Question

The Sun radiates energy at the rate of \(3.85 \cdot 10^{26} \mathrm{~W}\) a) At what rate, in \(\mathrm{kg} / \mathrm{s}\), is the Sun's mass converted into energy? b) Why is this result different from the rate calculated in Example \(40.6,6.02 \cdot 10^{11}\) kg protons being converted into helium each second? c) Assuming that the current mass of the Sun is \(1.99 \cdot 10^{30} \mathrm{~kg}\) and that it radiated at the same rate for its entire lifetime of \(4.50 \cdot 10^{9} \mathrm{yr}\), what percentage of the Sun's mass was converted into energy during its entire lifetime?

Short answer

The rate at which the Sun's mass is converted into energy is approximately \(4.28 \cdot 10^9 \mathrm{ kg / s}\). During its entire lifetime, approximately \(0.0305 \%\) of the Sun's mass has been converted into energy.

Step-by-step solution

Calculate the rate of mass conversion to energy

Using Einstein's mass-energy equivalence formula, E=mc^2, we can calculate the rate of mass conversion. We are given the rate of energy radiation, so we can rewrite the formula as: m = E/c^2. The speed of light, c, is approximately \(3.00 \cdot 10^8 \mathrm{m / s}\), and the energy radiation rate is \(3.85 \cdot 10^{26} \mathrm{W}\). m = \(\frac{3.85 \cdot 10^{26} \mathrm{W}}{(3.00 \cdot 10^8 \mathrm{m / s})^2} = 4.28 \cdot 10^9 \mathrm{ kg / s}\)

Explain the difference between this result and Example 40.6

The result we calculated in step 1 is higher than the rate calculated in Example 40.6, which is \(6.02 \cdot 10^{11} \mathrm{ kg / s}\). This difference can be attributed to the fact that the nuclear fusion process occurring in the Sun is not perfectly efficient. Not all of the mass of the protons in the Sun is converted into energy, and some energy is lost in the form of neutrinos.

Calculate the percentage of the Sun's mass converted into energy during its entire lifetime

Now, we need to find the total amount of mass converted into energy during the Sun's lifetime. We are given the current mass of the Sun, \(1.99 \cdot 10^{30} \mathrm{ kg}\), and its radiated energy rate for its entire lifetime of \(4.50 \cdot 10^9 \mathrm{yr}\). First, we need to convert the given lifetime in years to seconds. Total_lifetime_seconds = \(4.50 \cdot 10^9 \mathrm{yr} \cdot \frac{365 \mathrm{days}}{1 \mathrm{yr}} \cdot \frac{24 \mathrm{h}}{1 \mathrm{day}} \cdot \frac{60 \mathrm{min}}{1 \mathrm{h}} \cdot \frac{60 \mathrm{s}}{1 \mathrm{min}} = 1.42 \cdot 10^{17} \mathrm{s}\) Now, we can calculate the total amount of mass converted into energy during this time. Total_mass_converted = \(4.28 \cdot 10^9 \mathrm{ kg / s} \times 1.42 \cdot 10^{17} \mathrm{s} = 6.08 \cdot 10^{26} \mathrm{ kg}\) Finally, we can find the percentage of the Sun's mass that was converted into energy during its entire lifetime. Percentage = \(\frac{6.08 \cdot 10^{26} \mathrm{ kg}}{1.99 \cdot 10^{30} \mathrm{ kg}} \times 100 = 0.0305 \%\) Approximately \(0.0305 \%\) of the Sun's mass has been converted into energy during its entire lifetime.

Key concepts

Solar Energy

Solar energy is the radiant light and heat that the Sun emits. It is a key source of clean and renewable energy on Earth. Solar energy is harnessed through technologies such as solar panels and solar thermal systems. Understanding solar energy begins with recognizing its vast scale and efficiency:

• The Sun produces approximately \(3.85 \cdot 10^{26} \mathrm{~W}\) of energy each second.
• This energy powers natural processes on Earth, including photosynthesis.
• Solar panels convert sunlight directly into electricity, adding to our renewable energy resources.

The reliability and abundance of solar energy make it a vital component of future energy systems.

Nuclear Fusion

Nuclear fusion is a reaction where two light atomic nuclei combine to form a heavier nucleus, releasing energy in the process. This reaction powers the Sun and is the primary source of its energy output. Here's a closer look at why nuclear fusion is so important:

• In the Sun, hydrogen nuclei fuse to form helium, releasing substantial energy.
• Nuclear fusion reactions provide over 99% of the energy output from the Sun.
• The process involves converting mass to energy, as explained by Einstein's mass-energy equivalence principle.

Nuclear fusion has potential for sustainable energy generation on Earth if we can replicate controlled fusion reactions.

Energy Conversion

Energy conversion is the process of changing one form of energy into another, such as converting solar energy into electrical energy. This concept is essential in understanding how energy is utilized and transformed:

• In the Sun, mass is converted into energy via nuclear fusion, maintaining the Sun's energy output.
• On Earth, technologies convert solar energy into heat or electrical power.
• Efficient energy conversion reduces the dependency on non-renewable energy sources.

Energy conversion technologies are continually evolving to improve efficiency and minimize environmental impact.

Einstein's Theory of Relativity

Einstein's theory of relativity introduced a new understanding of the interconnectedness of mass and energy. The famous equation \(E=mc^2\) illustrates this relationship, where:

• \(E\) represents energy.
• \(m\) represents mass.
• \(c\) is the speed of light in a vacuum, approximately \(3.00 \cdot 10^8 \mathrm{m/s}\).

This equation shows that a small amount of mass can be converted into a large amount of energy, explaining the vast energy release in nuclear reactions. Mass-energy equivalence is a crucial concept in nuclear physics and cosmology, providing insight into both everyday and cosmic phenomena.