Final answer:
To calculate the energy of a photon needed to move an electron between two energy levels, you can use the equation E = hc/λ. For example, when an electron falls from the n=5 to the n=3 level in a helium ion, the energy of the emitted photon is approximately 2.99 x 10^-19 joules.
Step-by-step explanation:
To calculate the energy of a photon needed to move an electron between two energy levels, you 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.
For example, let's calculate the energy of the photon when an electron falls from the n=5 to the n=3 level in a helium ion (Z=2 for He+). The transition corresponds to an emission of light, so we need to calculate the energy of the emitted photon.
First, find the wavelength of the emitted light:
The electron transition can be represented as 5→3. Using the formula for the wavelength of the photon, we can calculate:
1/λ = RZ^2(1/nf^2-1/ni^2)
where R is the Rydberg constant (1.097 x 10^7 m^-1), Z is the atomic number, and nf and ni are the final and initial energy levels, respectively.
Plugging in the values:
1/λ = (1.097 x 10^7 m^-1)(2^2)(1/5^2 - 1/3^2)
This gives us 1/λ = 1.5 x 10^6 m^-1.
So, λ = 1/(1.5 x 10^6 m^-1) = 6.67 x 10^-7 m.
Now, calculate the energy of the photon:
E = hc/λ = (6.626 x 10^-34 J*s)(3.00 x 10^8 m/s)/(6.67 x 10^-7 m) = 2.99 x 10^-19 J.
Therefore, the energy of the photon is 2.99 x 10^-19 joules.