Ionization Potential of Hydrogen Atom

The ionization potential of an atom is the amount of energy required to remove an electron from its ground state. In the case of the hydrogen atom, the ionization potential is given as 2.17 x 10^-11 ergs per atom.

Bohr Model of Hydrogen Atom

The Bohr model of the hydrogen atom describes the electron orbits as discrete energy levels. The electron can transition between these levels by absorbing or emitting photons of specific energies.

Transition of Electron in Hydrogen Atom

When an electron in the hydrogen atom transitions from a higher energy level to a lower energy level, it emits a photon with energy equal to the difference in energy between the two levels. In this case, we are considering the transition of an electron from the 3rd Bohr orbit to the 1st Bohr orbit.

Calculating the Energy Difference

The energy difference between the 3rd and 1st Bohr orbits can be calculated using the formula:

ΔE = Eₙ - Eₖ = -13.6eV * (1/n² - 1/k²)

where Eₙ is the energy of the final orbit (1st orbit), Eₖ is the energy of the initial orbit (3rd orbit), and n and k are the principal quantum numbers of the final and initial orbits, respectively.

In this case, n = 1 and k = 3. Plugging these values into the formula, we get:

ΔE = -13.6eV * (1/1² - 1/3²)
= -13.6eV * (1 - 1/9)
= -13.6eV * (8/9)
= -12.18eV

Converting Energy to Wavelength

To calculate the wavelength of the emitted photon, we can use the equation:

E = hc/λ

where E is the energy of the photon, h is the Planck's constant (6.626 x 10^-34 J·s), c is the speed of light (2.998 x 10^8 m/s), and λ is the wavelength of the photon.

Converting the energy from electron volts (eV) to joules (J), we get:

E = -12.18eV * 1.602 x 10^-19 J/eV
= -1.952 x 10^-18 J

Plugging in the values for Planck's constant and the speed of light, we can solve for the wavelength:

-1.952 x 10^-18 J = (6.626 x 10^-34 J·s)(2.998 x 10^8 m/s)/λ

Simplifying the equation, we find:

λ = (6.626 x 10^-34 J·s)(2.998 x 10^8 m/s)/-1.952 x 10^-18 J
= 1.01 x 10^-7 m

Conclusion

The wavelength of the photon emitted when an electron in the 3rd Bohr orbit returns to the 1st orbit in a hydrogen atom is approximately 1.01 x 10^-7 meters (or 101 nm).

1.94*10^11