Radius of n^{th} stationery orbit for Hydrogen atom.

Bohr's assumed that hydrogen atom consist of unipositive charge concentrated on a nucleus and unit
negative
charge concentrated on a nucleus and unit negative charge.

The electrons which revolve around the nucleus is balanced by centripetal force as shown in above
figure.
The electrostatic force of attraction between electron and nucleus is given by:

F_{e}= (1/4πε₀)(z_{e}^{2}/r^{2}) <------- (i)
where, ε₀ is permitivity of free space.
Let, 'm' be the mass of an electron moving with velocity 'v' then centripetal force is given by,
F_{c}= mv^{2}/r <------- (ii)
The electron revolve around the nucleus,
F_{e} = F_{c}
mv^{2}/r = (1/4πε₀)(z_{e}^{2}/r^{2})
mv^{2} = (1/4πε₀)(z_{e}^{2}/r) <------ (iii)
From bohr's quantization condition, we have
mvr= nh/2π
v= nh/2πmr <------- (iv)
putting the value of v in equation (iii)
m(nh/2πmr)^{2}= (1/4πε₀)(z_{e}^{2}/r)
[(nh)^{2}/πmr]= (ze^{2}/ε₀)
r= [ε₀. (nh)^{2}]/(ze^{2}πm)

For hydrogen atom
z=1
So,
r= (ε₀/πm) (nh/e)^{2}
<------- (vi)
This is the required equation for radius of n^{th} stationery orbit for
H atom.

Velocity of n^{th} stationery orbit for Hydrogen atom.

Bohr's assumed that hydrogen atom consists of unipositive charge concentrated on a nucleus and unit
negative charge.

The electrons which revolve around the nucleus is balanced by centripetal force as shown in above
figure.

F_{c}= mv_{n}^{2}/r_{n}
<------- (i)
The electrostatic force of attraction between electron and nucleus is given by,
F_{e}= (1/4πε₀)(z_{e}^{2}/r^{2}) <------- (ii)
If v_{n} be the velocity of electron in n^{th} stationery orbit, the
condition for electron to revolve in the circular orbit,
the electrostatic force of attraction between electron and nucleus is provided by necessary
centripetal force.
F_{e}= F_{c}
(1/4πε₀)(z_{e}^{2}/r^{2}) = mv_{n}^{2}/r_{n}
mv_{n}^{2} = (1/4πε₀)(z_{e}^{2}/r_{n}) <------ (iii)
From bohr's quantization condition, the angular momentum of an electron is given by,
mv_{n}r_{n}= nh/2π <------- (iv)
dividing equation (iii) by equation (iv) we get,
v_{n}/r_{n}=[(z_{e}^{2}2π)/(4πε₀r_{n}nh)]
v_{n}= (z_{e}^{2}2π/4πε₀r_{n}nh) <------ (v)

For hydrogen atom , z=1
so,
v_{n}= e^{2}/2nhε₀ <------ (vi)
which is required expression of velocity of an electron in n^{th}
stationery orbit for hydrogen atom.
v_{n}= (e^{2}/2nε₀).(1/n)
v_{n}= k. 1/n, where k=(e^{2}/2nε₀)
v_{n} ∝ 1/n
Hence, velocity of electron in an stationery orbit is inversely proportional
to principle quantum number and decreases with
ratio 1:2:3.

NOTE: v_{n}= (C/137).1/n
when n=1
v_{1}= C/137
which is velocity of electron in ground state.

Total Energy of an electron in n^{th} stationery orbit for
hydrogen atom:

The total energy of an electron in a given orbit is the sum of
Kinetic energy K.E. and Potential energy P.E. Let, E_{k}
and E_{p} be the kinetic and potential energy of an electron
from total energy of an electron is given by,

E_{n}= E_{k} + E_{p}
<------- (i)
Let v_{n} be the velocity of an electron in an
stationery orbit then K.E. of an electron is
given by,
E_{k}= (1/2)mv_{n}^{2}
<-------(ii)
where 'm' is mass of electron but,
mv_{n}^{2}=
(1/4πε₀)(z_{e}^{2}/r_{n}) <------
(iii)
E_{k} =
1/2(1/4πε₀)(z_{e}^{2}/r_{n})
<------- (iv)
The potential energy of an electron is the amount of
work done to bring an electron from infinity to
r_{n} i.e.
E_{p}= _{∞}∫^{rn}
F_{e}. dr
E_{p}= _{∞}∫^{rn}
(1/4πε₀)(z_{e}^{2}/r^{2}).dr
E_{p}=(z_{e}^{2}/4πε₀)
_{∞}∫^{rn}
r^{-2}.dr
E_{p}=
-(1/4πε₀)(z_{e}^{2}/r_{n}>)
<---------- (v)
By using equation (iv) and equation (v) in
equation (i) we get,
E_{n}=
1/2(1/4πε₀)(z_{e}^{2}/r_{n})
-(1/4πε₀)(z_{e}^{2}/r_{n}>)
(1/4πε₀)(z_{e}^{2}/r_{n})
[(1/2)-1]
i.e. E_{n}=
z_{e}^{2}/8πε₀r_{n}
<------ (vi)
But r_{n}=
(n^{2}h^{2}ε₀)/mπz_{e}^{2}
So, equation (vi) becomes,
E_{n}=
-z_{e}^{2}/(8πε₀)x[(n^{2}h^{2}ε₀)/mπz_{e}^{2}]

For Hydrogen atom z=1 so,
E_{n}=
(-me^{4})/(8ε₀^{2}n^{2}h^{2})

It is the required expression for total
energy of an electron in n^{th}
stationery orbit for hydrogen atom
where '-ve' sign indicates that electron are
bonded by nucleus.
E_{n}=
-[(me^{4}/8ε₀^{2}h^{2})].1/n^{2}
where,
[(me^{4}/8ε₀^{2}h^{2})]
is constant for all.

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