Ground State Energy of an Electron

[jbowers9]

This is what I've tried to work out and I'm not getting -13.7 eV. What am I doing wrong?


E 2 Π m e^4 / (4 Π ε0 )^2 h^2 6.90E-19 J=4.31eV

m 9.11 x 10-31 kg 9.11E-31
e 1.60 x 10-19 C 1.60E-19
ε0 8.85 x 10-12 C2/Nm2 8.85E-12
h 6.63 x 10-34 J S 6.63E-34

1 joule = 6.24150974 × 10^18 electron volts

 

[muppet]

Sorry but I have absolutely no idea what that equation is supposed to be, or where you got it from. Try TeXing it perhaps? How did you arrive at it?

 

[Glenn Rudge]

Need more information, have no idea what you're saying.

 

[jbowers9]

The equation is based on the Bohr atom for energy levels. The version I wrote above is for n=1. The development in the text I'm using:

AUTHOR Mortimer, Robert G.
TITLE Physical chemistry / Robert G. Mortimer.
PUB INFO San Diego, Calif. : Academic Press, c2000.
pgs. 511-520 roughly

uses E_n = 2 Π m e⁴ / n² (4 Π ε₀ )² h²

When I plug in the constants, n=1, the value is off from 13.7 eV, after conversion from Joules, by a factor of 3.14, as if Pi doesn't belong in the denominator. I'm thinking that it is already included in the permitivity constant ε₀.

 

[DeShark]

The equation is based on the Bohr atom for energy levels. The version I wrote above is for n=1. The development in the text I'm using:

AUTHOR Mortimer, Robert G.
TITLE Physical chemistry / Robert G. Mortimer.
PUB INFO San Diego, Calif. : Academic Press, c2000.
pgs. 511-520 roughly

uses E_n = 2 Π m e⁴ / n² (4 Π ε₀ )² h²

When I plug in the constants, n=1, the value is off from 13.7 eV, after conversion from Joules, by a factor of 3.14, as if Pi doesn't belong in the denominator. I'm thinking that it is already included in the permitivity constant ε₀.

I presume you mean $$E_n = \frac{2 \pi m_e e^4}{n^2 (4 \pi \epsilon_{0}) h^2}$$

Incidentally, my quantum mech book gives the equation for the energy according to the Bohr model as $$\frac{m_e Z^2 e^4}{(4 \pi \epsilon_0)^2 2 \hbar^2}\frac{1}{n^2}$$

so you're missing a factor of $$2 \pi$$ up top and you're missing a 2 from down below... in other words, you're missing a factor of pi. Which is what you say you're missing. =)