Explanation Given data: Charge in electron, q = e = 1.6 × 10 − 19 C Charge in typical nucleus, q n = + 20 e The wavelength λ of the electron in terms of the nuclear radius R nuc is λ = 4 R nuc Formula used: The relativistic momentum p of an object of mass m and velocity v is: p = γ v m v Where, γ v is the Lorentz factor, given by γ v = 1 1 − ( v c ) 2 Where, c is the speed of light in vacuum. The relativistic kinetic energy of a body of mass m and velocity v is: K E = ( γ v − 1 ) m c 2 De Broglie's law states that the wavelength λ of an object is inversely related to its momentum P λ = h p Here, h being Planck's constant ( 6.63 × 10 − 34 J ⋅ s ) Calculation: The wavelength λ of the electron in terms of the nuclear radius R nuc λ = 4 R muc Substitute, P = m v in De Broglie's law states that the wavelength λ equation λ = h m v 4 R n u c = h m v v = h 4 m R n u c Divide each side by c v c = h 4 m c R nuc Substitute, ( 6.63 × 10 − 34 J ⋅ s ) for h , 9.11 × 10 − 31 kg for m , 3 × 10 8 m / s forc and 4 × 10 − 15 m for R nuc : v c = h 4 m c R n u c = ( 6.63 × 10 − 34 J ⋅ s ) 4 ( 9.1 × 10 − 31 kg ) ( 3 × 10 8 m s ) ( 4 × 10 − 15 m ) = 152 The velocity electron more than 150 times the speed of light, which is not possible So, we use the relativistic equation for kinetic energy. De Broglie's law states that the wavelength λ , with relativistic momentum. λ = h m v γ v = h m v 1 1 − ( v c ) 2 m v λ = h 1 − ( v c ) 2 m 2 v 2 λ 2 = h 2 [ 1 − v 2 c 2 ] m 2 v 2 λ 2 = h 2 − h 2 v 2 c 2 v 2 ( m 2 λ 2 + h 2 c 2 ) = h 2 v 2 ( m 2 c 2 λ 2 + h 2 c 2 ) = h 2 ( v 2 c 2 ) ( h 2 + m 2 c 2 λ 2 ) = h 2 ( v c ) 2 = h 2 h 2 + m 2 c 2 λ 2 Substitute ( v c ) 2 valueinto the Lorentz factor γ v = 1 1 − ( v c ) 2 = 1 1 − ( h 2 h 2 + m 2 c 2 λ 2 ) = 1 ( h 2 + m 2 c 2 λ 2 h 2 + m 2 c 2 λ 2 ) − ( h 2 h 2 + m 2 c 2 λ 2 ) = h 2 + m 2 c 2 λ 2 m 2 c 2 λ 2 = h 2 m 2 c 2 λ 2 + m 2 c 2 λ 2 m 2 c 2 λ 2 = h 2 m 2 c 2 λ 2 + 1 = 1 + ( h m c λ ) 2 Substituting the value of λ = 4 R muc γ v = 1 + ( h m c [ 4 R n u c ] ) 2 = 1 + ( h 4 m c R n u c ) 2 Then just insert that ( 6

2nd Edition

ISBN: 9780805303087

Author: Randy Harris

Publisher: Addison Wesley

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Chapter 4, Problem 73CE

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To find:Kinetic energy of electron standing wave confined in nucleus. And show that electron would escape the nucleus.

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Chapter 4, Problem 72CE

Chapter 5, Problem 1CQ

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