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Chapter 19: Problem 48

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 \(24-\mathrm{h}\) 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.)

Short Answer

Expert verified
The mass of solar material converted to energy over a 24-hour period to provide the daily amount of solar energy to the Earth is approximately \( 4.831 \times 10^{9} \mathrm{kg} \). To provide the same amount of energy, you would need to burn approximately \( 1.556 \times 10^{14} \mathrm{g} \) or \( 1.556 \times 10^{11} \mathrm{kg} \) of coal.

Step by step solution

01

Calculate the total solar energy received by Earth in a 24-hour period

To calculate the total solar energy received by Earth in a 24-hour period, we can simply multiply the given energy received per second by the total number of seconds in a 24-hour period. Energy_received_per_second = 1.8 * 10**14 kJ/s Seconds_in_24_hours = 24 * 3600 s Total_solar_energy = Energy_received_per_second * Seconds_in_24_hours Total_solar_energy = 1.8 * 10**14 kJ/s * 24 * 3600 s
02

Calculate the mass of solar material converted to energy using E=mc²

We know the total solar energy received by Earth in a 24-hour period. To find the mass of solar material converted to energy, we can rearrange the mass-energy equivalence equation and solve for mass (m): m = E/c² where E is the total solar energy, c is the speed of light (about 3 * 10**8 m/s). First, we need to convert the total solar energy from kJ to J (1 kJ = 1000 J): Total_solar_energy_J = Total_solar_energy * 1000 J m = Total_solar_energy_J / c² Mass_solar_material = Total_solar_energy_J / (3*10**8)**2
03

Determine the mass of coal needed to produce the same amount of energy

Now we have the mass of solar material that is equivalent to the total solar energy received by Earth in a 24-hour period. To determine the mass of coal needed to produce the same amount of energy, we can use the given energy content of coal (32 kJ/g) and set up a proportion: Mass_coal = (Total_solar_energy_J * Mass_coal) / (32 * 1000) or simply, Mass_coal = Total_solar_energy_J / (32 * 1000) Now you can plug in the values calculated above for Total_solar_energy_J and compute Mass_solar_material and Mass_coal to find the masses of solar material and coal needed to provide the daily amount of solar energy to the Earth.

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