PURPOSE

In this experiment you will use a grating spectrometer to measure the wavelengths of the emission lines in the visible part of the spectrum for hydrogen and helium atoms.

EQUIPMENT  diffraction grating, double-arm spectrometer, high voltage supply, H and He Geissler tubes, Na spectrum lamp, magnifying glass, flashlight.

RELEVANT EQUATIONS

Hydrogen energy spectrum
Wavelength of transition from state n to ground state
Transition from state n to state i

DISCUSSION

When an electric current passes through a gas, the atoms of the gas emit light at specific discrete wavelengths. The spectrum of this light is characteristic of the atoms involved, and gives important information about the arrangement of the electrons in the atom. The hydrogen spectrum, for example, is different from that of helium or any other atom. The Bohr model of the atom pictures an atom as a positively charged nucleus with a swarm of negatively charged electrons orbiting around it. However, it was possible for Bohr to explain the discrete nature of atomic spectra only after he imposed a quantum condition on the allowed orbits. This condition restricts the electrons in an atom to specific energy states. The transition of an electron from one state to another involves a specific change in the energy of the atom.

Bohr showed that the energy states of the hydrogen atom are given by the simple formula:

(1)

where n is an integer (n = 1,2,3,...) called the principal quantum number.

The lowest-energy state (called the ground state) has the energy E₁ = −13.6 electron volts (eV). When an electron in the ground state is excited by electrical discharge in the gas, it is raised to one of the higher energy states with n = 2, 3,...... The atom quickly returns to the ground state by emitting its excess energy in the form of light.

According to the quantum theory, light of a given frequency f consists of discrete particles called photons, each having energy E = h f, where h is Planck's constant. In addition, light obeys the usual wave relation c = f λ where c is the speed of light, and λ is the wavelength, so that for a photon E_p = hc/λ. The wavelength of the photon emitted in an atomic transition from the state n to the ground state is thus obtained by equating the energy of the photon to the energy given up by the atom. In the case of hydrogen this yields

(2)

Of course, the atom need not return directly to the ground state, but can first make a transition to an intermediate state of energy E_i and then another transition to the ground state. The general relation for the energy emitted in an atomic transition from state n to state i is therefore given by

(3)

The wavelength of the light emitted in such a transition is given by

(4)

In this experiment, the visible spectrum of hydrogen as well as the spectra of other atoms will be studied with an instrument called a grating spectrometer. The spectrometer uses the dispersive power of a diffraction grating to measure the wavelengths of the spectral lines. The relation between the diffraction angle and the wavelength for a grating with a line spacing of d is given by

(5)