How Can the Bohr Radius Help Calculate Energy Levels in Hydrogen?

[mIKEjONES]

I'm currently reading a book which says that by knowing by the Bohr radius, in this example that of hydrogren, the energy of each orbit level can be calculated by deriving the appropriate Rydberg formula constant using Coulomb's law and Newtonian mechanics.

Following their description I tried deriving the energy possessed by each orbit level for hydrogen, my final solution should then be $$E = -\frac{13.6eV}{n^{2}}$$ (E here is energy expressed in eV)
knowing the Bohr radius of hydrogen, the radius in terms of the energy level is
$$r=n^{2} a_0$$, where $$a_0 = 5.29*10^{-11} m$$ .

I then substituted r into Coulomb's law
$$F = {1 \over 4\pi\varepsilon_0}\frac{q_1 * q_2}{r^2}= 9*10^9 mF^{-1} * {1.60217646*10^{-19}C * -1.60217646*10^{-19}C \over \left(n^{2} * 5.29*10^{-11} m)^2$$

After not really having accomplished much, I don't know how to proceed. Any suggestions, explanations are welcome.

Thank you :)

 

[0xDEADBEEF]

The Coulomb Force field is conservative (no curl no dependence on t) thus there is a potential $$V(r) \propto - \frac{1}{r}$$ that you receive from integration. The Energy of an electron is the energy it needs to escape from the nucleus. If I were you I would calculate the Coulomb potential, plug in the radius which should give you a value for V(r). But Energy is potential plus kinetic energy (E = T + V) so you calculate the kinetic energy T from the speed of the electron. When the electron has escaped the energy is zero T+V=0, by definition of our integration constant.
Have fun!

 

[Entropia]

I can confirm that the approach you are using to calculate the energy levels of hydrogen is correct. By using Coulomb's law and Newtonian mechanics, you can derive the Rydberg formula constant and then use it to calculate the energy levels of hydrogen. However, I understand that you may be struggling with the calculation and may need some guidance.

Firstly, it is important to note that the negative sign in the formula for energy (E = -13.6eV/n^2) indicates that the energy is bound and the electron is in a stable orbit. This is because the electron is attracted to the positively charged nucleus and therefore has a negative potential energy.

To proceed with your calculation, you can use the equation for force (F) in terms of Coulomb's law and the formula for the radius in terms of the energy level (r = n^2a_0). This will give you the force experienced by the electron at each energy level.

Next, you can use the definition of work (W) which is equal to the force (F) multiplied by the distance (r). In this case, the distance is the radius of the orbit (r = n^2a_0). This will give you the work done by the force at each energy level.

Finally, you can use the formula for energy (E) which is equal to the work (W) divided by the charge of the electron (e). This will give you the energy at each energy level, which can then be expressed in terms of eV.

I hope this helps. Keep in mind that these calculations can be complex and may require multiple steps. If you need further assistance, don't hesitate to consult a textbook or seek guidance from a fellow scientist. Good luck with your calculations!