Answer:
The energy is [tex]4.57x10^{-19} J[/tex] and the wavelength is [tex]4.34x10^{-7}m[/tex] for the line in the spectrum of hydrogen that represents the movement of an electron from Bohr orbit with n = 2 to the orbit with n = 5.
In what part of the electromagnetic spectrum do we find this radiation?
In the Ultraviolet part of the electromagnetic spectrum.
Explanation:
The energy of the absorbed photon can be known by the difference in energy between the two states in which the transition is happening (In this case from n = 2 to n = 5):
[tex]E = E_{upper}-E_{lower}[/tex] (1)
The permitted energy for the atom of hydrogen, according with the Bohr's model, is defined as:
[tex]E_{n} = -\frac{13.606 eV}{n^{2}}[/tex] (2)
Or it can be expressed in Joules, since [tex]1eV = 1.60x10^{-19}J[/tex]
[tex]E_{n} = -\frac{2.18x10^{-18} J}{n^{2}}[/tex] (3)
Where the value [tex]-2.18x10^{-18}[/tex] represents the energy of the ground state¹ and n is the principal quantum number.
For the case of n = 2:
[tex]E_{2} = -\frac{2.18x10^{-18} J}{(2)^{2}}[/tex]
[tex]E_{2} = -5.45x10^{-19} J[/tex]
For the case of n = 5:
[tex]E_{5} = -\frac{2.18x10^{-18} J}{(5)^{2}}[/tex]
[tex]E_{5} = -8.72x10^{-20} J[/tex]
Replacing those values in equation (1) it is gotten:
[tex]E = -8.72x10^{-20} J-(-5.45x10^{-19} J )[/tex]
[tex]E = 4.57x10^{-19} J[/tex]
The wavelength can be determined by means of the Rydberg formula:
[tex]\frac{1}{\lambda} = R(\frac{1}{n_{l}^{2}}-\frac{1}{n_{u}^{2}})[/tex] (4)
Where R is the Rydberg constant, with a value of [tex]1.097x10^{7}m^{-1}[/tex]
For this particular case [tex]n_{l} = 2[/tex] and [tex]n_{u} = 5[/tex]:
[tex]\frac{1}{\lambda} = 1.097x10^{7}m^{-1}(\frac{1}{(2)^{2}}-\frac{1}{(5)^{2}})[/tex]
[tex]\frac{1}{\lambda} = 1.097x10^{7}m^{-1}(0.21)[/tex]
[tex]\frac{1}{\lambda} = 2303700m^{-1}[/tex]
[tex]\lambda = \frac{1}{2303700m^{-1}}[/tex]
[tex]\lambda = 4.34x10^{-7}m[/tex]
So the energy is [tex]4.57x10^{-19} J[/tex] and the wavelength is [tex]4.34x10^{-7}m[/tex] for the line in the spectrum of hydrogen that represents the movement of an electron from Bohr orbit with n = 2 to the orbit with n = 5.
In what part of the electromagnetic spectrum do we find this radiation?
In the Ultraviolet part of the electromagnetic spectrum.
Key terms:
¹Ground state: State of minimum energy.
