Question

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

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

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

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

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.

Key concepts

Solar Energy

The Earth receives an immense amount of energy from the sun in the form of solar radiation. This energy is essential for sustaining life on our planet, as it drives essential processes like photosynthesis in plants.
This daily reception of energy is quantified in kilojoules per second, indicating how much energy reaches the Earth every second.
The solar energy received by Earth in a 24-hour period is calculated by multiplying the energy received per second by the number of seconds in a day. This results in a colossal amount of energy that overshadow any energy humans currently produce.
Understanding how solar energy impacts our planet helps us appreciate the significance of renewable energy sources, which can reduce reliance on fossil fuels and help mitigate climate change impacts.

Energy Conversion

Energy conversion refers to changing energy from one form to another, such as transforming solar energy into electricity.
This conversion often involves transferring energy from its raw state. The sun's energy, for example, is converted into usable forms through processes like photovoltaics.
- Photovoltaic cells directly convert sunlight into electrical energy using semiconductors that create electricity upon exposure to light. - Solar thermal systems use sunlight to heat fluids, which then produce steam to drive turbines that generate electricity. - Photosynthesis in plants converts solar energy into chemical energy stored in glucose. Understanding these processes provides insight into developing efficient renewable energy systems. This energy form conversion is crucial as it enables us to use solar energy sustainably and reduce dependence on coal and other non-renewable resources.

E=mc²

Mass-energy equivalence, succinctly expressed with the famous equation \[ E=mc^2 \], reveals a profound connection between mass and energy.
In this equation, \(E\) represents energy, \(m\) is mass, and \(c\) is the speed of light in a vacuum.
According to this principle, mass can be converted into energy and vice versa, illustrating that a small amount of mass can be transformed into a large amount of energy due to the speed of light's large value.
Einstein's theory provides the foundation for understanding phenomena such as nuclear reactions and energy production in the sun, where nuclear fusion in the sun's core converts mass into energy, part of which travels to Earth as solar energy.
This concept helps explain how solar energy is generated and how we can calculate the mass of solar material converted into energy during nuclear reactions.

Coal Combustion

Coal combustion is a traditional method of generating energy by burning coal to produce heat. This heat then converts water into steam, driving turbines to generate electricity.
Coal has been a dominant source of energy for decades due to its abundance and relatively low cost. However, the combustion process releases significant carbon dioxide and other pollutants into the atmosphere, contributing to environmental issues like air pollution and climate change.
During coal combustion, energy is released as heat. This process is measured in terms of how much energy can be obtained from burning a given mass of coal, specifically grams.
The problem calculates the amount of coal needed to produce the same energy equivalent that Earth receives from the sun. This highlights stark contrasts between fossil fuels and renewable sources in terms of efficiency and environmental impact.
Ultimately, understanding these processes supports transitioning to cleaner energy sources to meet energy demands sustainably.