Question

A long time ago, in a galaxy far, far away, electric charge had not yet been invented, and atoms were held together by gravitational forces. Compute the Bohr radius (a0) and the n = 5 to n = 4 transition energy (E5 − E4) in a gravitationally bound hydrogen atom.

a0 = ______ m

E5 − E4 = ______ eV

A long time ago, in a galaxy far, far away, electric charge had not yet been invented, and atoms were held together by gravitational forces. Compute the Bohr radius (ao) and the n = 5 to n = 4 transition energy (E5 - E4) in a gravitationally bound hydrogen atom. ao = E5 – E4 = eV Additional Materials eBook

Answer 1

Boh's postulate will be,

m*v*r = n*h/(2*pi)

now by energy conservation,

0.5*m*v^2 = k*e^2/(2*r)

0.5*m*(n*h/(2*pi*m*r))^2 = e^2/(8*pi*$\varepsilon$0*r)

r = [($\varepsilon$0*h^2)/(m*e^2*pi)]*n^2

For bohr radius,

n = 1

then, r = a0 = ($\varepsilon$0*h^2)/(m*e^2*pi)

a0 = [(8.85*10^-12)*(6.63*10^-34)^2]/[pi*(9.1*10^-31)*(1.60*10^-19)^2]

a0 = 0.0529 nm

given, gravitational Force(Fg) = Electrostatic force(Fe)

G*m*m'/r^2 = e^2/(4*pi*$\varepsilon$*r^2)

G*m*m' = e^2/(4*pi*$\varepsilon$)

So, a0 = h^2/[4*pi^2*G*m^2*m']

here,

h = plank's constant = 6.63*10^-34

m = mass of electron = 9.1*10^-31 kg

m' = mass of proton = 1.67*10^-27 kg

then, a0 = [(6.63*10^-34)^2]/[4*pi^2*(6.67*10^-11)*(9.1*10^-31)^2*(1.67*10^-27)]

a0 = 1.19*10^29 m

Since, E(n+1) - En = -[2*pi^2*m*e^4//h^2]*(1/(n+1)^2 - 1/n^2)

E(n+1) - En = -[2*pi^2*m*(G*m*m')^2/(h^2)]*(1/(n+1)^2 - 1/n^2)

here, n = 4

So, E5 - E4 = -(2*pi^2*(9.1*10^-31)*((6.67*10^-11)*(9.1*10^-31)*(1.67*10^-27))^2/(6.63*10^-34)^2)*(1/5^2 - 1/4^2)

E5 - E4 = 9.45*10^-99 J

E5 - E4 = (9.45*10^-99)/(1.60*10^-19)

E5 - E4 = 5.9*10^-80 eV

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