Calculate the wavelength, λ, in meters of a photon capable of exciting an electron in from the ground state to n = 4.
To calculate the wavelength, λ, of a photon capable of exciting an electron from the ground state to a higher energy level (in this case, n = 4), you can use the Rydberg Formula:
1/λ = R * (1/n1² - 1/n2²)
Where:
- λ is the wavelength of the photon in meters,
- R is the Rydberg constant (approximately 1.097 × 10^7 m⁻¹),
- n1 is the initial energy level (in this case, the ground state, which is n1 = 1),
- n2 is the final energy level (in this case, n2 = 4).
First, plug in the values:
1/λ = 1.097 × 10^7 m⁻¹ * (1/1² - 1/4²)
Now calculate the fraction within the parentheses:
1/λ = 1.097 × 10^7 m⁻¹ * (1 - 1/16)
Simplify the fraction:
1/λ = 1.097 × 10^7 m⁻¹ * (15/16)
Multiply the terms:
1/λ = 1.022 × 10^7 m⁻¹
To find the value of λ, take the reciprocal of both sides:
λ = 1/(1.022 × 10^7 m⁻¹)
Simplify:
λ = 9.78 × 10⁻⁸ meters
Therefore, the wavelength of the photon capable of exciting the electron from the ground state to n = 4 is approximately 9.78 × 10⁻⁸ meters.
1/wavelength = R(1/n^2 - 1/n^2)
R = 1.09737E9
The first n above is 1 and the second n is 4.