An electron in the n = 7 level of the hydrogen atom relaxes to a lower-energy level, emitting light of 397 nm. What is the value of n for the level to which the electron relaxed?

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Welcome back, everyone. An electron in the N equals nine level of the hydrogen atom relaxes to a lower energy level. Emitting light of 923 nanometers. What is the value of M or the level to which the electron relaxed? For this problem, we're going to use the Reberg equation which tells us that the inverse of the wavelength or one divided by lambda must be equal to the Reberg constant. Rh multiplied by the difference between a one divided by N final squared and one divided by an initial squared. Immediately, we want to think if there must be a negative sign in front, if the atom will access to a lower energy level and final is lower. And therefore, the difference will be positive because M final is a lower value than the N initial, right. So the answer will be positive, there's no necessity to add a negative sign because we're looking for a photon emission and not absorption. So let's solve for DNF the final M valley. How do we do that? Well, essentially, first of all, let's divide both sides by Rh, we get a one divided by a Lambda multiplied by Rh is equal to one divided by N final squared minus 1/1 divided by an initial square. Let's isolate the first term we can say that one divided by NF squared is equal to one divided by Lambda Rh plus one divided by an initial squared. Let's take the reciprocal of the right hand side to get and final squared. So it is equal to one divided by one divided by Lambda Rh plus one divided by an initial squared. And because we're looking for the value of NF we can take the square root of the right hand side, it will be positive, right? So we are going to only use the positive value and that's it. We have our equation, we can simply substitute the givens. So first of all, let's include the square root sign. Let's put one on top. And now let's begin with our terms, one divided by lambda. Rh what is lambda? Well, it's 923 nanometers and we want to convert that into meters. So we're multiplying by, since the power of negative ninth, we need to multiply that by the value of our wage, the value of the reb constant would be 1.097 multiplied by 10 to the power of seventh meters to the negative first. So that's our first term. And then we want to add plus a one divided by N initial square and the initial N is nine. If we perform the calculation, we get three. Therefore, the final value of N is equal to three. Let's label it. That's our final answer. Thank you for watching.

An electron in the n = 7 level of the hydrogen atom relaxes to a lower-energy level, emitting light of 397 nm. What is the value of n for the level to which the electron relaxed?

Verified Solution

Key Concepts

Energy Levels in Hydrogen Atom

Wavelength and Energy Relationship

Rydberg Formula