### Bohr Frequency Condition

Posted:

**Wed Sep 28, 2016 10:16 pm**Can someone explain the Bohr frequency condition? I do not understand its application or where the upper and lower energy values come from for the formula.

Created by Dr. Laurence Lavelle

https://lavelle.chem.ucla.edu/forum/

https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=17&t=15422

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Posted: **Wed Sep 28, 2016 10:16 pm**

Can someone explain the Bohr frequency condition? I do not understand its application or where the upper and lower energy values come from for the formula.

Posted: **Wed Sep 28, 2016 10:53 pm**

In order to understand the Bohr frequency condition both conceptually and mathematically, it is necessary to establish a working knowledge of the theory that preceded it. A researcher by the name of Ernest Rutherford had previously performed a gold foil experiment, showing that, when helium atoms were shot at a piece of gold foil, most went through, but some bounced back. He inferred from this heuristic experience that the atom had to be a dense, positively charged nucleus with electrons surrounding it; this makes sense, as atoms went through the foil most of the time, but bounced back every once in a while.

When reviewing Rutherford's theory, Neils Bohr objected to it. The electrons orbiting around the nucleus were orbiting in a circle, and thus had a constantly changing velocity (since velocity is a vector quantity depending not only on speed, but on direction). So, each electron possessed a centripetal acceleration toward the nucleus. Bohr knew that when a charged particle (in this case an electron) accelerates, it radiates, or emits energy in the form of light. Thus, the electron loses kinetic energy to light energy in a death spiral towards the nucleus, and the hydrogen atom ceases to exist in very little time.

As a workaround, Bohr proposed that electrons did not simply exist in random circular orbits about the nucleus. Rather, they were segregated into discrete orbits represented by n, which can only take on positive integer values. This is consistent with what we know: energy is quantized. If photons of only certain particular energy levels were to excite an electron, it could move up an energy level, temporarily resisting the pull of the nucleus, until the electron moved back to its original orbital and emitted a photon of energy. This can be mathematically expressed as such:

Ep = E2 - E1

where Ep is the energy of the photon, E2 is the energy of the excited electron, and E1 is the energy of the electron before excitation. In concert with our knowledge of E=hv, we can mathematically iterate this phenomenon known as Bohr's Frequency Condition that solves for the frequency of incident light:

hv = (E1 - E2)

v = (E1 - E2)/h

Note that it was experimentally determined that the energy En at a given energy level is equal to -hR/n^2, where h is Planck's constant and R is the Rydberg constant (3.29*10^15 s^-1). The convention is that an electron with no relation to the atom would be at an infinitely high energy level n and thus its energy would approximate 0. Therefore, electrons in orbit are said to have negative energy since their energy is lower than that of a freed electron. Thus we can extend upon Bohr's Frequency condition as follows:

v= (-hR/(n1)^2 - -hR/(n2)^2)/h [replace E with -hR/n^2]

v= -hR(1/(n1)^2 - 1/(n2)^2)/h [factor out -hR]

v=-R(1/(n1)^2 - 1/(n2)^2) [h cancels, we are left with the Rydberg eqn]

This form of the Rydberg equation mathematically elucidates the Lyman, Paschen, and Balmer series. From empirical data, Lyman, Paschen, and Balmer concluded that transitions of electrons down to certain energy levels consistently produced light in the same areas of the electromagnetic spectrum. Lyman showed that transitions of electrons from a higher energy level down to n=1 produced UV light, Balmer showed that transitions down to n=2 produced visible light, and Paschen showed the same for transitions to n=3 producing infrared light. You can test these ideas out with the Rydberg equation to see for yourself!

Hope this helps and happy chemistry!

-D. Callos

When reviewing Rutherford's theory, Neils Bohr objected to it. The electrons orbiting around the nucleus were orbiting in a circle, and thus had a constantly changing velocity (since velocity is a vector quantity depending not only on speed, but on direction). So, each electron possessed a centripetal acceleration toward the nucleus. Bohr knew that when a charged particle (in this case an electron) accelerates, it radiates, or emits energy in the form of light. Thus, the electron loses kinetic energy to light energy in a death spiral towards the nucleus, and the hydrogen atom ceases to exist in very little time.

As a workaround, Bohr proposed that electrons did not simply exist in random circular orbits about the nucleus. Rather, they were segregated into discrete orbits represented by n, which can only take on positive integer values. This is consistent with what we know: energy is quantized. If photons of only certain particular energy levels were to excite an electron, it could move up an energy level, temporarily resisting the pull of the nucleus, until the electron moved back to its original orbital and emitted a photon of energy. This can be mathematically expressed as such:

Ep = E2 - E1

where Ep is the energy of the photon, E2 is the energy of the excited electron, and E1 is the energy of the electron before excitation. In concert with our knowledge of E=hv, we can mathematically iterate this phenomenon known as Bohr's Frequency Condition that solves for the frequency of incident light:

hv = (E1 - E2)

v = (E1 - E2)/h

Note that it was experimentally determined that the energy En at a given energy level is equal to -hR/n^2, where h is Planck's constant and R is the Rydberg constant (3.29*10^15 s^-1). The convention is that an electron with no relation to the atom would be at an infinitely high energy level n and thus its energy would approximate 0. Therefore, electrons in orbit are said to have negative energy since their energy is lower than that of a freed electron. Thus we can extend upon Bohr's Frequency condition as follows:

v= (-hR/(n1)^2 - -hR/(n2)^2)/h [replace E with -hR/n^2]

v= -hR(1/(n1)^2 - 1/(n2)^2)/h [factor out -hR]

v=-R(1/(n1)^2 - 1/(n2)^2) [h cancels, we are left with the Rydberg eqn]

This form of the Rydberg equation mathematically elucidates the Lyman, Paschen, and Balmer series. From empirical data, Lyman, Paschen, and Balmer concluded that transitions of electrons down to certain energy levels consistently produced light in the same areas of the electromagnetic spectrum. Lyman showed that transitions of electrons from a higher energy level down to n=1 produced UV light, Balmer showed that transitions down to n=2 produced visible light, and Paschen showed the same for transitions to n=3 producing infrared light. You can test these ideas out with the Rydberg equation to see for yourself!

Hope this helps and happy chemistry!

-D. Callos

Posted: **Thu Oct 12, 2017 2:03 pm**

I'm a bit confused at the basics. Let's say an electron has been excited to the n=3 state from its original n=1 state. When it's returning to its original energy level, does it stop at n = 2 to give off radiation and then go to n = 1 to give off radiation? Or does the electron go directly from n = 3 to n = 1 and release radiation once?