After determining that the Sun has existed for hundreds of...

After determining that the Sun has existed for hundreds of millions of years, but before the discovery of nuclear physics, scientists could not explain why the Sun has continued to burn for such a long time interval. For example, if it were a coal fire, it would have burned up in about 3000 yr. Assume the Sun, whose mass is $1.99 \times 10^{30} \mathrm{kg}$ , originally consisted entirely of hydrogen and its total power output is $3.85 \times 10^{96} \mathrm{W}$ . (a) Assuming the energy- generating mechanism of the Sun is the fusion of hydrogen into helium via the net reaction $$ 4\left(_{1}^{1} \mathrm{H}\right)+2\left(\mathrm{e}^{-}\right) \rightarrow_{2}^{4} \mathrm{He}+2 v+\gamma $$ calculate the energy (in joules) given off by this reaction. (b) Determine how many hydrogen atoms constitute the Sun. Take the mass of one hydrogen atom to be $1.67 \times 10^{-27} \mathrm{kg} .$ (c) If the total power output remains converted into helium, making the Sun die? The actual projected lifetime of the Sun is about 10 billion years because only the hydrogen in a relatively small core is available as a fuel. Only in the core are temperatures and densities high enough for the fusion reaction to be self-sustaining.

After determining that the Sun has existed for hundreds of millions of years, but before the discovery of nuclear physics, scientists could not explain why the Sun has continued to burn for such a long time interval. For example, if it were a coal fire, it would have burned up in about 3000 yr. Assume the Sun, whose mass is $1.99 \times 10^{30} \mathrm{kg}$ , originally consisted entirely of hydrogen and its total
power output is $3.85 \times 10^{96} \mathrm{W}$ . (a) Assuming the energy- generating mechanism of the Sun is the fusion of hydrogen into helium via the net reaction
$$
4\left(_{1}^{1} \mathrm{H}\right)+2\left(\mathrm{e}^{-}\right) \rightarrow_{2}^{4} \mathrm{He}+2 v+\gamma
$$
calculate the energy (in joules) given off by this reaction. (b) Determine how many hydrogen atoms constitute the Sun. Take the mass of one hydrogen atom to be $1.67 \times 10^{-27} \mathrm{kg} .$ (c) If the total power output remains converted into helium, making the Sun die? The actual projected lifetime of the Sun is about 10 billion years because only the hydrogen in a relatively small core is available as a fuel. Only in the core are temperatures and densities high enough for the fusion reaction to be self-sustaining.

Key Concepts

Nuclear Fusion

Nuclear fusion is the process by which light atomic nuclei merge to form a heavier nucleus, releasing energy as a result. In stars, such as the Sun, hydrogen nuclei fuse to form helium, and the resulting mass difference (mass defect) is converted into energy according to fundamental physical principles. This process is the primary energy source of stars.

Mass?Energy Equivalence

Mass-energy equivalence, encapsulated in Einstein’s equation E=mc², is the principle that mass can be converted into energy. In stellar nuclear fusion, a small amount of the mass of the reacting particles is converted into a significant amount of energy, which is what powers the star over long periods.

Energy Yield of a Fusion Reaction

The energy yield of a fusion reaction refers to the amount of energy released when specific nuclei fuse. This can be calculated from the mass difference between the reactants and products, and is critical in understanding how protracted energy generation in stars occurs despite the relatively small conversion efficiency per reaction.

Stellar Mass and Atomic Composition

Determining the number of atoms in a star by dividing its total mass by the mass of a single atom is a key concept. This approach relates macroscopic properties of a star (like its total mass) to its microscopic constituents, and is essential for quantifying the amount of fuel available for nuclear fusion.

Stellar Evolution and Lifetime

The concept of stellar evolution involves understanding how stars change over time, including the phases during which they can sustain nuclear fusion. The available fuel (typically confined to the hot, dense core) and the rate of fusion determine a star's lifetime. This underscores the idea that even if a star has a massive total mass, only a fraction is available for energy generation during the fusion process.

Transcript

00:01 Over here we have the equation of the fusion reaction that is occurring in the sun.

00:07 You're going to find us the q value of this reaction.

00:13 But before we do that, we can simplify this reaction a bit by adding two electrons on both sides.

00:24 And the reason why we do this is such that we will have the nucleus to be a neutral atom instead.

00:34 So we can use the atomic masses to calculate the q value.

00:40 Because the q value simply is looking at the mass difference between the initial and the final products.

00:50 And if you were to add same amount of electrons on both sides, it wouldn't affect the difference in mass, but it will help us to simplify the nuclides into neutral atoms.

01:02 Over here we have four hydrogen and four electrons so they all become atoms and so the reaction simplifies a bit we have the helium atom plus right so we can ignore the neutrino and the gamma ray when we're calculating the q value since they are massless so we take four times the mass of the hydrogen atom minus away the mass of our hedium 4 atom, multiply by c square, which is nitr1 .5 mvv per u.

02:20 So putting it into the calculator, you should get about 26 .73 mbv.

02:40 But to facilitate the subsequent questions, we will need to convert this into joules, so we multiply by 1 .6 times 10 power minus 13 joules per m .ev.

02:57 And we should get 4 .28 times 10 power minus 12 joules.

03:09 Now for the second part, we want to find what is the total number of hydrogen atoms or protons, assuming that the sun is completely made out of them.

03:21 Right, so we are given the total mass of the sun is 1 .99 times 10 power 30 kilograms...

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