Consider the three electronic transitions in a hydrogen atom shown here, labeled A, B, and C. (b) Calculate the energy of the photon emitted for each transition.
Calculate the energy of the photon emitted for transition C.

Question

Consider the three electronic transitions in a hydrogen atom shown here, labeled A, B, and C. (b) Calculate the energy of the photon emitted for each transition.

Energy level diagram of a hydrogen atom showing transitions A, B, and C.

Calculate the energy of the photon emitted for transition C.

Accepted Answer

Welcome back everyone labeled A. B, C and D. In the diagram below are four electronic transitions in a hydrogen atom. Find the energy and jewels of the photon emitted for transitions A through D. We're going to begin by recalling that we need to utilize our Rydberg equation where we have our energy change for a photon transition represented by delta E. The units are going to be in jewels. This is set equal to negative one times are Rydberg constant represented by R C. B, which is multiplied by the following bracket. Where we have one over our final energy level squared subtracted from one over our initial energy level squared. So let's begin with also noting that whenever we have an electron emission, our delta, the value should be negative. Beginning with transition A. We go from an equals three as our initial energy level to n equals two as our final energy level because that's where the arrowhead points. So let's start with the calculating delta E. For a transition A. So we want the energy change of our photon delta E. We have negative one times are right there constant. Which we should recall is the value 2.178 times 10 to the negative 18th power jewels from our textbooks. This is multiplied by the following brackets where we have one over our final energy level squared and in this case for transition a Our final energy level is two squared. This is subtracted from one over our initial energy level being three squared. So now in our next line we would still have a Rydberg constant 2.178 times 10 to the negative 18th power jewels multiplied by the following difference between our two fractions where we have one over four subtracted from 1/9. And so now we just would take the difference between our bracket, we have negative 2.178 times 10 to the negative 18 power jewels for Rydberg constant multiplied by the result of the difference in our brackets, which gives us a value of 0.138889. So taking the product here, we would get our energy change for the photon emission and transition A equal to negative 3.25 times 10 to the negative 19th power And we still have units of jewels, which is what we want for our photon energy. And so we have our first answer so far for our delta E value for transition A. Now let's move on to transition B. So transition B will use the color red or rather we'll use the color black. We can see that it's initial energy level of the electron is at N I equals four. And for the final energy level for transition be we're at an F equal to three. So doing our work for delta E. For transition be, we have delta equal to negative one times are Rydberg constant 2.178 times 10 to the negative 18th power jewels multiplied by our bracket where we have one over our final energy level for transition be given as three, this is squared in the prompt or in our equation subtracted from one over our initial energy level being the fourth energy level squared for transition be. This will give us negative 2.178 times 10 to the negative 18th power jewels in our next line, multiplied by the falling bracket where we have We have 1 9th subtracted from 1/16 and we can simplify this actually in one step to make things faster in our calculators and this will give us a result equal to a value of negative one point oh 5875 times 10 to the negative 19th power jewels. And we can round this to about negative 1.59 times 10 to the negative 19 power jewels as 466. So this would be our second answer so far for our energy change of our photon for transition be and note that to get this value, to be sure to use brackets in your calculators. So now we just need to move on to transition see now. So going back to our diagram, we have transition, see in blue we have our initial energy level at N. I equal to five and our final energy level equal to the fourth energy level. So doing our work for transition C. We have our energy change of our photon equal to negative one, times are Rydberg constant from our textbooks 2.178 times 10 to the negative 18th power jewels multiplied by our brackets where we have one over our initial or sorry our final energy level for transition. See according to the prompt is the fifth energy level, so that's five squared subtracted from and sorry I was getting ahead of myself. So one over our final energy level. And looking at our diagram, that is the fourth energy level squared. Since that is where our arrowhead points to. This is subtracted from one over our initial energy level according to our prompt, which is five squared. So this completes our bracket here and we can simplify fractions so that we have negative 2.178 times 10 to the negative 18th power jewels multiplied by our brackets where we have 1/16 subtracted from one 25th And in our calculators we're going to type everything in that line exactly with the brackets and parentheses included and we should get a result equal to so we have a value of negative 4.9005 times 10 to the negative 20th power. And our units are jewels. And so this would be we just need to round this 246 fix as negative 4.901 times 10 to the negative 20th power jewels as our third final answer so far for the energy change of our photon for transition C. And lastly we have transition D. Given in green. So we have the initial energy level at an equal six and the final energy level at N. F. Equal to five for transition D. So we'll do the work below. So our energy change delta E. Is equal to negative one times are Rydberg constant 2.178 times 10 to the negative 18 power jewels multiplied by our bracket. Where we have one over our final energy level for transition D. As we stated above, that is at the arrowhead being at the fifth energy level that is squared and then subtracted from one over our initial energy level for transition D. Which is six. And this is squared. So finishing off our bracket and simplifying, we have negative 2.178 times 10 to the negative 18 power jewels multiplied by we have the following fractions one 25th subtracted from one 36th. And then this will complete our brackets in our calculators. We're going to type everything accordingly with brackets and parentheses and we'll get a result equal to negative 2.662 and sorry negative 2.662 times 10 to the negative 20th power our units are jewels and this would be already as four sig figs here. So this would be our fourth final answer as the energy change of our photon for transition D. And jewels and all of our answers which are highlighted in yellow correspond to choice A and the multiple choice as the correct answer. So I hope that everything I reviewed made sense. And if you have any questions, just let us know.

Consider the three electronic transitions in a hydrogen atom shown here, labeled A, B, and C. (b) Calculate the energy of the photon emitted for each transition.

Calculate the energy of the photon emitted for transition C.

Verified Solution

Key Concepts

Energy Levels in Hydrogen Atom

Photon Energy Calculation

Transition Notation