Suppose we detect red photons at 674 nanometers emitted by electrons dropping from the n = 3 to n = 2 orbital in hydrogen. The hydrogen is in an interstellar cloud at 5,000 K. If the cloud were heated to 10,000 K, what would be the wavelength of the photons emitted by the transition?
To determine the wavelength of the photons emitted by the electron transition from the n = 3 to n = 2 orbital in hydrogen, we can use the formula for the change in energy for an electron transition in the hydrogen atom:
ΔE = E_final - E_initial = -13.6 eV * (Z^2 / n_final^2 - Z^2 / n_initial^2)
Where:
ΔE is the change in energy
E_final is the energy of the final state
E_initial is the energy of the initial state
Z is the atomic number of the nucleus (which is 1 for hydrogen)
n_final is the principal quantum number of the final state
n_initial is the principal quantum number of the initial state
In this case, the initial state is n = 3, and the final state is n = 2. Plugging these values into the equation, we get:
ΔE = -13.6 eV * (1^2 / 2^2 - 1^2 / 3^2)
Now we can calculate the change in energy:
ΔE = -13.6 eV * (1/4 - 1/9)
= -13.6 eV * (9/36 - 4/36)
= -13.6 eV * (5/36)
= -1.89 eV
The energy change (ΔE) is negative because the electron is transitioning to a lower energy level.
Now, let's find out the wavelength of the red photons emitted by this transition. We can use the equation:
ΔE = h * c / λ
Where:
ΔE is the change in energy
h is Planck's constant (6.626 x 10^-34 J·s)
c is the speed of light (2.998 x 10^8 m/s)
λ is the wavelength of light
Rearranging the equation to solve for wavelength (λ), we have:
λ = h * c / ΔE
Plugging in the values:
λ = (6.626 x 10^-34 J·s * 2.998 x 10^8 m/s) / (-1.89 eV * 1.602 x 10^-19 J/eV)
≈ 656.46 nm
So, for the original transition in the interstellar cloud at 5,000 K, the wavelength of the photons emitted is approximately 656.46 nm (which falls in the red region of the visible spectrum).
To determine the wavelength of the photons emitted by the transition if the cloud were heated to 10,000 K, we can assume that the energies of the states are proportional to the temperature. Therefore, the change in energy for the transition will double compared to the original case at 5,000 K.
ΔE_new = 2 * ΔE_original
= 2 * (-1.89 eV)
Now, we can use the same formula to calculate the new wavelength:
λ_new = (6.626 x 10^-34 J·s * 2.998 x 10^8 m/s) / (2 * (-1.89 eV) * 1.602 x 10^-19 J/eV)
≈ 327.61 nm
Therefore, if the interstellar cloud were heated to 10,000 K, the wavelength of the photons emitted by the transition would be approximately 327.61 nm.