Question

Consider the following fusion reaction, through which stars produce progressively heavier elements: \({ }_{2}^{3} \mathrm{He}+{ }_{2}^{4} \mathrm{He} \rightarrow{ }_{4}^{7} \mathrm{Be}+\gamma\). The mass of \({ }_{2}^{3} \mathrm{He}\) is \(3.016029 \mathrm{u},\) the mass of \({ }_{2}^{4} \mathrm{He}\) is \(4.002603 \mathrm{u},\) and the mass of \({ }_{4}^{7} \mathrm{Be}\) is \(7.0169298 \mathrm{u}\) The atomic mass unit is \(1 \mathrm{u}=1.66 \cdot 10^{-27} \mathrm{~kg}\). Assuming that the Be atom is at rest after the reaction and neglecting any potential energy between the atoms and the kinetic energy of the He nuclei, calculate the minimum possible energy and maximum possible wavelength of the photon, \(\gamma\), that is emitted in this reaction.

Short answer

Answer: The minimum energy of the emitted photon is approximately 8.504 x 10^-11 J, and the maximum wavelength is approximately 2.345 x 10^-8 m.

Step-by-step solution

Calculate the mass defect

To find the mass defect, we need to determine the difference in mass between the initial state (helium-3 and helium-4) and the final state (beryllium-7). The mass difference can be calculated as follows: Mass defect = (mass of helium-3 + mass of helium-4) - mass of beryllium-7 Mass defect = (3.016029 u + 4.002603 u) - 7.0169298 u Mass defect = 7.018632 u - 7.0169298 u Mass defect = 0.0017022 u

Calculate the energy of the emitted photon

Now we can convert the mass defect into energy using the mass-energy equivalence formula: E = (mass defect) * c^2 where E is the energy, c is the speed of light (\(3 \cdot 10^8 m/s\)), and the mass defect is converted into kilograms using the conversion factor \(1 u = 1.66 \cdot 10^{-27} kg\). Energy of the photon = \((0.0017022 \text{ u})(1.66 \cdot 10^{-27} \text{ kg/u})(3\cdot 10^8 \text{ m/s})^2\) Energy of the photon = 8.50402664 × 10^2 × 10^-27 × 10^16 J Energy of the photon = 8.50402664 × 10^-11 J

Calculate the maximum wavelength of the emitted photon

We can use the energy-wavelength relation to calculate the maximum wavelength of the emitted photon: E = h * c / λ where E is the energy of the photon, h is the Planck's constant (\(6.63 \cdot 10^{-34} J \cdot s\)), c is the speed of light, and λ is the wavelength. So, we have: λ = h * c / E λ = (\(6.63 \cdot 10^{-34} J \cdot s\)) (\(3 \cdot 10^8 m/s\)) / (8.50402664 × 10^-11 J) λ = 2.345 ~\text{x}~ 10^{-8} m The maximum possible wavelength of the emitted photon is approximately \(2.345 \times 10^{-8} m\).

Key concepts

Stellar Nucleosynthesis

At the core of stars, a fascinating process known as stellar nucleosynthesis takes place. This process involves the fusion of lighter atomic nuclei into heavier ones, a transformation that powers the luminosity of stars and produces a rich array of elements that constitute everything in the universe, including planets and life itself. The fusion reaction provided in the example, where helium-3 and helium-4 combine to form beryllium-7, is part of this complex dance of atomic synthesis that occurs in the intensely hot and dense cores of stars.

Simplifying the complex interplay of nuclear forces and extreme conditions, we can focus on the core concept of stellar nucleosynthesis - the idea that the fusion of lighter elements into heavier ones is not just about creating new atomic nuclei but also about the transformation of mass to energy, which is observable as both heat and light emanating from stars. One interesting aspect to highlight is that without this process, the heavier elements necessary for life, such as carbon, nitrogen, and oxygen, would not exist naturally.

Mass-Energy Equivalence

A cornerstone in understanding nuclear reactions is mass-energy equivalence, famously encapsulated by Einstein's equation, E=mc^2. This equation tells us that mass can be converted into energy, and vice versa. In the context of the fusion reaction provided, a tiny amount of mass known as the mass defect is not present in the end product (beryllium-7) and is instead converted into energy, which is emitted as a photon.

When students calculate the energy of the emitted photon, effectively, they are tracing the journey of the lost mass as it transforms into energy. This principle also lays the groundwork for exactly why stars can shine for billions of years; the conversion of mass into vast amounts of energy is incredibly efficient, significantly more than any chemical reaction like burning. This understanding provides a deep insight into not just stellar workings, but also the fundamental principles governing nuclear reactors and even atomic bombs.

Photon Energy and Wavelength Calculation

The characteristics of a photon released in nuclear reactions can be determined by the photon energy and wavelength calculation. The energy of a photon is inversely proportional to its wavelength, as described by the formula E = h*c / λ, where E is the energy of the photon, h is Planck's constant, c is the speed of light, and λ is the wavelength. Higher energy photons will have shorter wavelengths, and vice versa.

In our fusion reaction example, students calculate the maximum wavelength by finding the energy released when mass is converted during fusion and then applying it to the energy-wavelength relationship. This is crucial in astrophysics to analyze the light from stars which can reveal what reactions are happening within. This same principle is used in technologies like lasers and LED lights, linking the physics of the cosmos to practical applications in our everyday lives.