Explanation of spectra In 1885 Johann Balmer, a Swiss physicist, discovered, by trial and error, that the energies in the emission spectrum of hydrogen were given by the formula: Johann Balmer ⎛1 1⎞ (1825-1898) ∆E = −Rz gZ ⎜ 2 − 2 ⎟ 2 ⎝ n f ni ⎠ where n may take integer values 3, 4, 5, … and R is a constant number
Explanation of spectra Since the emitted light from a gas carries energy, it is reasonable to assume that the emitted energy is equal to the difference between the total energy of the atom before and after the emission. Since the emitted light consists of photons of a specific wavelength, it follows that the emitted energy is also of a specific amount since the energy of a photon is given by: hc E = hf = λ This means that the energy of the atom is discrete, that is, not continuous.
The “electron in a box” model If the energy of the atom were continuous the emission of light wouldnt always be a set of specific amounts. The first attempt to explain these observations came with the “electron in a box” model. Imagine that an electron is confined in a box of linear size L. If the electron is treated as a wave, it will have a wavelength given by: h λ= the electron can only be found somewhere p along this line x=0 x=L
The “electron in a box” model If the electron behaves as a wave, then: The wave is zero at the edges of the box The wave is a standing wave as the electron does not lose energy This means that the wave will have nodes at x=0 and x=L. This implies that the wavelength must be related to the size of the box through: 2L λ= n Where n is an integer
The “electron in a box” model Therefore, the momentum of the electron is: h h mh p= = = λ 2L 2L n The kinetic energy is then: 2 mh 2 2 2 p 2L = n h Ek = = 2m 2m 8mL2
The “electron in a box” model This result shows that, because the electron was treated as a standing wave in a “box”, it was deduced that the electron’s energy is quantized or discrete: h2 1× n =1 8mL2 h2 Ek = 4× 8mL2 n=2 h2 9× n=3 8mL2 However, this model is not correct but because it shows that energy can be discrete it points the way to the correct answer.
The Schrödinger theory In 1926, the Austrian physicist Erwin Schrödinger provided a realistic quantum model for the behaviour of electrons in atoms. The Schrödinger theory assumes that there is a wave associated to the electron (just like de Bröglie had assumed) Erwin Schrödinger This wave is called wavefunction and (1887-1961) represented by: ψ( x, t ) This wave is a function of position x and time t. Through differentiation, it can be solved to find the Schrödinger function: ∂ 2i ψ( r , t ) = − ∇ ψ( r , t ) +V ( r ) ψ( r , t ) 2 ∂t 2m
The Schrödinger theory The German physicist Max Born interpreted Schrödingers equation and suggested that: 2 ψ ( x, t ) can be used to find the probability of finding an electron near position x at time t. This means that the equation cannot tell exactly where to find the electron. This notion represented a radical change from classical physics, where objects had well-defined positions.
The Schrödinger theory Solving for Hydrogen, it is found that: 13.6 E = − 2 eV n In other words, this theory predicts that the electron in the hydrogen atom has quantized energy. The model also predicts that if the electron is at a high energy level, it can make a transition to a lower level. In that process it emits a photon of energy equal to the difference in energy between the levels of the transition.
The Schrödinger theoryBecause the energy of the photon is given by E = hf, knowing theenergy level difference, we can calculate the frequency andwavelength of the emitted photon.Furthermore, the theory also predicts the probability that aparticular transition will occur.This high n is essential to energy energyunderstand why levels very 0 eV close tosome spectral lines each otherare brighter than n=5others. n=4 n=3Thus, theSchrödinger theory n=2explains atomic -13.6 eVspectra. n=1