The Rydberg constant for Hydrogen is $1.097\times10^7$ m$^{-1}$. Calculate the short wavelength limit of Lyman series.

The Rydberg constant for Hydrogen is 1.097×10⁷ m⁻¹. Calculate the short wavelength limit of Lyman series.

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1.097\times10^7

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Concept understanding — Hydrogen Spectral Series

When an individual gaseous element such as hydrogen is excited (by heating or an electrical discharge) and its emitted light is examined through a spectrometer, the result is not a smooth, continuous rainbow of colours but rather a set of sharp, discrete bright lines at very specific wavelengths - a line spectrum, unique to that particular element, functioning almost like a fingerprint that lets scientists identify a gas's composition from its light alone (including the composition of distant stars and the Sun).

For hydrogen specifically, every observed spectral-line wavelength is given by a single Rydberg-type formula,

\frac{1}{\lambda}=R\left(\frac{1}{n^2}-\frac{1}{m^2}\right),\qquad m>n\ \text{positive integers}

where $R=1.09737\times10^7\ \text{m}^{-1}$ is the Rydberg constant. Fixing the lower level $n$ and letting $m$ range over every larger integer generates one complete named series of spectral lines: $n=1$ gives the Lyman series (ultraviolet region); $n=2$ gives the Balmer series (visible region - the only series bright enough to be seen directly with the eye); $n=3$ gives the Paschen series (near infrared); $n=4$ gives the Brackett series (mid infrared); and $n=5$ gives the Pfund series (far infrared). Within any one series, successive lines (as $m$ increases) crowd progressively closer together, approaching a limiting wavelength - the series limit - as $m\to\infty$ .

This same formula also lets the total number of distinct spectral lines an excited hydrogen sample can emit be counted: if an electron has been excited up to level $n$ , it can subsequently make a transition down to any lower level, and the total number of distinct transitions (and hence distinct emitted wavelengths) possible works out to $n(n-1)/2$ . …

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The Lyman-series short-wavelength limit is the wavelength corresponding to a transition from $n=\infty$ to $n=1$ ; substituting $n\to\infty$ into the Rydberg formula gives $\lambda=1/R$ , which evaluates to about $912\,\text{Å}$ .

The Rydberg formula for hydrogen

The wave number (reciprocal of wavelength) of the radiation emitted when an electron in a hydrogen atom jumps from an outer level $n$ to an inner level $n_1$ is given by

$\frac1\lambda=R\left(\frac1{n_1^2}-\frac1{n^2}\right)$

where $R=1.097\times10^7\,m^{-1}$ is the Rydberg constant for hydrogen.

The Lyman series

The Lyman series corresponds to all transitions that terminate on the ground state, $n_1=1$ , with the initial level $n=2,3,4,\ldots$ . These transitions lie in the ultraviolet region and have the formula

$\frac1\lambda=R\left(\frac1{1^2}-\frac1{n^2}\right),\qquad n=2,3,4,\ldots$

Short-wavelength (series) limit

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Q.The Rydberg constant for Hydrogen is 1.097×1071.097\times10^71.097×107 m−1^{-1}−1. Calculate the short wavelength limit of Lyman series.

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Q.The Rydberg constant for Hydrogen is $1.097\times10^7$ m $^{-1}$ . Calculate the short wavelength limit of Lyman series.

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