What is the Wavelength( in nm) of the least energetic spectral line in the infrared series of the H atom?
I don't understand how to find the least energetic spectral line?
I don't know if the question wants the least energetic line (IN THE INFRARED) or the least energetic line (where ever that may be) in the spectrum in which in infrared line appears. Perhaps this link will help you.
http://en.wikipedia.org/wiki/Hydrogen_spectral_series
The 1094 nm line is an infrared line in my book.
To find the least energetic spectral line in the infrared series of the hydrogen atom, you need to determine the transition that corresponds to the lowest energy level change. In this case, we are looking at the infrared series, which is associated with electron transitions to the n = 2 energy level.
In the hydrogen atom, the energy of an electron in a specific energy level is given by the equation:
E = -13.6 eV / n^2
Where n is the principal quantum number.
To find the least energetic spectral line, we need to find the transition where the electron moves closest to the nucleus, so we would look for the transition from a higher energy level to the n = 2 energy level.
The formula for calculating the wavelength (λ) of a spectral line is:
λ = c / ν
Where c is the speed of light (approximately 3.00 x 10^8 m/s) and ν is the frequency of the spectral line.
Since we are interested in the infrared series, we know that the spectral lines will have frequencies less than that of visible light.
To find the least energetic spectral line, we need to find the transition with the lowest frequency, which corresponds to the smallest energy difference between energy levels. In this case, the transition from the n = 3 energy level to the n = 2 energy level will have the smallest energy difference.
To find the wavelength (in nm), we can use the relationship between frequency (ν) and wavelength (λ):
ν = c / λ
Rearranging the equation, we get:
λ = c / ν
Now we can substitute the values and calculate the wavelength for the least energetic spectral line in the infrared series of the hydrogen atom.
To find the least energetic spectral line in the infrared series of the hydrogen atom, we can use the Rydberg formula. The Rydberg formula relates the wavelength of a spectral line to the energy levels of the hydrogen atom.
The formula is given by:
1/λ = R * (1/n1^2 - 1/n2^2),
where λ represents the wavelength of the spectral line, R is the Rydberg constant, and n1 and n2 are positive integers representing different energy levels of the hydrogen atom. For the infrared series, the energy levels involved are in the range of n1 = 4 to n2 = ∞.
The least energetic spectral line corresponds to the transition between the energy levels with the highest and lowest values of n1 and n2, respectively. In this case, n1 = 4 and n2 = ∞.
Now, we need to substitute these values into the Rydberg formula to find the wavelength. The Rydberg constant for hydrogen is approximately 1.097 × 10^7 m^-1. However, the wavelength is usually given in nanometers (nm), so we need to convert the answer to the desired unit.
Let's plug in the values:
1/λ = (1.097 × 10^7 m^-1) * (1/4^2 - 1/∞^2).
Since 1/∞^2 is negligible compared to 1/4^2, we can consider it equal to zero.
1/λ ≈ (1.097 × 10^7 m^-1) * (1/4^2)
Simplifying further,
1/λ ≈ (1.097 × 10^7 m^-1) * (1/16)
≈ 6.85625 × 10^5 m^-1.
Now, we can calculate the wavelength by taking the reciprocal:
λ ≈ 1/(6.85625 × 10^5 m^-1)
Converting into nanometers (nm):
λ ≈ 1.459 × 10^-6 nm.
Therefore, the wavelength of the least energetic spectral line in the infrared series of the hydrogen atom is approximately 1.459 nm.