An excited hydrogen atom emits light with a wavelength of 397.2nm to reach the energy level for which n=2. In which principle quantum level did the electron did begin?

##DeltaE=-R_H*(Z)^2[1/n_f^2-1/n_i^2] ##

In the hydrogen atom, the energy of the electron in a given energy level is given by : ##E_n= -R_H*(Z/n)^2##

##E_f= -R_H*(Z/n_f)^2##

##E_i= -R_H*(Z/n_i)^2##

##DeltaE=E_f-E_i ##

##DeltaE=[-R_H*(Z/n_f)^2]-[ -R_H*(Z/n_i)^2] ##

take ## -R_H*(Z)^2## as a common factor,

##DeltaE=-R_H*(Z)^2[1/n_f^2-1/n_i^2] ##

The energy of the photon emitted is given by:

##DeltaE=-hc/lambda##

please note that a negative sign must be introduced to the energy expression since energy is released.

combining the two equations gives:

##-hc/lambda=-R_H*(Z)^2[1/n_f^2-1/n_i^2] ## (equation 1)

##h" is Planck's constant" = 6.626*10^-34 J.s ## ##R_H" is Rydberg constant" = 2.178*10^-18 J ## ##Z" is the atomic number of the hydrogen atom" = 1## ##n" is principle quantum number"##

##n_i " is the initial quantum state of the electron."## ##n_f =2##

plugging the numbers in (equation 1)

##-(6.626*10^-34 J.s*2.998*10^8 m/s) /(397.2*10^-9 m)=-2.178*10^-18 J*(1)^2[1/2^2-1/n_i^2] ##

##-(6.626*10^-34 cancel(J).cancel(s)*2.998*10^8cancel(m)/cancel(s)) /(397.2*10^-9 cancel(m))=-2.178*10^-18 cancel(J)*(1)^2[1/2^2-1/n_i^2] ##

solve for ##n_i##,

##n_i= 7##