What is the wavelength of light (in nanometers, nm) emitted when the electron in a hydrogen atom moves from the n=5 energy level to the n=3 energy level?
1/λ = R(1/n^2 - 1/m^2)
R = 1.1•10^7 m^-1 is the Rydberg constant
n = 3, m = 5
λ= 7.99•10^-8 =79.9 nm
To find the wavelength of light emitted when an electron in a hydrogen atom moves from one energy level to another, we can use the Rydberg formula.
The Rydberg formula is given by:
1/λ = R * (1/n₁² - 1/n₂²)
Where:
- λ is the wavelength of light emitted
- R is the Rydberg constant (1.097 x 10^7 m⁻¹)
- n₁ is the initial energy level
- n₂ is the final energy level
In this case, the electron is moving from n=5 to n=3. By plugging these values into the formula, we can solve for the wavelength.
1/λ = (1.097 x 10^7 m⁻¹) * (1/5² - 1/3²)
1/λ = (1.097 x 10^7 m⁻¹) * (1/25 - 1/9)
1/λ = (1.097 x 10^7 m⁻¹) * (9/225 - 25/225)
1/λ = (1.097 x 10^7 m⁻¹) * (-16/225)
1/λ = -0.0776 x 10^7 m⁻¹
Now, we just need to take the reciprocal of both sides to solve for λ:
λ = 1/(-0.0776 x 10^7 m⁻¹)
λ = -12.9 x 10⁻⁸ m
To convert this value to nanometers (nm), we multiply it by 10⁹:
λ = (-12.9 x 10⁻⁸ m) * (10⁹ nm/1 m)
λ ≈ -12.9 nm
Since the wavelength of light cannot be negative, we take the magnitude of the value:
λ ≈ 12.9 nm
Therefore, the wavelength of light emitted when the electron in a hydrogen atom moves from the n=5 energy level to the n=3 energy level is approximately 12.9 nm.