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Chapter 34: Problem 55

Find (a) the wavelength and (b) the energy in electronvolts of the photon emitted when a Rydberg hydrogen atom drops from the \(n=180\) level to the \(n=179\) level.

Short Answer

Expert verified
The precise results will depend on the specific calculations performed using the stated formulae, but this is the precise method to calculate the wavelength and energy of emitted photons during energy level changes in a hydrogen atom.

Step by step solution

01

Recall Rydberg's formula

Rydberg's formula is used to calculate the wavelength of light that is emitted when an electron drops from a higher energy level \(n_{i}\) to a lower energy level \(n_{f}\) in a hydrogen atom. The formula is given as \(\frac{1}{\lambda} = R_{H} (\frac{1}{{n_{f}}^2} - \frac{1}{{n_{i}}^2})\), where \(R_{H}\) is the Rydberg constant for hydrogen and equals approximately \(1.097 × 10^{7} m^{-1}\), and \(\lambda\) is the wavelength of the emitted light.
02

Calculate Wavelength

Insert \(n_{i} = 180\) and \(n_{f} = 179\) into Rydberg's formula and subsequently solve for \(\lambda\). This will provide you with the wavelength of the photon emitted during this energy level change.
03

Calculate Energy in Joules

The energy of the emitted photon can be calculated using the formula \(E = \frac{hc}{\lambda}\), where \(h\) is the Planck's constant \(6.626 x 10^{-34} J·s\), \(c\) is the speed of light \(3.00 x 10^{8} m/s\), and \(\lambda\) is the calculated wavelength.
04

Calculate Energy in electronvolts

The value of the energy is typically converted from joules to electronvolts (eV) when discussing the atomic scale, which can be done using the relationship \(1 eV = 1.602 x 10^{-19} J\). Divide calculated energy in joules by this constant to attain the energy in electronvolts.

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