Hydrogen's H line is the Balmer transition

PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure · One-electron atoms and the origins of atomic structure

Reading the Spectrum, Ch The nuclear atom and the Bohr-Sommerfeld model

Problem

Hydrogen's H

\alpha

line is the

n=3\to n=2

Balmer transition. He

^+

has a line, from

n=6\to n=4

, that falls close to the same wavelength. Take the reduced-mass-corrected Rydberg constants

R_{\rm H}=109677.58\,\mathrm{cm^{-1}}

and

R_{\rm He}=109722.3\,\mathrm{cm^{-1}}

(NIST ASD), and use

\tilde\nu = Z^2R_M\left(\dfrac{1}{n_f^2}-\dfrac{1}{n_i^2}\right)

for each line. Chapter 1's teaching grating, used near H

\alpha

, resolves two lines as separate only if their vacuum wavelengths differ by at least

\Delta\lambda_{\min}=0.022\,\mathrm{nm}

. ((a)) Calculate the term values

T_n=Z^2R_M/n^2

for hydrogen's

n=2

and

n=3

levels and for He

^+

n=4

and

n=6

levels, all in

\mathrm{cm^{-1}}

. Explain why the He

^+

and hydrogen term diagrams nearly overlap at these particular levels, and say whether this is because He

^+

"has one electron, like hydrogen" or for a different, more specific reason. ((b)) Compute the vacuum wavelength of hydrogen's H

\alpha

line and of He

^+

n=6\to4

line (to three decimal places, in nm), and hence their wavelength separation

\Delta\lambda

. ((c)) Using the teaching grating's resolving limit, decide whether a hydrogen lamp contaminated with helium would show H

\alpha

and He

^+

n=6\to4

line as one blended line or as two resolved lines, and state the separation as a multiple of the grating's limit.

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