Recoil Energy of Helium Atom due to Emission of Photon

When a helium atom emits a photon, it experiences a recoil due to the conservation of momentum. The recoil energy can be calculated using the following steps:

Step 1: Calculate the Momentum of the Photon
The momentum of a photon can be calculated using the formula: p = h/λ, where p is the momentum, h is the Planck's constant (6.626 x 10^-34 J·s), and λ is the wavelength of the photon.

Given that the wavelength of the photon emitted by the helium atom is 0.1 A (Angstrom), we need to convert it to meters:
λ = 0.1 A = 0.1 x 10^-10 m

Using the formula, we can calculate the momentum of the photon:
p = (6.626 x 10^-34 J·s) / (0.1 x 10^-10 m) = 6.626 x 10^-24 kg·m/s

Step 2: Calculate the Mass of the Helium Atom
The mass of a helium atom (m) is approximately 4 atomic mass units (u) or 6.64 x 10^-27 kg.

Step 3: Calculate the Recoil Energy
The recoil energy (E) can be calculated using the formula: E = (p^2) / (2m), where E is the recoil energy, p is the momentum of the photon, and m is the mass of the helium atom.

Substituting the values into the formula, we get:
E = ((6.626 x 10^-24 kg·m/s)^2) / (2 x 6.64 x 10^-27 kg)
E = (4.38 x 10^-47 kg^2·m^2/s^2) / (1.33 x 10^-26 kg)
E = 3.29 x 10^-21 J

Therefore, the recoil energy of the helium atom due to the emission of a photon with a wavelength of 0.1 A is approximately 3.29 x 10^-21 J.