Question

Be sure to show all calculations clearly and state your final answers in complete sentences. In daylight, Earth's surface absorbs about 400 watts per square meter. All of Earth's internal radioactivity produces a total of 3 trillion watts, which leak out through the surface. Calculate the internal heat flow (watts per square meter) averaged over Earth's surface. Compare this internal heat flow quantitatively to solar heating, and comment on why internal heating drives geological activity.

Short answer

Internal heat flow is 0.006 watts/m²; it's much smaller than solar heating, less than 0.0015% of it. Yet, it's crucial for geological activity.

Step-by-step solution

Calculate Earth's Surface Area

To find the internal heat flow of the Earth (in watts per square meter), we need to know the total surface area of the Earth. The formula to calculate the surface area of a sphere is given by the formula: \[ A = 4\pi R^2 \]where \( R \) is the radius of the Earth. The average radius of the Earth is approximately 6,371 kilometers, or 6,371,000 meters. Substituting this value in:\[ A = 4\pi (6,371,000)^2 \]Calculating this gives us approximately \(5.1 \times 10^{14} \) square meters.

Calculate Internal Heat Flow per Square Meter

The total internal heat generation by Earth's internal radioactivity is given as 3 trillion watts, or \( 3 \times 10^{12} \) watts. To find out how many watts leak through each square meter of Earth's surface:\[ \ ext{Internal heat flow per square meter} = \frac{3 \times 10^{12} \text{ watts}}{5.1 \times 10^{14} \text{ square meters}} \]Solving this equation gives approximately \( 0.006 \, \text{watts per square meter} \).

Compare Internal Heat Flow to Solar Heating

The solar heating on Earth is approximately 400 watts per square meter. We can compare the internal heat flow and solar heat flow by calculating their ratio:\[ \ ext{Ratio} = \frac{0.006 \, \text{watts per square meter}}{400 \, \text{watts per square meter}} = 0.000015 \]This shows that the internal heat flow is much smaller than the solar heat flow; it is less than 0.0015% of the solar heating.

Comment on Geological Activity

Even though the internal heat flow is small compared to solar heating, it is still significant because it is responsible for driving geological activity such as plate tectonics, volcanism, and mountain building. This heat supports convection currents in the mantle, leading to these and other geological processes.

Key concepts

Geothermal Energy

Geothermal energy is a fascinating source of heat that originates from within the Earth. It stems from the decay of radioactive elements inside the Earth's core, mantle, and crust. The term "geothermal" is derived from "geo" meaning Earth and "thermal" meaning heat. This energy continuously seeps through the planet's surface, albeit at a much slower rate compared to energy received from the sun.
The internal heat flow is measured in watts per square meter, a unit indicating how much energy passes through a unit area over time. In our specific context, Earth's internal heat contributes only about 0.006 watts per square meter. While this might seem negligible compared to solar energy, it plays a vital role in geological and environmental processes.
You might wonder why we don’t notice geothermal energy in our daily lives. It’s because the majority of Earth’s internal heat is used up in driving geological activity rather than affecting the surface temperature, which is overwhelmingly controlled by solar radiation.

Solar Radiation

Solar radiation is the primary driver of Earth's surface temperature and climate. It provides the Earth with approximately 400 watts per square meter during daylight. This energy supports most life on Earth by driving photosynthesis in plants and regulating the climate and weather patterns that affect all organisms.
Unlike geothermal energy, solar radiation is much more potent on a per square meter basis. The sun's energy affects everything from the movement of winds and ocean currents to the growth of crops and the generation of solar power.
In quantitative terms, solar radiation is about 66,667 times stronger than the geothermal heat flow. This emphasizes its importance over internal heat sources in sustaining surface temperatures and biological ecosystems on Earth.

Geological Activity

Geological activity refers to the various processes powered by the Earth's internal heat. These include plate tectonics, volcanic eruptions, and the formation of mountains. Despite internal heat making up a tiny fraction of the Earth's energy budget compared to solar energy, it is essential for sustaining the dynamic nature of our planet.
The Earth's mantle, which lies beneath the crust, experiences convection currents due to geothermal energy. These currents are similar to the boiling effect in a heated pot of water, causing the movement of tectonic plates. This movement is what leads to earthquakes, volcanic activity, and the reshaping of Earth's surface over millions of years.
The importance of geological activity cannot be understated. It plays a critical role in shaping the Earth, creating fertile soils, forming vital natural resources, and even affecting long-term climatic conditions.

Surface Area of Earth

The Earth's surface area is pivotal in determining how energy, both from within and from the sun, interacts with the planet. The formula for calculating the surface area of a sphere is \( A = 4\pi R^2 \), where \( R \) is the radius, roughly given as 6,371 kilometers for Earth.
By substituting this radius into the formula, we get an approximate surface area of \(5.1 \times 10^{14} \) square meters. This large area is why the heat flow from the Earth's interior is so diffused—averaging out to just 0.006 watts per square meter globally.
Understanding the Earth's surface area helps in many fields such as climatology, where it plays a role in calculating how solar radiation affects different regions. It's also crucial in engineering and environmental science for planning structures and managing natural resources efficiently.