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?

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 = \(\frac{3.85 \cdot 10^{26} \mathrm{W}}{(3.00 \cdot 10^8 \mathrm{m / s})^2} = 4.28 \cdot 10^9 \mathrm{ kg / s}\)

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.

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.