Calculating Wavelengths & Energy Levels of Hydrogen Atom

[HjGanap]

Homework Statement

the wavelengths of the emission lines produced by the hydrogen atom are given by the formula:

1/lambda = 1.096776 x 10^-2 ((1/n2^2) - (1/n1^2)) nm^-1

(a) what are the wavelengths of the first two lines in the Balmer series, H-alpha and H-beta (involving transitions to level n=2) ?

(b) calculate the energy in eV required to raise an electron from the ground state to level 2.

(c) electrons of energy 12.9 eV are fired at H atoms in a gas discharge tube. if initially all the atoms are in the ground state, what is the highest level to which the electrons in the atom can be excited? what is the longest wavelength of the possible level transitions that may then follow?

Homework Equations

as mentioned above

The Attempt at a Solution

by using data from energy-level diagram, but, confused which 1 to use for the values of n.

 

[Redbelly98]

(a)
Since these are emission lines, the electron goes from a higher n1 to a lower n2. The lower n2, as it says, is 2.

 

[nlightner]

I would like to clarify that the formula provided is for the calculation of the wavelengths of the emission lines produced by the hydrogen atom, which is known as the Balmer series. The Balmer series involves transitions to level n=2, which is the second energy level of the hydrogen atom. The values of n1 and n2 in the formula represent the initial and final energy levels, respectively.

(a) For the first two lines in the Balmer series, H-alpha and H-beta, the values of n1 and n2 are 1 and 2, respectively. Plugging these values into the formula, we get the following wavelengths:

H-alpha: 656.3 nm
H-beta: 486.1 nm

(b) To calculate the energy required to raise an electron from the ground state to level 2, we can use the formula E = -13.6/n^2 eV, where n represents the energy level. Plugging in n=2, we get E = -13.6/2^2 = -3.4 eV.

(c) If electrons of energy 12.9 eV are fired at H atoms in a gas discharge tube, the highest level to which the electrons can be excited is level 3. This is because the energy of the electrons (12.9 eV) is not enough to excite them to level 4, which requires 16.2 eV. The longest possible wavelength of the level transitions that may then follow is given by the formula mentioned in the problem, with n1=3 and n2=4. Plugging these values in, we get a wavelength of approximately 1640 nm.