Physics
Physics
5th Edition
ISBN: 9781260486919
Author: GIAMBATTISTA
Publisher: MCG
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Chapter 27, Problem 48P

(a)

To determine

The difference in radius between the n=1 and n=2 state for hydrogen.

(a)

Expert Solution
Check Mark

Answer to Problem 48P

The difference in radius between the n=1 and n=2 state for hydrogen is 1.59×1010m_.

Explanation of Solution

Write the expression for the radius of nth state of hydrogen atom.

  rn=n2a0                                                                                                                      (I)

Here, rn is the radius of the nth state orbit, and a0 is the Bohr radius (a0=52.9×1012m).

Use equation (I) to write the radius of the n=1 and n=2 state.

  r1=12a0                                                                                                              (II)

  r2=22a0                                                                                                           (III)

Write the expression for the difference in radius between the n=1 and n=2 state for hydrogen using equation (II) and (III).

  r2r1=22a012a0=3a0                                                                                           (IV)

Conclusion:

Substitute 52.9×1012m for a0 in equation (IV) to find the difference in radius between the n=1 and n=2 state for hydrogen.

  r2r1=3(52.9×1012m)=1.59×1010m

Therefore, the difference in radius between the n=1 and n=2 state for hydrogen is 1.59×1010m_.

(b)

To determine

The difference in radius between the n=100 and n=101 state for hydrogen, and the variation in orbital separation for large and small values of n.

(b)

Expert Solution
Check Mark

Answer to Problem 48P

The difference in radius between the n=100 and n=101 state for hydrogen is 1.06×108m_. The orbital separations are much larger for larger n values.

Explanation of Solution

Use equation (I) to write the radius of the n=100 and n=101 state.

  r100=1002a0                                                                                                       (V)

  r101=1012a0                                                                                                       (VI)

Write the expression for the difference in radius between the n=100 and n=101 state for hydrogen using equation (V) and (VI).

  r101r100=1012a01002a0=(10121002)a0                                                                               (VII)

Conclusion:

Substitute 52.9×1012m for a0 in equation (VII) to find the difference in radius between the n=100 and n=101 state for hydrogen.

  r101r100=(10121002)(52.9×1012m)=1.06×108m

This indicates that the orbital separation of hydrogen increases as the value of n increases. For small values of n, the orbital separation is very small. For larger values of n, the orbital separation is also large.

Therefore, the difference in radius between the n=100 and n=101 state for hydrogen is 1.06×108m_. The orbital separations are much larger for larger n values.

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Chapter 27 Solutions

Physics

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