In the Bohr model of atomic structure, electrons are constrained to orbit a nucleus at specific distances, given by the equation

where r is the radius of the orbit, Z is the charge on the nucleus, a0 is the Bohr radius and has a value of 5.292 * 10-11 m, and n is a positive integer (n = 1, 2, 3...) like a principal quantum number. Furthermore, Bohr concluded that the energy level E of an electron in a given orbit is

where e is the charge on an electron. Derive an equation that will let you calculate the difference ∆E between any two energy levels. What relation does your equation have to the Balmer–Rydberg equation?

Video transcript

Welcome back everyone. The Bohr model of atomic structure is described by the following equation where R is equal to n squared times a not over Z. Where Z is the charge on the nucleus a knot is the Bohr radius R is the radius of orbit and N is a positive integer. Similar to the principle quantum number. Based on this model, electrons are bound to orbit the nucleus at specific distances and board concluded that the energy level E of an electron in a certain orbit is equal to negative Z times E squared over two. R. Where E is the electron charge, illustrate the energy difference between any two energy levels by creating your own equation and comparing it to the Balmer Rydberg equation. So we're going to begin by recognizing that in this first equation we have our terms are associated by the following relationship which we see are also in the second equation given. So we can substitute this relationship for our denominator are here in the second equation. So we can actually say that E. Is equal to negative Z times E squared over two, which is multiplied by what we're substituting for our as n squared times A. Not divided by Z. And so now we just want to simplify this. So because we have this two terms multiplied by a parentheses we're going to have. So what we have is to over one multiplied by N squared A. Not over Z. Which we will simplify to Z. Or sorry, two times n squared times A. Not over our term Z. And we need to remember that we still have our numerator that this is being divided by. So we're going to recognize that we can actually make this a reciprocal. So we'll say make reciprocal so that we have are negative Z. And that's actually we have negative Z times E squared. Which is now going to be multiplied by the reciprocal fraction, which is now Z over two times and squared times a not. And so what we can simplify too now is that we'll have and let's actually use the color black. We'll have negative Z times Z, which is just going to be negative Z. This is squared and then multiplied by E squared. And then in our denominator we have two times a, not times and squared. And so we can ultimately say that therefore E is equal to negative Z squared times E squared divided by two times a not times and squared. And we need to make an association between the difference between energy levels of the electrons in a certain orbit. And so we're going to need to write out E with a delta sign in the beginning so that we can now say that we have final energy minus the initial energy level. So we would say that we can say where delta E is E two for our higher energy level -11 are lower energy level. So E two again is higher And E one lower. So now with this noted down, let's clearly define this relationship so that we can say that our term N. In our equation here represents our energy level because according to the prompt, N is an integer similar to the principle quantum number. And recall that our principal quantum number relates the energy level of an electron within an atom that can go up all the way to infinity. So we're gonna say that instead we have delta E equal to rolled right out E two first. So we have negative Z squared times E squared divided by two times a knot. And now we're going to take our principal quantum number term N. And we're gonna make this N two squared since this is for the Higher energy level. E two we end our parentheses off. And this is subtracted from our second relation of this expression where we have again negative Z squared times E squared divided by two times a. Not times our second energy level which is the lower one E one. So we would have N one squared and end off our parentheses. And since we technically have minus a negative here we can actually simplify this and rewrite it so that we have delta equal to E one written first which is going to be again Z squared times E squared divided by two times a. Not times and one squared which is now still subtracted from the E two or higher energy of Z squared times E squared divided by two times a not times N two squared and sub two squared. So before I continue I just want to reiterate that E one, the lower energy level is represented by N one, N sub one squared and E to the higher energy level is represented by n sub two squared. And we just rearrange things at this point. So now we can further simplify this by recognizing that we have constant terms. Here we have Z are constant term for the charge of a nucleus. We have a knot which is our constant term for Bohr radius and we have our third constant E, which is the electrons charge according to the prompt. So E Z and a not all constant. So we can say that therefore delta E. The difference in energy is directly proportional to everything except our constants, which would just be our energy level terms. So it's directly proportional to one over n sub one squared minus one over n sub two squared. And this is going to be written in parentheses And we want to make note of the fact that because we're subtracting from from n sub two squared, we would say that And sub two is going to be less than one. It has to be if we're subtracting from it now because we see that the energy level difference is directly proportional to the difference between our inverse of our lower energy level minus the inverse of our higher energy level. We want to recall that our Rydberg equation relates are inverse wavelength represented by lambda here in the denominator. So this is lambda, which is equal to our Rydberg constant. Are multiplied by the charge of a nucleus Z squared, multiplied by our parentheses here. Where we have our universe initial energy level squared minus the inverse of our final energy level squared. And in this case we still have our term and sub two for the higher energy level, which should be greater than n sub one, the initial or lower energy level. So just to make a correction, this also should be the higher energy level greater than the initial energy. So apologies for that mistake because we need this to be a positive term when we're subtracting. And so with the Freedberg equation, we can see that we have also a direct relationship between our inverse wavelength and the difference between our energy levels. So one over N sub one squared minus one over and sub two squared. So we have essentially successfully defined a relationship between energy difference between two energy levels, which is quite similar to our Balmer Rydberg equation. Because we can ultimately say that the difference in our energy levels is equal to Z, the charge of our nucleus times e the electron's charge squared divided by two times a not our boards radius then multiplied by our difference in our energy levels. So one over n sub one squared. Our lower energy level minus one over n sub two squared are higher energy level. And just to make some more room, Let's scoop this over. So we're going to conclude that this is similar to Balmer Rydberg equation because both equations are proportional to the difference between the higher and lower energy levels. For our final answer, we have our outlined relationship for the difference between any two energy levels and I hope that everything I reviewed was clear. If you have any questions, please leave them down below and I'll see everyone in the next practice video.