The energy levels the electron can occupy in the H atom can be calculated using the energy level equation. Calculate the wavelength (nm) of the photon absorbed when the electron goes from a n= 1 energy level to a n= 3 energy level. Express answer in scientific notation.
understanding the equation will help
http://mooni.fccj.org/~ethall/rydberg/rydberg.htm
To calculate the wavelength (λ) of the photon absorbed when the electron transitions from an energy level of n=1 to an energy level of n=3 in a hydrogen atom, we can use the energy level equation:
E = -13.6 eV * (Z^2 / n^2)
where:
E is the energy of the electron transition
Z is the atomic number (1 for hydrogen)
n is the principal quantum number of the energy level
To find the energy difference (ΔE) between the final and initial energy levels:
ΔE = Ef - Ei = E3 - E1
Now, let's calculate the energy difference:
E3 = -13.6 eV * (1^2 / 3^2) = -13.6 eV * (1/9) = -1.511 eV
E1 = -13.6 eV * (1^2 / 1^2) = -13.6 eV
ΔE = -1.511 eV - (-13.6 eV) = 11.089 eV
To convert this energy difference into wavelength, we can use the relation:
E = hc / λ
where:
E is the energy of the photon
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 the photon
First, we need to convert the energy difference from electron volts (eV) to joules (J):
1 eV = 1.602 x 10^(-19) J
ΔE (J) = 11.089 eV * (1.602 x 10^(-19) J/eV) = 1.778 x 10^(-18) J
Now we can rearrange the energy-wavelength equation to solve for λ:
λ = hc / E
Substituting the known values:
λ = (6.626 x 10^(-34) J s * 2.998 x 10^8 m/s) / (1.778 x 10^(-18) J)
λ = 1.178 x 10^(-6) m
Finally, we can convert this wavelength from meters (m) to nanometers (nm):
1 nm = 1 x 10^(-9) m
λ (nm) = 1.178 x 10^(-6) m * (1 x 10^9 nm/1 m) = 1.178 x 10^3 nm
Therefore, the wavelength of the photon absorbed when the electron transitions from n=1 to n=3 energy levels in a hydrogen atom is 1.178 x 10^3 nm.