Optical Spectroscopy and Atomic Structure

[sheri1987]

Homework Statement

Calculate the energy and radius for each of the five lowest (n = 1,2,3,4,5) electron orbits in a Helium atom with only one electron. Enter the energy as a positive value in units of "eV" and the radius in units of "nm". Hint: The Helium nucleus has 2 protons so Z = 2 and the ground state energy of Helium will be E0 = k(Ze)2/2r.

If n=2, find E and r1

Homework Equations

The Attempt at a Solution

I know how to answer the question, but I need help with finding the ground Energy E0 of Helium. I keep ending up with incorrect answers using that equation given in the problem, can anyone help me solve for the ground energy of Helium?

 

[G01]

Your using the correct formula.

In order to find your mistake, I'll have to see your calculations.

 

[thomate1]

The ground state energy of Helium (E0) can be calculated using the equation given in the problem: E0 = k(Ze)^2/2r. In this case, Z=2 and the charge of a proton (e) is equal to 1.602 x 10^-19 C. The constant k is the Coulomb constant with a value of 8.987 x 10^9 Nm^2/C^2. The radius (r) can be calculated using the Bohr radius equation: r = n^2h^2/4π^2mke^2, where n is the energy level, h is the Planck's constant (6.626 x 10^-34 J.s), m is the mass of an electron (9.109 x 10^-31 kg), and e is the charge of an electron.

For n=1, we have E0 = (8.987 x 10^9 Nm^2/C^2)(2)(1.602 x 10^-19 C)^2/2(5.291 x 10^-11 m) = -109.7 eV. This is the ground state energy of Helium.

For n=2, we can find the radius (r1) using the Bohr radius equation: r1 = (2^2)(6.626 x 10^-34 J.s)^2/4π^2(9.109 x 10^-31 kg)(8.987 x 10^9 Nm^2/C^2)(1.602 x 10^-19 C)^2 = 0.5291 nm. Then, we can calculate the energy (E) using the equation given in the problem: E = (8.987 x 10^9 Nm^2/C^2)(2)(1.602 x 10^-19 C)^2/2(0.5291 x 10^-9 m) = -27.4 eV.

Similarly, for n=3,4,5, we can follow the same steps to calculate the energy and radius for each energy level.

Note: The negative sign indicates that the electron is bound to the nucleus, and the energy values will be negative for all energy levels in a Helium atom with only one electron.