Marks: 1
In which of the following cases is the wavelength of the emitted radiation greatest?
Choose one answer.
a. Electron jumps from third orbit to second orbit.
b. Electron jumps from third orbit to first orbit.
c. Electron jumps from fourth orbit to first orbit.
d. Electron jumps from fifth orbit to second orbit.
To determine the case in which the wavelength of the emitted radiation is greatest, we need to use the concept of energy levels and the equation relating energy and wavelength.
The energy levels of an atom are quantized and can be represented by specific orbit numbers. The energy of an electron in an orbit is given by the equation E = -13.6Z^2/n^2, where Z is the atomic number and n is the principal quantum number representing the orbit.
According to the equation, as the electron jumps from a higher energy level to a lower energy level, it emits energy in the form of radiation. The wavelength of this radiation can be calculated using the equation λ = c/f, where λ is the wavelength, c is the speed of light, and f is the frequency.
Higher energy level jumps correspond to larger differences in energy, which result in the emission of radiation with longer wavelengths. Therefore, to find the case with the greatest wavelength of emitted radiation, we need to compare the energy differences for each option.
Let's analyze each option:
a. Electron jumps from the third orbit to the second orbit:
The energy emitted can be calculated by taking the difference in energy levels: ΔE = E3 - E2.
Here, ΔE = (-13.6Z^2/3^2) - (-13.6Z^2/2^2) = -13.6Z^2/9 + 13.6Z^2/4.
b. Electron jumps from the third orbit to the first orbit:
ΔE = E3 - E1.
ΔE = (-13.6Z^2/3^2) - (-13.6Z^2/1^2) = -13.6Z^2/9 + 13.6Z^2.
c. Electron jumps from the fourth orbit to the first orbit:
ΔE = E4 - E1.
ΔE = (-13.6Z^2/4^2) - (-13.6Z^2/1^2) = -13.6Z^2/16 + 13.6Z^2.
d. Electron jumps from the fifth orbit to the second orbit:
ΔE = E5 - E2.
ΔE = (-13.6Z^2/5^2) - (-13.6Z^2/2^2) = -13.6Z^2/25 + 13.6Z^2/4.
By comparing the energy differences ΔE for each option, we can determine the largest difference and hence the greatest wavelength of emitted radiation.