Question

A hydrogen atom transitions from the ni =4 to the nf =2 state. What wavelength (in nm ) of light is emitted? A hydrogen atom transitions from the ni =5 to the nf =3 state. What wavelength (in nm ) of light is emitted?

Answer 1

For the hydrogen atom transitioning from ni = 4 to nf = 2, the wavelength of emitted light can be calculated using the Rydberg formula. The result is approximately 486 nm. Similarly, for the transition from ni = 5 to nf = 3, the calculated wavelength is approximately 656 nm.

Step-by-step explanation:

The wavelength of light emitted during a transition in a hydrogen atom can be determined using the Rydberg formula:

`\[ (1)/(\lambda) = R_H \left( (1)/(n_f^2) - (1)/(n_i^2) \right) \]`

Here,
`\( R_H \)` is the Rydberg constant for hydrogen
`(\( R_H \approx 1.097 * 10^7 \, \text{m}^(-1) \)), \( \lambda \)` is the wavelength,
`\( n_f \)` is the final energy level, and
`\( n_i \)` is the initial energy level.

For the transition from ni = 4 to nf = 2:

`\[ (1)/(\lambda) = 1.097 * 10^7 \left( (1)/(2^2) - (1)/(4^2) \right) \]`

`\[ \lambda \approx 486 \, \text{nm} \]`

For the transition from ni = 5 to nf = 3:

`\[ (1)/(\lambda) = 1.097 * 10^7 \left( (1)/(3^2) - (1)/(5^2) \right) \]`

`\[ \lambda \approx 656 \, \text{nm} \]`

These calculations provide the wavelengths of light emitted during the specified transitions. The results align with the principles of atomic spectroscopy and the quantized nature of electron energy levels in atoms. The emitted light corresponds to specific spectral lines, and the calculated wavelengths represent the characteristic colors associated with these transitions.