Question

What is the energy (in joules) and the wavelength (in meters) of 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?

Answer 1

Answer:

The energy is $4.57x10^{-19} J$ and the wavelength is $4.34x10^{-7}m$ 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):

$E = E_{upper}-E_{lower}$   (1)

The permitted energy for the atom of hydrogen, according with the Bohr's model, is defined as:

$E_{n} = -\frac{13.606 eV}{n^{2}}$   (2)

Or it can be expressed in Joules, since $1eV = 1.60x10^{-19}J$

$E_{n} = -\frac{2.18x10^{-18} J}{n^{2}}$   (3)

Where the value $-2.18x10^{-18}$ represents the energy of the ground state¹ and n is the principal quantum number.

For the case of n = 2:

$E_{2} = -\frac{2.18x10^{-18} J}{(2)^{2}}$

$E_{2} = -5.45x10^{-19} J$

For the case of n = 5:

$E_{5} = -\frac{2.18x10^{-18} J}{(5)^{2}}$

$E_{5} = -8.72x10^{-20} J$

Replacing those values in equation (1) it is gotten:

$E = -8.72x10^{-20} J-(-5.45x10^{-19} J )$

$E = 4.57x10^{-19} J$

The wavelength can be determined by means of the Rydberg formula:

$\frac{1}{\lambda} = R(\frac{1}{n_{l}^{2}}-\frac{1}{n_{u}^{2}})$  (4)

Where R is the Rydberg constant, with a value of $1.097x10^{7}m^{-1}$

For this particular case $n_{l} = 2$ and $n_{u} = 5$:

$\frac{1}{\lambda} = 1.097x10^{7}m^{-1}(\frac{1}{(2)^{2}}-\frac{1}{(5)^{2}})$

$\frac{1}{\lambda} = 1.097x10^{7}m^{-1}(0.21)$

$\frac{1}{\lambda} = 2303700m^{-1}$

$\lambda = \frac{1}{2303700m^{-1}}$

$\lambda = 4.34x10^{-7}m$

So the energy is $4.57x10^{-19} J$ and the wavelength is $4.34x10^{-7}m$ 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.