An electron drops from the fourth energy level, in an atom to the third level and then to the first level. Two frequencies of light are emitted. How does their combined energy compare with the energy of the single frequency that would be emitted if the electron dropped from the fourth level directly to the first level?
The combined energies of the two-photon process equals that of the single-photon process.
(E4 - E3) + (E3 - E1) = E4 - E1
1st photon 2nd photon single photon
To understand the comparison of the combined energy of two frequencies of light emitted versus the energy of a single frequency, we need to consider the energy levels involved and the relationship between energy levels and light frequencies.
In atoms, electrons exist in specific energy levels or orbitals. When an electron drops from a higher energy level to a lower energy level, it releases energy in the form of light. The energy difference between the two levels determines the frequency (and therefore the color) of the emitted light.
In this case, the electron drops from the fourth energy level (E4) to the third level (E3), and then from the third level (E3) to the first level (E1). To find the energy for each transition, we subtract the initial energy level from the final energy level:
(E4 - E3) gives the energy for the first photon emitted, and
(E3 - E1) gives the energy for the second photon emitted.
To compare this with the energy of a single photon emitted if the electron dropped directly from the fourth level (E4) to the first level (E1), we subtract the initial and final energy levels directly:
(E4 - E1) gives the energy for the single photon emitted.
The statement given in the question suggests that the combined energies of the two-photon process equal that of the single-photon process:
(E4 - E3) + (E3 - E1) = E4 - E1
This equation shows that the combined energy of the two photons emitted in two successive transitions is equal to the energy of the single photon emitted in one direct transition between the fourth and first energy levels.
In summary, the combined energies of the two frequencies of light emitted during two successive transitions from the fourth to the first energy level are equal to the energy of a single frequency of light emitted if the electron dropped directly from the fourth to the first energy level.