\) Where: E is the energy in joules (we'll convert from kJ to J), m is the mass of the solar material in kilograms, c is the speed of light in m/s, which is approximately 3 × 10^8 m/s. First, let's convert the energy from kJ to J: \(1.5552 \times 10^{19} \mathrm{kJ} = 1.5552 \times 10^{22} \mathrm{J}\) Now, we'll rearrange the formula to find the mass, m: \(m = \frac{E}{c^2}\) Insert the values: \(m = \frac{1.5552 \times 10^{22} \mathrm{J}}{(3 \times 10^8 \mathrm{m/s})^2}\) \(m \approx 1.727 \times 10^9 \mathrm{kg}\) So, the mass of solar material converted to energy over a 24-hour period is approximately \(1.727 \times 10^9 \mathrm{kg}\).

We are given that coal releases 32 kJ of energy per gram when burned, so we can find the mass of coal needed to produce the same amount of energy as follows: Mass of coal × Energy per gram of coal = Total solar energy in kJ Mass of coal = \(\frac{\mathrm{Total\,solar\,energy\,in\,kJ}}{\mathrm{Energy\,per\,gram\,of\,coal}}\) Mass of coal = \(\frac{1.5552 \times 10^{19} \mathrm{kJ}}{32 \mathrm{kJ/g}}\) Mass of coal = \(4.860 \times 10^{17} \mathrm{g}\) Since 1 kg = 1000 g, we can convert the mass of coal to kg: Mass of coal = \(4.860 \times 10^{14} \mathrm{kg}\) To provide the same amount of energy as the Sun, approximately \(4.860 \times 10^{14} \mathrm{kg}\) of coal would have to be burned.