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### Course: AP®︎/College Physics 2>Unit 8

Lesson 2: Atoms and electrons

# Bohr model radii (derivation using physics)

Learn about the Bohr model of the hydrogen atom and the physics behind it. Use equations such as Coulomb's law and Newton's second law, along with the assumption that angular momentum of an electron is quantized, to calculate the radius of the atom for a ground state electron. Created by Jay.

## Want to join the conversation?

• At , "Bohr thought this angular momentum should be quantized." Why does he get to do that? How does one know that "quantizing" a parameter is appropriate, and how does one know HOW to quantize it?
• it comes from quantum physics. This is also done with momentum, not only angular momentum
• why coulomb's law is really similar to the law that describes the gravitation between two object F=G*(m1*m2)/r^2
• Nice observation - it simply is. Charge and graviational attraction are two very similiar concepts!
• When we multiplies e by negative e we should have got -e squared right? Where did that negative sign go?
• If you are referring to the calculation at time , he pointed out that the interest is in the magnitude of the charge; hence, he took the absolute values (as indicated by the parallel vertical lines on the left side of the equation).
• What is n in the equation?? I did not quite get it
• Bohr and the science community at the time already knew that energy from the H atom was emitted at specific, discrete values (referred to as the wavelength or frequency of the emitted light energy). He and his mentor, Rutherford, had already conceived of the "orbital model" of atoms (as opposed to the earlier plum pudding model where all electrons and and protons are mixed together in the nucleus like raisins in a pudding). So, based on these he took an insightful hypothesis that the orbits that the electrons were in were orbits that were in incremental, or quantum if you prefer, steps. So his hypothesis involved asserting that the electrons were in an orbital distance of 1 radius, or 2 radii or 3 radii from the nucleus. It was an ingenious insight, but a hypothesis is an educated guess based on the observations. He needed to express it mathematically and compare that mathematical result with the observations of the energy emissions.

Like many hypotheses, it worked very well to describe the phenomena in a set of instances, but it failed to explain the in the broader and more general cases. It still provides a good basis for understanding effects, but in more complex atoms it did not work out. Rather than take a Ptolemy approach and add more and more conditions onto the hypothesis to explain the more complicated cases, the Bohr model was replaced with a fuller quantum explanation, in a sense like Copernicus replace the Ptolemaic hypotheses of a geocentric system.
• How did Bohr arrive at the conclusion that L(angular momentum)=mvr=nh/2pi? I tried hard to make sense but still didn't get it. Please help
• L is defined to be r x p, which is r*p*sin(theta), where theta is the angle between the radius vector and the momentum vector. Since they are moving in a circle, that means that p and r are perpendicular, so sin(theta) is just 1, leaving rp. Since p is just mv, that means that L=mvr.

The way he arrived at the conclusion L=nh/2pi is experimentally. The predicted emission spectrum of Hydrogen would not agree with reality unless he quantized L like that.
• does first orbit of all elementns has same radius as in hydrogen?
• Nice question

I dont think so. Each time to add an electron, the energy level of the system changes and, as far as I recall, the orbits get smaller.
• What is angular momentum ?
• It is the equivalent of momentum (mass*velocity) but because velocity is in one direction, and circular movement changes direction continuously (going in circles, not in a straight line) and therefore you use angular momentum instead.
• how was the bohr's angular momentum quantization derived??
(1 vote)
• Bohr guessed that angular momentum might be quantized, and then he calculated the consequences of that in order to see if they matched up with experimental observations. His guess did a pretty good job of explaining the emission spectrum of hydrogen atoms.