According to the Bohr model for the hydrogen atom, how much energy is necessary to excite an electron from n=1 to n=2?

This energy can be determined from the Rydberg equation, which is

#\mathbf(1/lambda = R_H(1/(n_i^2) - 1/(n_j^2)))#

where:

• #lambda# is the wavelength in units of #"m"#.
• #R_H# is the Rydberg constant, #"10973731.6 m"^-1#.
• #n_i# is the principal quantum number #n# for the lower energy level.
• #n_j# is the principal quantum number #n# for the higher energy level.

In this case, #n_i = 1# and #n_j = 2#, since the excitation from #n = 1# to #n = 2# is upwards. Thus:

#1/lambda = "10973731.6 m"^-1(1/1^2 - 1/2^2)#

#= "10973731.6 m"^-1(1/1^2 - 1/2^2)#

#= "8230298.7 m"^(-1)#

Now, the wavelength is

#color(green)(lambda) = 1/("8230298.7 m"^(-1))#

#= color(green)(1.215xx10^(-7) "m"),#

or #"121.5 nm"#. Let's relate this back to the energy. Recall the equation:

#\mathbf(DeltaE = hnu = (hc)/lambda)#

where:

• #DeltaE# is the change in energy in #"J"#.
• #h# is Planck's constant, #6.626xx10^(-34) "J"cdot"s"#.
• #nu# is the frequency in #"s"^(-1)#. Remember that #c = lambdanu#.
• #c# is the speed of light, #2.998xx10^8 "m/s"#.
• #lambda# is the wavelength in #"m"# like before.

So, the absorption of energy into a single hydrogen atomic system associated with this process is:

#color(blue)(DeltaE) = ((6.626xx10^(-34) "J"cdotcancel"s")(2.998xx10^(8) cancel"m/s"))/(1.215xx10^(-7) cancel"m")#

#= color(blue)(1.635xx10^(-18) "J")#

(absorption is an increase in energy for the system, thus #DeltaE > 0#.)