Question about the Atomic Model

[deleted647690]

In TBR gen chem, in the chapter about atomic theory, on page 66, a figure is given showing electronic energy levels, such as this one

In the example question, it asks "how would the photon from an n=4 to n=2 transition compare to the photon from a n=2 to n=1 transition?"
The answer was that the n=2 to n=1 transition is more than twice as energetic as the n=4 to n=2 transition.


I sort of understand this based on the explanation that as you get further from the nucleus, it becomes easier to move electrons to higher levels because they are less tightly bound. However, I do not see the relationship to wavelength. At lower energy transitions, shouldn't there be longer wavelengths? The n=1 to n=2 transition has the longest wavelength, so therefore the lowest energy? But shouldn't it be high energy since it is closer to the nucleus?

Oh wait, so if you drew a wavelength representation of the n=4 to n=2 transition, the wavelength would be longer than the one shown for n=2 to n=1 then?

And the diagram shows electronic energy levels as they are for absorptions. If the diagram showed wavelength values for ejections, would it show the n=2 to n=1 as shorter wavelength than n=3 to n=2?

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[betterfuture]

The transition from n=1 to n=2 takes more energy.

In the following equation you can see the relationship between energy and frequency.

E=hf

Frequency is directly related to Energy. Higher frequency means greater Energy.

Wavelength is the exact opposite. Since the transition from n=1 to n=2 is greater, the wavelength is smaller and frequency is bigger. The transition from n=2 to n=4 is smaller in comparison to n=1 to n=2. In this case, wavelength is bigger and frequency is smaller.

For absorption and emission, the energy is the same except your either giving off energy in the form of a photon or absorbing it. Either way, energy remains the same, in magnitude of course.

 

[deleted647690]

The transition from n=1 to n=2 takes more energy.

In the following equation you can see the relationship between energy and frequency.

E=hf

Frequency is directly related to Energy. Higher frequency means greater Energy.

Wavelength is the exact opposite. Since the transition from n=1 to n=2 is greater, the wavelength is smaller and frequency is bigger. The transition from n=2 to n=4 is smaller in comparison to n=1 to n=2. In this case, wavelength is bigger and frequency is smaller.

For absorption and emission, the energy is the same except your either giving off energy in the form of a photon or absorbing it. Either way, energy remains the same, in magnitude of course.

How is it that any transition n=2 to n=infinity is always going to be smaller than the transition between n=1 and n=2? Is it because the relative difference between each successive energy level gets infinitely smaller as you get higher in energy levels?

 

[betterfuture]

Here is what I've read up on:

As n, the energy level, gets bigger, the electron gets very far away from the nucleus. When it does so, the probability of finding electrons closer to the nucleus is small because of the increased distance from positive nucleus, that is, in that energy level. The electron would become separated from the atom easily because there is less of an attraction. So in this case, the atom would carry a charge with itself after losing an electron instead of being a neutral atom since the electron to proton is distribution is unequal.

 

[aldol16]

How is it that any transition n=2 to n=infinity is always going to be smaller than the transition between n=1 and n=2? Is it because the relative difference between each successive energy level gets infinitely smaller as you get higher in energy levels?

This is also a fundamental postulate of quantum mechanics - as n goes to infinity, energy no longer "appears" to be quantized and instead appears as though it is along a spectrum.

 

[deleted647690]

This is also a fundamental postulate of quantum mechanics - as n goes to infinity, energy no longer "appears" to be quantized and instead appears as though it is along a spectrum.

That sounds interesting, but I guess it's beyond what I would be able to understand then haha

 

[BerkReviewTeach]

Excellent responses and questions here. As explained above, the force holding an electron is reduced as it gets farther from the nucleus, which means it will take less energy to completely remove it. It just so happens that energy is exponential in terms of n, so the drop off in energy as it gets farther from the nucleus is 1 over exponential. They key thing to know here is that the energy depends on 1/nexp2, so think about final - initial in terms of that math.

from n=1 to n=2: 1/1exp2 - 1/2exp2 = 1 - 1/4 = 3/4 vs. from n=2 to n=∞: 1/2exp2 - 1/∞exp2 = 1/4 - 0 = 1/4​

So it takes three times as much energy to excite an electron from n=1 to n=2 than it takes from n=2 to n=∞. No matter what, it will take more energy for an electron on hydrogen to go from n=1 to n=2 than any other transition.

 

[deleted647690]

Thank you for the responses!