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Chapter 34: Problem 8

How many spectral lines are in the entire Balmer series?

Short Answer

Expert verified
Theoretically, there are infinite spectral lines in the entire Balmer series, but in practice, due to the series limit and experimental constraints, only a finite number can be observed.

Step by step solution

01

Understand spectral line generation

Spectral lines in a Hydrogen atom are generated when an electron transitions from a higher energy state to a lower energy state. The transition results in the emission of light, which corresponds to a spectral line. In the Balmer series, these transitions are into the second energy level (n=2).
02

Calculate possible transitions

The number of spectral lines corresponds to the number of possible transitions an electron can make into the second energy state. For an electron in the third energy state (n=3), there is 1 possible transition (3-->2). For an electron in the fourth energy state (n=4), there are 2 possible transitions (4-->3, 4-->2), and so on.
03

Understand the Balmer series limit

As the initial energy state before transition (n) approaches infinity, the frequency of the spectral lines approaches a limit. This is known as the series limit. This means that despite the energy state (n) theoretically stretching to infinity, the number of spectral lines does not. Due to this limit, there are a finite number of spectral lines in the Balmer series.
04

Sum up the transitions

To find out the total spectral lines in the Balmer series, sum up the transitions calculated in Step 2 till infinity, but considering the series limit. The series to sum up is 1+2+3+4+... and so on. This is an arithmetical progression which is infinite, but the Balmer series limit makes it finite. This infinite series sum can be calculated using the formula for the sum of natural numbers \(\frac{n(n+1)}{2}\), but considering the limit as \(n\) approaches infinity. As a result, the sum equals to infinity which means we have an infinite number of spectral lines. But practicality considering, we only observe a few due to constraints in experimental physics and the Balmer series limit.

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