A hydrogen atom initially in the ground level absorbs a photon,which excites it to the n = 4 level. Determine the wavelength andfrequency of photon.

To determine the wavelength and frequency of the photon, we can use the equation:

E = hc/λ

where E is the energy of the photon, h is Planck's constant (6.626 x 10^-34 J·s), c is the speed of light (3.00 x 10^8 m/s), and λ is the wavelength of the photon.

First, we need to find the energy difference between the n = 4 and ground levels of the hydrogen atom. The energy difference can be calculated using the formula:

ΔE = E_final - E_initial

where E_final is the energy of the n = 4 level and E_initial is the energy of the ground level.

The energy of a hydrogen atom in the nth energy level can be given by the formula:

E = -13.6 eV/n^2

where eV is electron volts.

For the ground level (n = 1), the energy is:

E_initial = -13.6 eV/1^2 = -13.6 eV

For the n = 4 level, the energy is:

E_final = -13.6 eV/4^2 = -0.85 eV

Now, we can calculate the energy difference:

ΔE = E_final - E_initial = -0.85 eV - (-13.6 eV) = 12.75 eV

To convert the energy difference from electron volts to joules, we can use the conversion factor:

1 eV = 1.602 x 10^-19 J

So, the energy difference in joules is:

ΔE = 12.75 eV x 1.602 x 10^-19 J/eV = 2.04 x 10^-18 J

Now, we can use the equation E = hc/λ to find the wavelength of the photon:

λ = hc/ΔE = (6.626 x 10^-34 J·s)(3.00 x 10^8 m/s)/(2.04 x 10^-18 J) = 9.71 x 10^-7 m

Therefore, the wavelength of the photon is 9.71 x 10^-7 meters.

To find the frequency of the photon, we can use the equation:

c = λν

where c is the speed of light and ν is the frequency of the photon.

Rearranging the equation, we get:

ν = c/λ = (3.00 x 10^8 m/s)/(9.71 x 10^-7 m) = 3.09 x 10^14 Hz

Therefore, the frequency of the photon is 3.09 x 10^14 hertz.