Bohr Model of the Hydrogen atom: Prove that Eo = 13.6 eV

Click For Summary

Discussion Overview

The discussion centers on verifying the equation for the ground state energy Eo of the Bohr atom, specifically showing how it simplifies to Eo = 13.6 eV. Participants explore the mathematical and unit conversions involved in this verification process.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant presents the equation Eo = (2pi²e⁴mek²)/h² and attempts to simplify it to show that Eo = 13.6 eV, but claims to arrive at Eo = 0, suggesting a potential error in calculations.
  • Another participant questions the arithmetic, noting that multiplying non-zero numbers should not yield zero, and suggests checking for rounding errors in calculations.
  • A third participant points out a mistake in the mass of the electron (me), correcting it from 9.10938291 x 10^-19 Kg to 9.10938291 x 10^-31 Kg.
  • This participant also clarifies that the constant k referenced is not the Boltzmann constant but rather the Coulomb's law constant (ke = 8.9875 Nm²/C²).
  • There is a correction regarding the charge e, emphasizing that it should not be replaced with 1 eV, as e is the elementary charge (1.6 x 10^-19 C) and 1 eV is a measure of energy.
  • Another participant advises on converting energy from Joules to eV by dividing by 1.6 x 10^-19 J/eV and suggests using scientific notation for calculations involving very small or large numbers.

Areas of Agreement / Disagreement

Participants generally agree that there are errors in the initial calculations presented, but there is no consensus on the final resolution of the problem, as the discussion remains focused on identifying and correcting mistakes rather than arriving at a definitive conclusion.

Contextual Notes

Limitations include potential misunderstandings of constants and units, as well as unresolved steps in the mathematical simplification process that could affect the outcome.

TRE
Messages
3
Reaction score
0

Homework Statement



Verify that the equation of the ground state energy Eo of the Bohr atom: Eo= (2pi2e4mek2)/h2
simplifies to Eo = 13.6 eV. Show clearly how the units of the different
quantities in the equation simplify to the eV.

This is all they give. Nothing more.



Homework Equations




Eo= (2pi2e4mek2)/h2



The Attempt at a Solution



me = 9.10938291 x 10-19 Kg

k = 1.3806488 x 10-23J/K

h = 6.62606957 x 10-34J.s

1.6 x 10-19J = 1 eV

2 pi2 = 19.7392088


e4 = 6.589333674 x 10-76 C4

k2 = 1.906195527 x 10-46J2/K2

h2 = 4.390478986 x 10-67J2.s2

when you multiply the above constants together : 2pi2.e4.me.k2 = 0 which means that the whole equation is equal to zero thus Eo = 0 and clearly Eo is not equal to 0 but to 13.6 eV

Also for the units I obtain for the answer when I multiply together and then divide is (C4.Kg/(K2.s2) which is not correct and I did not see any other units for the constants we have to use.
 
Physics news on Phys.org
when you multiply the above constants together : 2pi2.e4.me.k2 = 0
Does that make sense mathematically:
How can you multiply a lot of non-zero numbers together and get zero?

Perhaps your calculator has rounded off?
How should you go about multiplying very small numbers - hint: exploit the scientific notation.
 
Sorry you feel that way ... and yes, I have done this myself. Many times.

You have clearly made a mistake in the arithmetic since it is not possible to multiply a bunch of non-zero numbers together and get zero for the answer. If it is not a matter of your calculator deciding that 10^-70 or whatever is zero, then it is something else.

GO through the calculation carefully and see which step gets you the zero.

Note: if the dimensions do not match, then the equation is wrong.
 
TRE said:

Homework Statement



Verify that the equation of the ground state energy Eo of the Bohr atom: Eo= (2pi2e4mek2)/h2
simplifies to Eo = 13.6 eV. Show clearly how the units of the different
quantities in the equation simplify to the eV.


Homework Equations




Eo= (2pi2e4mek2)/h2



The Attempt at a Solution



me = 9.10938291 x 10-19 Kg
there is a mistake here: me = 9.10938291 x 10-31kg.

TRE said:
[STRIKE]k = 1.3806488 x 10-23J/K[/STRIKE]

That 'k' is not the Boltzmann constant, but the constant ke in Coulomb's Law: ke=8.9875 Nm2C-2 .

TRE said:
h = 6.62606957 x 10-34J.s

e=1.6 x 10-19[STRIKE]J = 1 eV[/STRIKE]
You can not replace e with 1 eV. eV is energy, e is charge. e=1.6 x 10-19 C and 1 eV= 1.6 x 10-19 J.

When you multiply the constants, you get the energy in Joules. You have to convert it to eV-s by dividing it 1.6x 10-19 J/eV

Do not forget to set your calculator to SCI mode, to use the normal form of numbers.

ehild
 
  • Like
Likes   Reactions: 1 person
To ehild

Thank You very much for your help, I appreciate it. I finally understand where I went wrong.

Regards
 
You are welcome.

Just a hint: when you have to calculate with very big or very small numbers, treat the exponents separately, add and subtract them.

ehild
 
  • Like
Likes   Reactions: 1 person

Similar threads

Energy of Li2+ atom according to Bohr's model
  • · Replies 2 ·
Replies
2
Views
2K
Rutherford model of atom and Bohr model
  • · Replies 1 ·
Replies
1
Views
2K
Ionization Energy of Helium Atom | 24.6 eV
  • · Replies 4 ·
Replies
4
Views
2K
What is the Orbit of Hydrogen Atom for an Electron at 734 km/s?
  • · Replies 3 ·
Replies
3
Views
2K
Energy transition (bohrs model)
  • · Replies 1 ·
Replies
1
Views
2K
A hydrogen atom transitions from n= 5 state to the ground state
  • · Replies 3 ·
Replies
3
Views
2K
Electrostatic Energy in the Hydrogen Atom
  • · Replies 4 ·
Replies
4
Views
4K
About The Bohr Model of the Hydrogen Atom
  • · Replies 2 ·
Replies
2
Views
3K
Calculating Electric Force in Bohr Model of Hydrogen Atom
  • · Replies 4 ·
Replies
4
Views
6K
Bohr's model of hydrogen like atoms question.
  • · Replies 7 ·
Replies
7
Views
3K