Boher.docx
- 1. The Bohr Model of the Hydrogen Atom Niels Bohr suggested a model for the hydrogen atom that predicted the existence of linespectra. he was ableto explain the spectrum of radiation emitted by hydrogen atoms in gas-dischargetubes. Based on the work of Planck and Einstein,Bohr made the revolutionary assumption thatcertain properties of the electron in a hydrogen atom – including energy, can have only certain specific values. Bohr proposed the followingthree postulates for his model. The hydrogen atom has only certain allowableenergy levels,called stationary states.Each of these states is associated with a fixed circular orbitof the electron around the nucleus. The atom does not radiateenergy whilein one of its stationary states. That is,even though it violates the ideas of classical physics,the atom does not change energy whilethe electron moves within an orbit. The electron moves to another stationary state(orbit) only by absorbing or emitting a photon whose energy equals the difference in the energy between the two states. Eph = Ef – Ei = hv The Bohr radius,denoted by ao (ao = 0.0529 nm) can be calculated usingthe Formula ao = 𝑛 𝑟 where n is a positiveinteger which is called quantum number. r is the radius of the orbitand is given by: r = 𝑛∈ ℎ 𝜋𝑚 𝑒 𝑧 where ∈o is the vacuum dielectric constant(∈o = 8.854 × 10–12 C V–1 m–1). A spectral lineresults fromthe emission of a photon of specific energy (and therefore, of specific frequency), when the electron moves from a higher energy state to a lower one. An atomic spectrum appears as lines rather than as a continuum becausethe atom’s energy has only certain discreteenergy levels or states. The lower the quantum number, the smaller is theradius of the orbitand the lower is the energy level of the atom. When the electron is in the orbitclosestto the nucleus (n = 1), the atom is in its lowest(first) energy level, which is called the ground state. By absorbinga photon whose energy equals the difference between the firstand second energy levels, the electron can move to the next orbit. This second energy level (second stationary state) and all higher levels are called excited states. ∆E = Ee – Eg where Eg and Ee represent the ground and the excited energy states, respectively. A larger orbitradius means a higher atomic energy level,the farther the electron drops,the greater is the energy (higher v, shorter ) of the emitted photon. Bohr ableto useclassical physicsto calculateproperties of the hydrogen atom. In particularand hederived an equation for the electron energy (En En = -RH Z2/n2 where RH is the Rydberg constant,has a valueof, RH = 2.18 × 10–18 J Z is the charge of the nucleus. ∆E= hv = Ef – Ei = -2.18 × 10–18 J( 1 𝑛 − 1 𝑛 ) Limitations of the Bohr Theory He failed to predictthe wavelengths of spectral lines of atoms more complicated than hydrogen, even that of helium,the next simplestelement. He could not explain the further splittingof spectral lines in thehydrogen spectra on application of magnetic field and electric fields. Exercise 1. Calculatethe energies of the states of the hydrogen atom with n = 2 and n = 3, and calculatethe wavelength of the photon emitted by the atom when an electron makes a transition between these two states. 2. What is the wavelength of a photon emitted duringa transition fromthe ni = 5 state to the nf = 2 state in the hydrogen atom? 3. How much energy, in kilojoulesper mole, is released when an electron makes a transition fromn = 5 to n = 2 in an hydrogen atom? Is this energy sufficientto break the H– H bond (436 kJ / mol is needed to break this bond)? 4.What is the characteristicwavelength of an electron (in nm) that has a velocity of 5.97 × 106 m s–1 (me = 9.11 × 10–31 kg)? 5. Calculatethe energy required for the ionization of an electron from the ground state of the hydrogen atom. 6. Calculatethe wavelength (in nm) of a photon emitted when a hydrogen atom undergoes a transition from n = 5 to n = 2. Quantum Numbers An atomic orbital is specified firstby three quantum numbers that are associated respectively,with the orbital's size(energy), shape, orientation and,later fourth quantum number is used to describethe spin of the electrons that occupy the orbital’s. There are four quantum numbers. These are: The principal quantumnumber (n) is a positiveinteger havingvalues n = 1,2, 3, ... . it Relativesizeof the orbital or the relativedistanceof the electron from the nucleus. The Size of orbital increases with the increaseof principal quantumnumber n. Higher the n value,greater is the energy. The Maximum number of electrons present in any shell is given by the formula 2n2. The azimuthal quantum number (l) is also known as angular momentum or subsidiary quantumnumber. It is an integer havingvalues from 0 to (n – 1).
- 2. For an orbital with n = 1, l can have a valueonly of 0. n = 2, l can have a valueof 0 or 1; n = 3, l can be 0, 1 or 2; For a given value of n, the maximum possible value of l is (n – 1). The azimuthal quantum number gives the followinginformation: o Number of sub shell present within any shell. o it describes the shapeof the orbital and is sometimes also called the orbital- shapequantum number. The magnetic quantum number (ml ) is also known as the orbital-orientation quantumnumber. It is an integer havingvalues from –l through 0 to +l. An orbital with l = 0 can have only ml = 0. l = 1, can have ml valueof –1, 0, or + 1; The number of possible ml values or orbital’s for a given l value is (2l + 1). The electron spin quantum number (ms ) has only two possiblevalues, +½ (represented by the arrow, ) and – ½ (represented by the arrow). The name electron spin quantum suggests that electrons have a spinningmotion. The quantum numbers specify the energy states of the atom. • The atom's energy levels or shells aregiven by the n value. • The atom's sublevels or subshells aregiven by the n and l values.Each level contains sublevels that designate the shapeof the orbital. • The atom's orbital’s arespecified by the n, l and ml values. Each sublevel is designated by a letter: l = 0, is an s sublevel l = 1, is a p sublevel l = 2, is a d sublevel l = 3, is a f sublevel The letters s, p, d, and f arederived from the names of spectroscopic lines: s, sharp;p, principal;d,diffuse;and f, fundamental. Sublevels are named by joining the n valueand the letter designation. Example:- the sublevel (sub shell) with n = 2, l = 0 is 2ssublevel; the only orbital in this sublevel has n = 2, l = 0 and ml = 0. The sublevel with n = 3, l = 1, is a 3p sublevel.It has three possibleorbital’s:onewith n = 3, l = 1 and ml = –1 ; another with n = 3, l = 1 and ml = 0 and the third n = 3, l = 1, and ml = +1. Number of orbital’s =n2 in a shell. Number of orbital’s in a sub shell =2l+1 Example:- What values of the angular momentum quantum number (l) and magnetic quantum number (ml ) areallowed for a principal quantumnumber (n) of 3? How many orbitals are allowed for n = 3? Solution: n = 3, l = 0, 1, 2 Determining ml for each l value: For l = 0, ml = 0 l = 1, ml = –1, 0, +1 l = 2, ml = –2, –1, 0, +1, +2 Number of orbital’s =n2, n = 3 and n2 = 32 = 9 orbital’s 3s : 1 orbital ,3p : 3 orbital’s and 3d : 5 orbital’s Total = 9 orbital’s