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What is the third-longest wavelength in the absorption spectrum of hydrogen?

Short Answer

Expert verified

The wavelength of the third longest wave is 97.3 nm

Step by step solution

01

Given information

The absorption line for hydrogen gas containing atoms in the ground state has the longest wavelength of 121.6nm.

02

Calculations

Our goal is to discover the hydrogen atom's absorption spectrum's third longest wavelength.

Assume we have a quantized system, such as the hydrogen atom, which has discrete energy levels rather than a continuum. When an electron absorbs a photon of a specific energy, it makes a quantum jump from a lower energy level Eito a higher energy level Ej. The amount of energy required to excite an electron from its ithto its jthstate is,

Eij=ΔEij=EjEiwhereEj>Ei=hcλijλij=hcEjEi(1)

Eijdenotes that the quantized system excites from the Eistate to the Ejstate by absorbing an amount of energy equal to role="math" localid="1650736975140" Eij. We may deduce from (1) that λijis inversely proportional to ΔEij, and since ΔE21<ΔE31<ΔE41λ12>λ13>λ14

As a result, the third longest wavelength is λ14. The energy of a particular state n for the hydrogen atom where Z=1 is calculated using Eq. (38.38).

En=13.6eVn22

We get by swapping (2) for (1),

λij=hc13.6eVj2+13.6eVi2wherej>iλ14=4.1357×1015eV.s×3×108m.s113.6eV42+13.6eV12=9.73×108m=97.3nm

The wavelength of the third longest wave is 97.3 nm

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