Question

How might the energy levels of an atom be measured directly—that is, without recourse to analysis of spectra?

Short answer

We can do experiments similar to Frank-Hertz experiment.

Step-by-step solution

Important Concepts Franck Hertz Experiment

A vacuum tube was used in the Franck Hertz experiment to analyse the powerful electrons that flew through a thin vapour of mercury atoms. It was observed that when electrons collide with the mercury atom, just a small portion of an atom's kinetic energy is lost.

Explanation  

Experiments similar to that of Frank and Hertz might be used to measure the energy levels of an atom directly. In Frank-Hertz experiment the electrons are conducted through vapour of the element of interest with some definite kinetic energy by applying the right amount of voltage across the vapour-discharge tube. The applied voltage (and hence the kinetic energy of the electrons) is varied until the electron-conduction current through the tube abruptly lowers; this occurs when the kinetic energy of the electrons drops dramatically. the electrons has just the right value to excite the vapour atoms from the ground state into the first excited state during an elastic-collision process. In this situation, the so-called excitation voltage corresponds to the amount of energy that the first excited state of this atom has above the ground state, or the first excitation energy. Increasing the applied voltage shows the higher excitation energy of such an atom in the same way. We require the energy value of the ground state of the studied atom to determine the real energy levels of the considered atom (not simply their energies relative to the ground state). this atom, which can be determined in terms of the ionization energy for this atom as measured from the electric discharge experiments.