Chemistry

A one-electron atom is an atom with Z protons in the nucleus and one electron. For example, Z = 2 for helium and Z = 3 for lithium. Use our class discussion of the allowed radii and energies of the hydrogen atom with Z =1 as a guide to answering the following questions where the atom has Z protons and one electron. Represent the electron mass as � and the distance between the nucleus and electron as r.

a) Write down an expression for the potential energy, U, between the nucleus with Z protons and the electron.

b) Write down the Virial Theorem. Use it and the potential energy from part a) to write down an expression for the kinetic energy, K, of the electron.

c) Use your answers to part a) and b) to write down an expression for the total energy, E, of the electron.

d) Use the fact that the angular momentum, L, of the electron has quantized values of �ℏ to write another expression for the kinetic energy, K, of the electron. Show your work on this part.

e) Equate the two expressions for kinetic energy from parts b) and d) to write an expression for the allowed radii, rn, of the electron. Show your work on this part.

f) Combine your expressions from parts c) and e) to write an expression for the allowed energies in the atom with Z protons and one electron. Give your answer in terms of the hydrogen ground state energy of -13.634 eV, Z, and n. Show your work on this part.

g) Compute the energy levels of an electron in an ionized helium atom (He II), i.e., a oneelectron atom with Z = 2 for n = 1 and n = 2. In other words, compute E1 and E2 in eV. Show your work in this part.

h) Compute the wavelengths of a photon that can excite the electron in a helium atom from n = 1 to n = 2. Give your answer to the nearest integer in units of nm. Show your work in this part.

i) Compare your answer from part h) to the 122 nm photon that can excite a hydrogen atom from n =1 to n =2.

can you give me the answer (f,g,h,i)