How Do You Calculate the Ionization Energy of Hydrogen Using Quantum Mechanics?

[erik05]

Hello everybody, just wondering if anyone could point me in the right direction for this question. Please and thanks.

1) Since the ionisation process takes an electron from its ground state to the zero level of the energy scale, the ground state energy calculated for the electron of the hydrogen atom can be compared to the negative of its ionization energy (IE). Use the equation E= -2.178 x 10^-18 (Z^2/n^2) for energy of a valence level n, to build a equation that can be used to calculate the ionisation energy of H in units of kJ/mol.

So far what I have is...

Since Z represents the atomic number and the atomic number for hydrogen is 1:

E= -2.178 x 10^-18 (1^2/n^2)

so equation is E= -2.178 x 10^-18 (1/n^2)

2) Use equation to calculate ionization IE for H and confirm that value obtained is consistent with the experimental value of 1310 kJ/mol.

E= -2.178 x 10^-18 (1/ (1-0)^2) since electron going from ground state to zero level therefore:
E= -2.178 x 10^-18 J per 1 atom. So then,

E= -2.178 x 10^-18 J/atom x (6.022 x 10^23 atom/mol) / 1000
= -1312 kJ/mol

 

[Gokul43201]

Your answer is correct, but there is a tiny error in the way you've substituted in the formula.

Where did you get 1/(1-0)^2 from ? The ionization energy is the energy difference between the first shell and an infinitely removed shell. So the shell dependence goes like $1/n_1^2 - 1/n_2^2 = 1/1^2 - 1/\infty^2 = 1 - 0 = 1$. The rest is good.

PS : Didn't I answer an almost identical question recently ?

EDIT : Found it : https://www.physicsforums.com/showthread.php?t=93440

 

[Blueshift5]

The equation you have constructed is correct. To calculate the ionization energy of hydrogen, we use the equation E= -2.178 x 10^-18 (Z^2/n^2), where Z=1 for hydrogen and n is the principal quantum number of the valence level. In this case, n=1 since we are considering the ground state of hydrogen.

To confirm the value obtained, we can convert the energy from Joules to kJ/mol by multiplying it by Avogadro's number (6.022 x 10^23 atoms/mol) and dividing by 1000 to get the energy in kJ/mol. This results in a value of -1312 kJ/mol, which is consistent with the experimental value of 1310 kJ/mol.

The negative sign in the equation indicates that the ionization energy is the amount of energy needed to remove an electron from the atom. This value is always positive, so we use the negative sign to indicate that energy is being released from the atom during ionization.

In summary, the ionization energy equation E= -2.178 x 10^-18 (Z^2/n^2) can be used to calculate the ionization energy of hydrogen in units of kJ/mol and the value obtained using this equation is consistent with the experimental value. This equation can also be used for other atoms, with Z representing the atomic number and n representing the principal quantum number of the valence level.