QUESTION :
Prove the bohr radius for hg and find its quantized energy levels by using modern physics
Answer
Bohr's radius = 0.538x10-10m
Explanation
Centripetal force :
F= mv2/r
Electrostatic force :
F = 1/4πε * ze2 /r
In a hydrogen atom,
Electrostatic force = Centrifugal force
mv2/r = 1/4πε * ze2 /r
v2 = 1/4πε * ze2 /mr -> eq1
From Bohr's quantum equation,
L = mvr = nh'-> eq2
Where h' = h/2π
Get v from eq2,
v = nh'/mr
Then substitute v from eq1,
1/4πε * ze2 /mr = (nh'/mr)2 ; get the radius , r
r = 4πε (nh')2/ mze2 -> eq3
As 4πε = 1.9 x109
And h' = h/2π ; h = 6.625x10-34
The radius of Hydrogen Atom has ;
n=1
m = 9.11 ✕ 10-31kg,
z=1
e = 1.609x10-19 C
Substitute this values in the Bohr's radius formula; eq3
r = 4πε (nh')2/ mze2
= (1.9 x109(1)2(6.625x10-34/2π)2)/(9.11 ✕ 10-31)(1)(1.609x10-19)
r = 0.538x10-10m
Prove the bohr radius for hg and find its quantized energy levels by using modern physics
Answer
Bohr's radius = 0.538x10-10m
Explanation
Centripetal force :
F= mv2/r
Electrostatic force :
F = 1/4πε * ze2 /r
In a hydrogen atom,
Electrostatic force = Centrifugal force
mv2/r = 1/4πε * ze2 /r
v2 = 1/4πε * ze2 /mr -> eq1
From Bohr's quantum equation,
L = mvr = nh'-> eq2
Where h' = h/2π
Get v from eq2,
v = nh'/mr
Then substitute v from eq1,
1/4πε * ze2 /mr = (nh'/mr)2 ; get the radius , r
r = 4πε (nh')2/ mze2 -> eq3
As 4πε = 1.9 x109
And h' = h/2π ; h = 6.625x10-34
The radius of Hydrogen Atom has ;
n=1
m = 9.11 ✕ 10-31kg,
z=1
e = 1.609x10-19 C
Substitute this values in the Bohr's radius formula; eq3
r = 4πε (nh')2/ mze2
= (1.9 x109(1)2(6.625x10-34/2π)2)/(9.11 ✕ 10-31)(1)(1.609x10-19)
r = 0.538x10-10m
