How much energy (in eV) is needed for an electron revolving in the second orbit of He+ ion inorder to double its angular momentum

To solve this problem, we need to use the principles of quantum mechanics and the Bohr model of the atom.

Step 1: Understand the problem The problem is asking for the energy needed to double the angular momentum of an electron in the second orbit of a He+ ion. According to the Bohr model, the angular momentum of an electron in an orbit is quantized and is given by L = nħ, where n is the principal quantum number (the number of the orbit) and ħ is the reduced Planck's constant.

Step 2: Calculate the initial and final angular momentum The initial angular momentum (L1) in the second orbit (n=2) is L1 = 2ħ. If we want to double this, the final angular momentum (L2) will be L2 = 2L1 = 4ħ.

Step 3: Calculate the initial and final energy levels The energy of an electron in an orbit is given by E = -13.6Z^2/n^2 eV, where Z is the atomic number and n is the principal quantum number. For a He+ ion, Z=2. So the initial energy (E1) in the second orbit (n=2) is E1 = -13.6*(2)^2/2^2 = -13.6 eV. The final energy (E2) in the fourth orbit (n=4, because the angular momentum is 4ħ) is E2 = -13.6*(2)^2/4^2 = -3.4 eV.

Step 4: Calculate the energy difference The energy needed to move an electron from one energy level to another is given by the difference in the energy levels. So the energy needed is ΔE = E2 - E1 = -3.4 - (-13.6) = 10.2 eV.

So, 10.2 eV of energy is needed for an electron revolving in the second orbit of a He+ ion in order to double its angular momentum.