According to Rutherford model of an atom, atom is made of outer electron cloud (-vely charged electron) and inner nucleus (neutral neutron and +vely charged proton).Niels Bohr agreed with this concept , he also agreed that electron revolve round the nucleus like a planet moves round sun. But when he applied the Plank's quantum theory to the electron revolving round the nucleus , he found some important facts that are disscused below.
Postulates of Bohr's theory--
1- An electron revolves around the nucleus in an orbit (like planet revolve around the sun) with a constant energy. During this motion electron neither loses nor gains energy.
In othert word the energy of an electron remains constant in a perticular orbit, it means each orbit is associoated with a definite energy i.e. with a definite whole number of quanta of energy.The orbits , there fore called as energy levels or energy shells. Bohr gave numbers 1, 2, 3, ...etc to these energy levels, there are they now called as Principal quantum numbers.these energy levels are also designted by k, l, m, n, etc...
The energy levels which are more far from the nucleus associated with greater amount of energy.2- Energy is emmited or absorbed by an atom only when an electron moves from one orbit(energy level) to another.
3- The angular momentum of an electron moving round the nucleus is quantised.
Angular momentum = mvr = n ( h/2π )( where n is any integer number( 1, 2, 3, ...etc ) and m is mass, v is linear velocity of electron , while r is the radius of the orbit.)
Thus the angular momentum of any electron may be h/2π or a whole number multiple of h/2π , 2h/2π , 3h/2π....nh/2π . This is called quantisation of angular momentum.
Bohr' equation--
Bohr was able to calculate energy of electron moving in different orbits round the nucleus in Hydrogen atom.
Electron contineus to remain in its orbit because the electrostatic force of attraction exerted by nucleus is balanced by the centrifugal force arising from its circular motion.
if the charge on the hydrogen atom nucleus = +Ze.
For hydrogen Z = 1 and hydrogen like atom such as He⁺ and Li⁺⁺ ,
Z= 2 and 3 respectively.
The electron with charge -e revolves round the nucleus in an orbit of radius r,with the tangential velocity v . If mass of electron is m
Then ,
Electrostatic force acting on electron by the nucleus = (Ze )x e/r² = Ze²/r²
centrifugal force acting on electron arising by circular motion = mv²/r
Both the forces balance to the electron during its motion thus,
mv²/r = Ze²/r² --------(1)
v²=Ze² / mr --------(2
according to bohr's postulate , angular momentum = mvr = n ( h/2π )
v = nh/2πmr --------(3)
This equation is called velocity of n electron.
Ze²/mr = n²h²/4π ²m²r² ----------(5)
Ze² = n²h²/4 π²mr ----------(6)
hence radius of nth orbit is = rn = n²h²/4 π²m Ze²
for H atom Z = 1 then ,
rn = n²h²/4 π²m e² ----------(7)
for H atom n = 1 in ground state , the radius r is designated as Bohr's radius (aₒ)
a ₒ = h²/4 π²m e² ----------(8)
The total energy of the revolving electron = k.energy =1/2mv²)+pot. Energy (-Ze²/r)
E = 1/2mv² + (-Ze²/r) ----------(9)
E = 1/2mv² - Ze²/r ----------(10)
thus ,
1/2mv² = Ze²/r ----------(11)
substituting the value of mv² in above equation
E = Ze²/2r - Ze²/r = -Ze²/2r ----------(12)
After substituting the value of r = n²h²/4 π²m Ze² we get ,
E = -Ze²/2 x 4 π²m Ze²/n²h² ---------(13)
Thus the energy of electron in nth orbital is ........
En = 2π²m Z²e⁴ / n²h² ----------(14)
This equation is Bohr equation . this equation is applicable to hydrogen and hydrogen like atoms such as He+ , Li++ , Be+++, etc.
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