Question

For He⁺ , a transition takes place from the orbit of radius 105.8 pm to the orbit of radius 26.45 pm. The wavelength (in nm) of the emitted photon during the transition is ___.

[Use: Bohr radius, a = 52.9 pm, Rydberg constant, H = 2.2 × 10⁻¹⁸ J, Planck’s constant, h = 6.6 × 10⁻³⁴ J s, Speed of light, c = 3 × 108 m s⁻¹]

Answer 1

To determine the wavelength of an emitted photon from a He⁺ ion, one should calculate the energy difference between the two orbits using the Rydberg formula, convert the energy to wavelength, and then express it in the desired units (nm).

Step-by-step explanation:

The student is asking about the wavelength of the photon emitted when there is an electron transition in a helium ion from a higher to a lower energy orbit. To calculate this, we use the Bohr model of the hydrogen-like atom, which also applies to helium ions with one electron (He⁺). We start by finding the energy difference (ΔE) between the two orbits using the formula ΔE = -H( 1/nf^2 - 1/ni^2) where H is the Rydberg constant, nf is the final orbit number, and ni is the initial orbit number. The Rydberg constant for the energy levels can be used directly since the energy levels of He⁺ are similar to those of hydrogen but with a higher effective nuclear charge. We then convert the energy difference to wavelength (λ) using the relationship λ = hc/ΔE, where h is Planck’s constant and c is the speed of light. Finally, we convert the wavelength from meters to nanometers by multiplying by 10^9.

To find the orbit numbers (n values), we use the radii given and the formula for the radius of the nth orbit of the hydrogen atom, which is rn = n^2 × Bohr radius. By comparing the given radii with the Bohr radius, we can determine the values of ni and nf.