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

[TRE]

Homework Statement

Verify that the equation of the ground state energy Eo of the Bohr atom: E_o= (2pi²e⁴m_ek²)/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

E_o= (2pi²e⁴m_ek²)/h²

The Attempt at a Solution

m_e = 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.m_e.k² = 0 which means that the whole equation is equal to zero thus E_o = 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.

 

[Simon Bridge]

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.

 

[Simon Bridge]

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.

 

[ehild]

Homework Statement

Verify that the equation of the ground state energy Eo of the Bohr atom: E_o= (2pi²e⁴m_ek²)/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

E_o= (2pi²e⁴m_ek²)/h²

The Attempt at a Solution

m_e = 9.10938291 x 10^-19 Kg

there is a mistake here: m_e = 9.10938291 x 10^-31kg.

[STRIKE]k = 1.3806488 x 10^-23J/K[/STRIKE]

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

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

 

[TRE]

To ehild

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

Regards

 

[ehild]

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