Calculate the wavelength, in nanometers, of the spectral line produced when an electron in a hydrogen atom undergoes the transition from the energy level n = 4 to the level n = 1.v
To calculate the wavelength of the spectral line produced when an electron in a hydrogen atom undergoes a transition between two energy levels, you can use the Rydberg formula. The Rydberg formula is given by:
1/λ = R_H * (1/n_f^2 - 1/n_i^2)
Where:
- λ is the wavelength of the spectral line
- R_H is the Rydberg constant for hydrogen, approximately equal to 1.097 x 10^7 m^-1
- n_f is the final energy level (in this case, n = 1)
- n_i is the initial energy level (in this case, n = 4)
Let's substitute the values into the formula and calculate the wavelength:
1/λ = (1.097 x 10^7 m^-1) * (1/1^2 - 1/4^2)
1/λ = (1.097 x 10^7 m^-1) * (1 - 1/16)
1/λ = (1.097 x 10^7 m^-1) * (15/16)
1/λ = 1.028 x 10^7 m^-1
Now, we can find the wavelength (λ) by taking the reciprocal of both sides of the equation:
λ = 1/(1.028 x 10^7 m^-1)
λ ≈ 9.71 x 10^-8 m
To convert meters (m) to nanometers (nm), we multiply by 10^9:
λ ≈ 9.71 x 10^-8 m * 10^9 nm/m
λ ≈ 97.1 nm
Therefore, the wavelength of the spectral line produced when an electron in a hydrogen atom undergoes the transition from n = 4 to n = 1 is approximately 97.1 nanometers.
To calculate the wavelength of the spectral line produced when an electron in a hydrogen atom undergoes a transition from the energy level n=4 to the level n=1, we can use the Rydberg formula.
The Rydberg formula is given by:
1/λ = R * (1/n1^2 - 1/n2^2)
where λ is the wavelength, R is the Rydberg constant (1.0973731568539 x 10^7 m^-1), n1 is the initial energy level, and n2 is the final energy level.
In this case, n1 = 4 and n2 = 1. Plugging the values into the formula:
1/λ = (1.0973731568539 x 10^7 m^-1) * (1/4^2 - 1/1^2)
Simplifying the equation:
1/λ = (1.0973731568539 x 10^7 m^-1) * (1/16 - 1/1)
1/λ = (1.0973731568539 x 10^7 m^-1) * (1/16 - 1)
1/λ = (1.0973731568539 x 10^7 m^-1) * (1 - 16)/16
1/λ = (1.0973731568539 x 10^7 m^-1) * (-15/16)
Multiplying both sides by λ:
λ = 16/(1.0973731568539 x 10^7 m^-1) * (-15/16)
λ = 15/(1.0973731568539 x 10^7 m^-1)
To convert the wavelength to nanometers, we multiply by 10^9:
λ = 15/(1.0973731568539 x 10^7 m^-1) * 10^9 nm/m
λ = (15 * 10^9 nm) / (1.0973731568539 x 10^7 m)
λ ≈ 1368.47 nm
Therefore, the wavelength of the spectral line produced is approximately 1368.47 nanometers.