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What is the radius of a hydrogen atom whose electron moves at7.3ร—105m/s ?

Short Answer

Expert verified

The radius of the hydrogen atom =0.476nm

Step by step solution

01

given information

An electron, in a certain state n, moves with a speed equals to 7.3ร—105m/saround the hydrogen's nucleus

02

simplification

Our aim is to find the radius of the hydrogen atom. An electron, moving in a circular orbit raround a certain nucleus, makes a standing waves where

2ฯ€rn=nฮป

Every particle, which has a certain momentum, has an associated wavelength where

ฮป=hp=hmv

By substituting (2) into (1), we get

2ฯ€rn=nhmcvnโ‡’rn=nh2ฯ€mcvn

From Fig.(38.27) in the book, we know that.

role="math" localid="1650742362891" rn=n2aB(whereaDis the bohr radius)

By equating (3) and (4), we get

role="math" localid="1650742483141" n2aB=yh2ฯ€mevnโ†’n=h2ฯ€mevnaHsโ‰ƒ3

Hence the radius of the hydrogen atom, in which an electron moves with a speed equals to 7.3ร—105m/sin the third state, is

role="math" localid="1650742623215" r3=32ร—5.29ร—10-2nm=0.476nm

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