Question

two deuterons each of mass \(\mathrm{m}\) fuse to form helium resulting in release of energy \(\mathrm{E}\) the mass of helium formed is (A) \(\mathrm{m}+\left(\mathrm{E} / \mathrm{C}^{2}\right)\) (B) \(\left[\mathrm{E} /\left(\mathrm{mC}^{2}\right)\right]\) (C) \(2 \mathrm{~m}-\left(\mathrm{E} / \mathrm{C}^{2}\right)\) (D) \(2 \mathrm{~m}+\left(\mathrm{E} / \mathrm{C}^{2}\right)\)

Short answer

The mass of the helium formed after the fusion of two deuterons is given by \(M_h = 2m - \frac{E}{c^2}\), which corresponds to option (C).

Step-by-step solution

Recall Mass-Energy Equivalence Principle

According to Einstein's mass-energy equivalence principle, the energy E released during the fusion can be described by the difference in mass between the initial deuterons and the helium formed. This difference in mass is converted into energy, as given by the equation: E = Δm * c^2 Where Δm is the difference in mass and c is the speed of light.

Calculate the Initial Mass

The initial mass of the two deuterons is given as 2m (since each of them has a mass m). As mass is conserved during the nuclear process, we will set up a relationship between the initial mass, mass of helium formed, and the difference in mass.

Define the Mass of Helium and Difference in Mass

Let M_h be the helium mass formed after the fusion process, then the difference in mass between the initial deuterons and helium formed can be represented as: Δm = 2m - M_h

Substitute the Difference in Mass Equation into Mass-Energy Equivalence Equation

Plug the above equation for Δm into the equation for mass-energy equivalence: E = (2m - M_h) * c^2

Solve for the Mass of Helium

Isolate the variable M_h by solving for it: M_h = 2m - (E/c^2) This is the expression for the mass of helium formed after the fusion of two deuterons. Now, we will match our result with the given options.

Choose the Correct Option

Comparing our expression for M_h with the given options, we can see that our result matches with option C: M_h = 2m - (E/c^2) Hence, the mass of the helium formed after the fusion of two deuterons is given by option (C).

Key concepts

Mass-Energy Equivalence

The concept of mass-energy equivalence was introduced by Albert Einstein and is encapsulated in the famous equation \[ E = mc^2 \]This equation tells us that mass can be converted into energy and vice versa. Here, \( E \) represents energy, \( m \) represents mass, and \( c \) is the speed of light in a vacuum, approximately \( 3 \times 10^8 \) meters per second. This equivalence plays a vital role in understanding nuclear reactions such as fusion.
In nuclear fusion, when two light nuclei, like deuterons, come together to form a heavier nucleus, there can be a small but significant loss of mass, which is converted into energy. This released energy is what powers the sun and other stars. Therefore, according to the mass-energy equivalence principle, the slight decrease in mass from the initial reactants to the final product accounts for the energy released during the fusion process.

Deuterons

Deuterons are the nuclei of deuterium atoms. They consist of one proton and one neutron. Deuterium is a heavy isotope of hydrogen, represented by the symbol \(^2H\) or simply "D." When two deuterons are involved in a fusion reaction, they approach each other with high kinetic energy, overcoming the electrostatic repulsive force due to their positive charges.
Through nuclear fusion, deuterons can combine to form a helium nucleus. The fusion of deuterons is an example of a thermonuclear reaction, as it requires very high temperatures and pressures to occur, like those found in the cores of stars. It is one of the simplest fusion reactions and is significant in the study of nuclear fusion and potential future energy sources.

Helium Formation

When deuterons fuse, they form helium, specifically an isotope known as helium-4, which consists of two protons and two neutrons. This process involves the deuterons overcoming the Coulomb barrier—the electrostatic repulsion between their positively charged protons—thanks to kinetic energy.
The fusion process can be represented as:\[ ^2H + ^2H \rightarrow ^4He + energy \]The mass of the resulting helium nucleus is slightly less than the mass of the two original deuterons. This loss of mass corresponds to the energy released during the fusion as described by the mass-energy equivalence equation. Helium formation in stellar environments is a crucial step in the lifecycle of stars and is part of the proton-proton chain reaction that powers many stars, including our own sun.

Energy Release in Fusion

The energy release in fusion is a result of the conversion of mass into energy, as stated by Einstein's equation. During fusion, when deuterons come together to form helium, some of the mass is lost and transformed into energy.
This energy release occurs because the combined mass of the deuterons is greater than the mass of the helium formed. The difference in mass is converted into energy, which can be calculated using the relation:\[ E = \Delta m \cdot c^2 \]In this formula, \( \Delta m \) is the change in mass, and this energy accounts for the stability of the resulting helium nucleus. Fusion energy is incredibly high and serves as the potential for creating a powerful and clean energy source if it can be harnessed efficiently for practical use on Earth.