Physics for Scientists and Engineers
Physics for Scientists and Engineers
6th Edition
ISBN: 9781429281843
Author: Tipler
Publisher: MAC HIGHER
Question
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Chapter 36, Problem 38P

(a)

To determine

The probability of finding the electron in the region between r and r+Δr .

(a)

Expert Solution
Check Mark

Answer to Problem 38P

The probability of finding the electron in the region between r and r+Δr is 0.0162 .

Explanation of Solution

Given:

The radius of Bohr bolt is r=a0 .

The value of Δr is 0.3a0 .

Formula used:

The expression for the normalized ground state wave function for an atom with number Z is given by,

  ψ(r)=1π(Z a 0 )3/2eZra0

The expression for the radial probability density of finding a particle at a position r is given by,

  P(r)=4πr2|ψ(r)|2

The expression probability of finding the particle with region Δr is given by,

  P=P(r)Δr

Calculation:

The atomic number of hydrogen atom is 1 .

The normalized ground state wave function is calculated as,

  ψ(r)=1π( 1 a 0 )3/2e ( 1 )( a 0 ) a 0 =1π( 1 a 0 )3/2e1=1ea0 π a 0 ψ2(a0)=1e2a03π

The radial probability density is calculated as,

  P(r)=4πr2|ψ(r)|2=4π( a 0)2(1 e 2 a 0 3 π)=4e2a0

The probability of finding the particle with region Δr is calculated as,

  P=P(r)Δr=4e2a0(0.03a0)=4 ( 2.718 )2(0.03)=0.0162

Conclusion:

Therefore, the probability of finding the electron in the region between r and r+Δr is 0.0162 .

(b)

To determine

The probability of finding the electron in the region between r and r+Δr .

(b)

Expert Solution
Check Mark

Answer to Problem 38P

The probability of finding the electron in the region between r and r+Δr is 0.00879 .

Explanation of Solution

Given:

The radius of Bohr bolt is r=2a0 .

The value of Δr is 0.3a0 .

Formula used:

The expression for the normalized ground state wave function for an atom with number Z is given by,

  ψ(r)=1π(Z a 0 )3/2eZra0

The expression for the radial probability density of finding a particle at a position r is given by,

  P(r)=4πr2|ψ(r)|2

The expression probability of finding the particle with region Δr is given by,

  P=P(r)Δr

Calculation:

The atomic number of hydrogen atom is 1 .

The normalized ground state wave function is calculated as,

  ψ(r)=1π( 1 a 0 )3/2e ( 1 )( 2 a 0 ) a 0 =1π( 1 a 0 )3/2e2=1e2a0 π a 0 ψ2(a0)=1e4a03π

The radial probability density is calculated as,

  P(2a0)=4πr2|ψ(r)|2=4π(2 a 0)2(1 e 4 a 0 3 π)=16e4a0

The probability of finding the particle with region Δr is calculated as,

  P=P(r)Δr=16e4a0(0.03a0)=16 ( 2.718 )4(0.03)=0.00879

Conclusion:

Therefore, the probability of finding the electron in the region between r and r+Δr is 0.00879 .

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