First Bohr Radius - Quantum and Atomic Model

In summary: Gm1m2/e^2This conversation discusses the calculation of the radius of the first Bohr orbit if the electron and proton were bound by gravitational force instead of electric force. The equation used is r = h^2 / 4∏^2*mke^2, but the gravitational force is substituted for the electric force. The force of gravity between two masses is used, with the mass of the proton and electron substituted and the gravitational constant denoted as "k." The calculation results in r = Gm1m2/e^2, with the centripetal force referring to the electron. In summary, the conversation explores the calculation of the first Bohr orbit radius for a hydrogen atom bound by gravitational force.
  • #1
PeachBanana
191
0

Homework Statement



Suppose an electron was bound to a proton, as in the hydrogen atom, but by the gravitational force rather than by the electric force. What would be the radius of the first Bohr orbit?

Homework Equations



r = h^2 / 4∏^2*mke^2

The Attempt at a Solution



I'm not sure how to manipulate the above equation to account for gravitational force. Should I start by looking at that fact

F = ma
kZe^2/r^2 = mv^2/r
 
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  • #2
Replace the electric force by the gravitational one. What is the force of gravity between two masses? (It is the mass of the proton and the mass of the electron now).

ehild
 
  • #3
Ah, so F = km1m2/r^2 ?

Which means...

F = ma
km1m2/r^2 = mv^2/r
 
  • #4
"k" must be the gravitational constant, usually denoted by G.
So what do you get for r?

ehild
 
  • #5
r = Gmm/mv^2

The question I now have is which mass will cancel as I have a mass in the numerator and denominator...
 
  • #6
Which particle orbits around the other?

ehild
 
  • #7
The electrons orbit around the protons so does this mean the electron mass will cancel out?
 
  • #8
PeachBanana said:
The electrons orbit around the protons so does this mean the electron mass will cancel out?

It needs the centripetal force, so the term mv^2/r refers to the
electron.


ehild
 

1. What is the First Bohr Radius?

The First Bohr Radius, denoted as a0, is a physical constant that represents the smallest possible distance between the nucleus and the electron in a hydrogen atom according to the Bohr model of the atom.

2. How is the First Bohr Radius calculated?

The First Bohr Radius can be calculated using the equation a0 = ℏ²/(me²), where ℏ is the reduced Planck's constant and me is the mass of the electron.

3. What is the significance of the First Bohr Radius?

The First Bohr Radius is significant because it provides a fundamental understanding of the structure of the hydrogen atom and serves as a basis for the development of quantum mechanics and the atomic model. It also helps in understanding the behavior of other atoms with only one electron, such as the helium atom.

4. How does the First Bohr Radius relate to the Bohr model of the atom?

In the Bohr model, the First Bohr Radius represents the first stable orbit of the electron around the nucleus. According to this model, the electron can only exist in specific energy levels, with the first level having a radius equivalent to the First Bohr Radius.

5. Can the First Bohr Radius be applied to other atoms?

Yes, the First Bohr Radius can be applied to other atoms with only one electron, such as the helium atom. However, for atoms with more than one electron, the concept of the First Bohr Radius is not applicable as the electrons interact with each other and do not have discrete energy levels like in the Bohr model.

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