# What is the difference in energy between the lowest energy state of the hydrogen atom, and the next higher energy level (i.e. the difference in energy between the 2 lowest energy states)?

##### 1 Answer

As a preface, since you have asked for the "difference in energy", I will give an answer that has no sign. That is, I will give

Since you are looking at the Bohr model of the hydrogen atom, recall that there exists something called the **Rydberg equation** (or formula):

#\mathbf(1/lambda = R_H(1/(n_f^2) - 1/(n_i^2)))# where:

#lambda# is thewavelengthin#"m"# ,corresponding to the difference in energy between quantum levels#n_i# and#n_f# .#n# is theprincipal quantum numberas usual.#R_H# is theRydberg constant,#"10973731.6 m"^(-1)# .(Normally there's a sign in the equation, but all we need to do is ensure that

#lambda > 0# .)

The lowest energy state is for hydrogen's electron in

To solve for the difference in energy, we will also need another equation.

If you recall, the **difference in energy** is related to the *frequency* (which is related to the wavelength) by *Planck's constant*.

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

#DeltaE# is the difference in energy you're looking for.#h = 6.626xx10^(-34) "J"cdot"s"# isPlanck's constant.#nu# is thefrequencycorresponding to the difference in energy.#c = 2.998xx10^8 "m/s"# is thespeed of light.

So, combining these two equations, we would proceed as follows:

#lambda = (hc)/(DeltaE),#

#1/lambda = 1/((hc)/(DeltaE)) = R_H(1/(n_f^2) - 1/(n_i^2))#

#(DeltaE)/(hc) = R_H(1/(n_f^2) - 1/(n_i^2))#

#\mathbf(DeltaE = hcR_H(1/(n_f^2) - 1/(n_i^2)))#

The **difference in energy**, *a single hydrogen atom*, is:

#color(blue)(|DeltaE|) = hcR_H|1/(n_f^2) - 1/(n_i^2)|#

#= (6.626xx10^(-34) "J"cdotcancel"s")(2.998xx10^8 cancel"m""/"cancel"s")("10973731.6" cancel("m"^(-1)))#

#xx|1/(2^2) - 1/(1^2)|#

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

Or, if you wanted this in

#color(blue)(|DeltaE|) = 1.635xx10^(-18) "J" xx "1 eV"/(1.602xx10^(-19) "J")#

#~~# #color(blue)("10.2 eV")#

And indeed, the lowest energy level of hydrogen atom (