How Do You Calculate the Orbit Radii in Bohr's Model of the Hydrogen Atom?

[McLaugh]

This is my problem in my chemistry class. It's a project we have to do over the weekend. I have been listening and taking notes of everything in our class that our teacher says or writes on the board and I still do not understand this problem.

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Using Models Niels Bohr proposed that electrons must occupy specific, quantized energy levels in an atom. He derived the following equations for hydrodgen's electron orbit energie (En) and radii (Rn).

Rn = (0.529 x 10 ^-10 m) n2

En = -(2.18x10 ^-18 J)/n2

* ^ indicates an exponent.

Analysis
Using the orbit radii equation, calculate hydrodgen's first 7 electron orbit radii and then construct a scale model of these orbits. Use a compass and a metric ruler to draw your scale model on two sheets of paper that have been taped together. Using the orbit energy equation, calculate the energy of each electron orbit and record the values on your model.

 

[RogerPink]

No problem. Basically you're going to draw an atom. The picture will look like this. You'll have a proton at the center, that's the nucleus of the Hydrogen Atom. Then you're going to have 7 circles, each larger then the next, all with the proton at their center. You will use the equation for Rn to find the radius of each circle. Simply substitute n=1, 2, 3, 4, 5, 6, and 7 into the equation to get the radius (1/2 the diameter) of each of the 7 orbits. Each orbit will have an energy value associated with it, use the same n in the energy equation to calculate the energy for each orbit.

In order to make a scale model, I recommend you calculate the highest orbit radius first and choose a scale that will allow it to fit on the two sheets of paper. Figure a scale around 1 angstrom = .25 inch or something.

Some things worth noticing when you make this model. The energy of the closest orbit is the highest, the one furthest away is the lowest. The energy of the orbit is actually the energy of an electron in the orbit. If the electron moves to another orbit, simply subtract the energy of the new orbit from the energy of the old one. If the difference in energy is positive, then energy has been released, if the difference in energy is negative (larger radius orbit to smaller radius orbit), then energy has been absorbed.

Hope this helps.

 

[Gokul43201]

Some things worth noticing when you make this model. The energy of the closest orbit is the highest, the one furthest away is the lowest.

Correct this. It's the other way round.

 

[Chronos]

What Gokul said. Electrons give up photons [lose energy] when they drop to lower orbital shells. A neat way to cheat on this problem is to look at spectral lines.

 

[RogerPink]

I think you misunderstood what I was saying

Correct this. It's the other way round.

I was saying that the orbit with the smallest radius has the highest energy. I think you agree with that right? The ionization of a 1s orbital is much higher than that of a 3s orbital right? Transitions from 1s to 2p would absorb a photon wheras transitions from 2p to 1s would emit a photon.

So don't reverse what I said. The closest (smallest radius) orbital will have the highest energy, but you don't have to take my word for it, just use your equations and you'll see that for increasing n, the energy gets smaller and the radius gets larger.

 

[Gokul43201]

I was saying that the orbit with the smallest radius has the highest energy. I think you agree with that right?

But it's not. Why is the innermost orbit called the ground state, and why are outer orbits called excited states?

Perhaps you forgot that there's a negative sign in front of the expression for energy?

 

[RogerPink]

But it's not. Why is the innermost orbit called the ground state, and why are outer orbits called excited states?

Perhaps you forgot that there's a negative sign in front of the expression for energy?

I see the disconnect, what I should have said is the orbit with the smallest radius has the largest "magnitude" (absolute value) of energy. The negative sign indicates an attractive potential. You were right to correct me as my earlier statement was misleading and techniquely incorrect. Notice if you were to allow n to become very large, then R=infinite and E=0, thus for a very very large radius, the energy is almost zero.

 

[Chronos]

Perhaps there is a terminology problem. When electrons occupy the inner most orbital shell, they can no longer emit photons. Perhaps an analogy would help clarify Gokul's point. A rock on top of the mountain has more potential energy than a rock at the foot of the same mountain. An electron occupying the inner most orbital shell is the quantum equivalent of the rock at the foot of the mountain.