Question

What is the binding energy of an \(\alpha\) particle (a \(^{4} \mathrm{He}\) nucleus)? The mass of an \(\alpha\) particle is \(4.00151 \mathrm{u}\).

Short answer

Question: Calculate the binding energy of an α particle with a mass of 4.00151 atomic mass units. Answer: To calculate the binding energy of the α particle, follow these steps: 1. Determine the mass of the individual nucleons in the α particle (two protons and two neutrons). 2. Calculate the mass defect by finding the difference between the total mass of the individual nucleons and the mass of the α particle. 3. Convert the mass defect into energy using Einstein's equation \(E = mc^2\). 4. Convert the energy to megaelectronvolts (MeV) using the appropriate conversion factors. By following these steps, you can find the binding energy of the α particle in MeV.

Step-by-step solution

Determine the mass of the individual nucleons

First, we need to find the mass of the individual nucleons in the α particle. In an α particle, there are two protons and two neutrons. The mass of a proton is approximately 1.00728 u, and the mass of a neutron is approximately 1.00867 u. So, we can calculate the total mass of the nucleons as follows: Total mass of nucleons = (2 × mass of a proton) + (2 × mass of a neutron)

Calculate the mass defect

Next, we will calculate the mass defect, which is the difference between the total mass of the individual nucleons and the mass of the α particle. Given that the mass of the α particle is 4.00151 u, we can calculate the mass defect as follows: Mass defect = Total mass of nucleons - Mass of α particle

Convert the mass defect into energy

Now we'll convert the mass defect we found in the previous step into energy using Einstein's equation \(E = mc^2\). Note that we must first convert the mass defect from atomic mass units (u) to kilograms (kg) using the following conversion factor: 1 u = 1.660539 x \(10^{-27}\) kg. Once converted, we can then calculate the energy in joules (J) as follows: Energy = Mass defect (in kg) × \((3.0 \times 10^8 m/s)^2\)

Convert the energy to MeV

Finally, we need to convert the energy found in the previous step from joules (J) to megaelectronvolts (MeV), which is a more commonly used unit for binding energy. To do this, we will use the following conversion factor: 1 J = 6.242 x \(10^{12}\) MeV. So, the binding energy will be: Binding energy = Energy (in J) × conversion factor Following these steps, you will be able to find the binding energy of the α particle in MeV.