What is the ratio of speed of an electron in the first orbit of a hydrogen atom to the speed of light?

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NEET 2023 - Target Batch - Aryan Raj Singh

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`(c)/(1.37)``(c)/(1370)``(c)/(13.7)``(c)/(137)`

Answer : D

Solution : Velocity of electron in the 1st orbit, `v=h//(2pimr)=2.189xx10^(8)cm//sec`, velocity of light `c=3xx10^(10)` cm/sec. <br> Ratio `c//v=137`

The ratio of speed of an electron in the ground state in the Bohr's first orbit of hydrogen atom to velocity of light (c) is ____________.

(h = Planck's constant, ε0 = permittivity of free space, e = charge on electron)

• `(2"e"^2ε_0)/("hc")`

• `"e"^3/(2ε_0"hc")`

• `"e"^2/(2ε_0"hc")`

• `(2ε_0"hc")/"e"^2`

The ratio of speed of an electron in the ground state in the Bohr's first orbit of hydrogen atom to velocity of light (c) is `underline("e"^2/(2ε_0"hc"))`.

Explanation:

The speed of an electron in the hydrogen atom's nth state is

vn = `("e"^2/(2"h"ε_0)) 1/"n" "m"//"s"`

For ground state (n = 1) v1 = `"e"^2/(2"h"ε_0)` m/s

∴ `"v"_1/"c" = "e"^2/(2ε_0"hc")`

Concept: Bohr’s Atomic Model

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