   Chapter 30, Problem 24PE

Chapter
Section
Textbook Problem

The wavelength of the four Balmer series lines for hydrogen are found to be 410.3, 434.2, 486.3, and 656.5 nm. What average percentage difference is found between these wavelength numbers and those predicted by 1 λ = R ( 1 n f 2 − 1 n i 2 ) ? It is amazing how well a simple formula (disconnected originally from theory) could duplicate this phenomenon.

To determine

What average percentage difference is found between these wavelength numbers and those predicted by1λ=R(1nf21ni2) ?

Explanation

Given Data:

The wavelength of the four Balmer series lines for hydrogen are found to be410.3 ,434.2 ,486.3 and656.5 nm.

Formula Used:

The expression to find the wavelength is as follows:

1λ=R(1nf21ni2)

Whereλ= Wavelength

R= Rydberg Constant

Calculation:

For Balmer series,

nf=2

Case 1: Forni=3

SubstituteR=1.097×107m1 ,nf=2 andni=3.

Wavelength is calculated as

1λ=(1.097×107m1)(122132)

1λ=(1.097×107m1)(1419)

1λ=1.524×106m1

λ=656.3nm

Also, the known value is656.5nm

Thus, percentage difference=656.5nm656.3nm656.5nm×100=0.030%

Case 2: Forni=4

SubstituteR=1.097×107m1 ,nf=2 andni=4.

Wavelength is calculated as

1λ=(1.097×107m1)(122142)

1λ=(1.097×107m1)(14116)

1λ=2.057×106m1

λ=486

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