   Chapter 30, Problem 23PE

Chapter
Section
Textbook Problem

Verify Equations r n = n 2 Z a B and a B = h 2 4 π 2 m e k q e 2 = 0.529 × 10 − 10 m using the approach stated in the text. That is, equate the Coulomb and centripetal forces and then insert an expression for velocity from the condition for angular momentum quantization.

To determine

To prove:

The equation:rn=n2/Z(aB) , whereaB=h2/4Π2mekqe2=0.529×1010 m.

Explanation

Given info:

Equate the coulomb and centripetal forces and insert the value for the velocity from the condition of quantization of angular momentum.

Formula used:

For Coulombs force:kZqe2/rn2

For Centripetal force:mev2/rn

The equationrn=n2/Z(aB) andaB=h2/4Π2mekqe2 are used to calculate the radii of allowed (quantized) electron orbits in the hydrogen like atoms. For example: for Hydrogen, Z= 1, r1 =aB.

Here,aB=h2/4Π2mekqe2 is called the 'Bohr Radius'.

Proof:

Equating the equations for coulomb interaction and centripetal force together, we get:

kqe2/rn2=mev2/rnrn=kZqe2/mev2rn=kZqe2/me(1/v2)

From:mevrn=n(h/2Π) (Quantization of angular momentum)

rn=(kZqe2/me)×(4Π2me<

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