Question

Using postulates of bohr's theory of hydrogen atom show that the radii of orbit increases as n²

Answer 1

According to Bohr's theory

Angular momentum= m × v × r = nh/2π

v = nh/ 2πmr

Squaring both sides

v² = n²h²/ 4 π²m²r²

We know that, centripetal force= electrostatic force

mv²/r = e²/ 4π∈₀r²

v² = e²/ 4π∈₀rm

∴ n²h²/ 4π²m²r² = e²/ 4π∈rm

  n²h²/ πmr = e²/∈₀

∴ r = n²h²∈₀/πme²

So we see radius increases with n²

n = principal quantum number

h = Planck's constant

r = radius

∈₀ = electrical permittivity of free space

e = charge on electron

m = mass

v = velocity

We know,

mvr =  nh/ 2π

v = nh/ 2πmr

Replacing the value of of r we get,

v = nhπme²/ 2n²h²∈₀ m

v = πe²/ 2nh∈₀

We know,

E = mv²

E = mπ²e⁴/ 4n²h²∈₀²

∴ We see that energy varies with 1/n²

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