* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 2.8 Atomic Spectra of Hydrogen For some time scientist had known
Survey
Document related concepts
Chemical bond wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
James Franck wikipedia , lookup
Wave–particle duality wikipedia , lookup
Auger electron spectroscopy wikipedia , lookup
Astronomical spectroscopy wikipedia , lookup
Mössbauer spectroscopy wikipedia , lookup
Electron scattering wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
X-ray fluorescence wikipedia , lookup
X-ray photoelectron spectroscopy wikipedia , lookup
Atomic orbital wikipedia , lookup
Tight binding wikipedia , lookup
Electron configuration wikipedia , lookup
Transcript
2.8 Atomic Spectra of Hydrogen For some time scientist had known that every atom, when subjected to high temperatures or an electrical discharge, emits electromagnetic radiation of characteristic frequencies or each atom has a characteristic emission spectrum. Because the emission spectra of atoms consists of only certain discrete frequencies, they are called line spectra. Hydrogen, the lightest and simplest atom, has the simplest spectrum. Since atomic spectra are characteristic of the atoms involved, it is reasonable to suspect that the spectrum depends on the electron distribution in the atom. A detailed analysis of the hydrogen atomic spectrum turned out to be a major step in the elucidation of the electronic structure of atoms. For many years, scientists had tried to find a pattern in the wavelengths or frequencies of the lines in the hydrogen atomic spectrum. Finally in 1885 the Swiss scientist, Johann Balmer, showed that, ̅𝜈 = 10973731.6 (1/22 – 1/n2) m-1 when n = 3,4,5…. Where ̅𝜈 = 1/λ = ν/c. This equation is called Balmer formula. Subsequent to the discovery of the Balmer series of lines, it was found that many other spectral lines are also present in the non visible regions of the electromagnetic spectrum. Thus Lyman series originates by the movement of electrons from higher energy states (n = 2,3,4 …..) to the first energy level for which n =1. Similarly, the Balmer series is accounted by the movement of electrons from n = 3,4,5,6…… to n =2. Hydrogen atom has a series of spectral lines ; the Lyman series in U V region, the Balmer series in the visible region, Paschen, Brackett and Pfund series in the I. R region. By the application of Balmer`s equation, the Swedish Physicist Johannes Rydberg showed that every line in the entire spectrum of hydrogen can be accounted by the general expression 1/λ = 𝜐̅ = R [1/n12- 1/n22], Where n1 and n2 represent integers with n2 > n1. R is the Rydberg constant equal to 10973731.6 m-1 for H atom and 𝜐̅ is the wave number. Fig.2.2 Spectral series for hydrogen atom It can be seen that the difference of energy values of electrons between successive orbits become smaller and smaller and ultimately tend to merge. The spectrum also shows that the line become closer together at high energy values. The higher energy end is that end where frequency is increasing or wave length is decreasing. 2.9 Bohr`s Theory of the Atom In 1913 the Danish physicist Niels Bohr presented a theory of the hydrogen atom that gave a beautifully simple explanation of the hydrogen spectrum. Bohr made his contribution by arbitrarily welding Planck`s quantum hypothesis to classical mechanics. 1. The electron in an atom can exist only in certain states and in these states they never radiate energy even if it has acceleration. This is the stationary state. Each stationary state has a definite amount of energy. 2. Energy is emitted only when an electron makes a transition between two stationary states and not continuously 3. The frequency of the radiation emitted when the transition occurs between two orbits that differ in energy ΔE 4. The angular momentum of the orbital electron can have only certain values and that are integral multiple of h/2п 5. The dynamical equilibrium that governs the stationary orbit is determined by classical mechanics, by balancing the electrostatic attraction against the centrifugal effect of the orbital motion. Bohr pictured the electron in a hydrogen atom moving in a circular orbit about the nucleus. Fig.2.3 Bohr atom model There is a nucleus charged Ze and an electron having mass m and charge e revolving at a distance r from the nucleus, with a velocity v. The electrostatic force of attraction between the nucleus and the electron = Ze2/4пε0r2 Where ε0 is permittivity of vacuum, ε0 = 8.85418 x 10-12 C2 N-1 m-2 The centripetal force acquired by the electron = mv2/r In the circular orbit, mv2/r = Ze2/4пε0r2 v= Ze2/4пε0r2.mvr According to Bohr mvr = nh/2п , n = 1,2,3…. So v = Ze2/2ε0nh This is the velocity of an electron in a circular orbit The radius of the orbit r can also be calculated. r = nh/2п.mv r= n2h2εo/Zmпe2 i.e r ∞ n2 Thus the radii of the allowed orbits or Bohr orbits are quantised. The electron can move around the nucleus in circular orbits only with radii given by the above eqn. The orbit with the smallest radius is one with n =1 for which r = (8.8541 x 10-12C2N-1m-2) (6.626 x 10-34Js)2 3.14 (9.109 x 10-31Kg) (1.6022 x 10-19C)2 = 5.292 x 10-11m = 0.529A0 The radius of the first Bohr orbit is often denoted by a In general we can write, rn = 0.529 x n2/z A0 Problem Calculate the radius ratio for 2nd orbit of He+ ion & 3rd orbit of Be+++ ion. r1 (radius of 2nd orbit of He+ ion) = 0.529 (22/2) A0 r2 (radius of 3rd orbit of Be+++ ion) = 0.529 (32/4) A0 r1 r2 = 0.529 (22/2) A0 0.529 (32/4) A0 = 8/9 Energy of Electron in Hydrogen and Hydrogen-like Atoms Total energy of an electron = Kinetic energy + Potential energy mv2/r = Ze2/4пε0r2 In the circular orbit, mv2 = Ze2/4пε0r Kinetic energy = ½ mv2 = Ze2/8пε0r r Potential energy = ∫∞ Ze2/4пε0r2 = - Ze2/4пε0.r Total energy = ½ (Ze2/4пε0r2) - Ze2/4пε0r2 = - Ze2/8пε0r Putting the value of r E = - Ze2.Zmпe2/8пε0n2h2ε0 = - mZ2e4/8ε02n2h2 The negative sign in this equation indicates that the energy states are bound states. n = 1 corresponds to the state of lowest energy. This energy is called the ground state energy. At ordinary temperature, hydrogen atom, as well as most other atoms and molecules are found almost exclusively in their ground electronic states. The states of higher energy are called excited states and are generally unstable with respect to the ground state. An atom or a molecule in an excited state will usually relax back to the ground state and give off the energy as electromagnetic radiation. Bohr assumed that the observed spectrum of the hydrogen atom is due to transition from one allowed energy state to another and he predicted that the allowed energy difference is given by ΔE = hυ = me4Z2/8ε02h2 [1/n12 - 1/n22] Thus, the simple explanation for the line spectrum is that the atoms are constrained to emit quanta of only a few specific energies depending on the energy difference between the orbits involved in the movement of electrons. Energetically excited atoms are not able to emit light of continuously varying wavelengths and therefore not a continuous spectrum is produced from atoms. Merits of Bohr Theory 1. Bohr theory explains well the features of hydrogen spectrum and the spectra of hydrogen like atoms He+, Li2+ etc 2. It helps to calculate radii of various energy levels 3. It also helps to calculate energy associated with various orbits and the velocity of electrons in various orbits The Spectrum of the Hydrogen Atom Bohr was able to derive a correct expression for the energy levels of hydrogen like atoms (i.e atoms with one electron). Bohr`s formula for the Rydberg constant was in perfect agreement with the experimental data. However, Bohr`s theory could not be extended successfully to helium or other atoms with more than one electron. According to Bohr, the spectrum arises when the electron in the initial higher energy state of principal quantum number n2 get transmit to the final lower energy state of principal quantum n1 so that the difference of the energies associated with these levels is emitted as photon of frequency υ ΔE = hυ = E2 – E1 = - m Z2 e4/8 ε02 n22 h2 – (- m Z2 e4/8 ε02 n12 h2) υ = m Z2 e4/8 ε02 h3 (1/n12 – 1/n22) υ̅ = 1/λ = m Z2 e4/8 ε02c h3 (1/n12 – 1/n22) since c =υλ υ̅ = RH (1/n12 – 1/n22) RH = Rydberg constant = m Z2 e4/8 ε02c h3 = 10973731.6 m-1 for H atom Using this equation, the wave numbers of photons of the various spectral series can be calculated. Table 2.1 shows the different spectral series of hydrogen with their n1 and n2 values. The shortest wavelength line any series occurs when n2 is infinitely large so that 1/n22 is zero. That is, if n2 = ∞, then 1/n22 = 0 Name of series Value of n1 Value of n2 Region Lyman 1 2, 3, 4…… U. V Balmer 2 3, 4, 5……. Visible Paschen 3 4, 5, 6…… I.R Brackett 4 5, 6, 7……. I.R Pfund 5 6, 7, 8…….. I.R Table 2.1 Different spectral series of hydrogen Problem 1. Calculate the ionization potential of the hydrogen atom The ionization potential of an atom or molecule is the energy needed to completely remove an electron from the atom or molecule in its ground state E = RH [1/n12 - 1/n22] n1 =1, n2 = ∞ = 2.179 x 10-11 erg 2. Calculate the value of the Rydberg constant of hydrogen atom in eV. RH = 10973731.6 m-1 for H atom and 1eV = 8.0655 x 105m-1 RH = 13.6 eV Fig.2.4 Energy levels for the hydrogen atom as calculated from the Bohr theory Defects of Bohr model Though, Bohr model could explain the stability of atoms and the origin of line spectrum, the model was not satisfactory for many observations. (1). It could not explain the fine spectrum- The spectrum with additional lines when observed with a high resolution instrument. (2). It could not explain the splitting of the spectral lines when the source emitting the spectral lines were placed in a magnetic field (Zeeman effect) or electric field (Stark effect) (3). The model could not explain why atoms react It is to be noted that Sommerfield later modified the Bohr model by adding elliptical orbits in addition to the circular orbits by which a more appropriate explanation was feasible.