Physics · Atoms · NEET
Total energy E = KE + PE. The KE is positive (+13.6 Z^2/n^2 eV) but the PE is negative and twice as large in size (-27.2 Z^2/n^2 eV). Adding them: E = 13.6 - 27.2 = -13.6 Z^2/n^2 eV. The negative sign means the electron is BOUND to the nucleus. You must ADD energy to pull it free (to infinity, where E = 0). If total energy were positive, the electron would not stay in a closed orbit.
Start from KE = e^2/(8 pi e0 r) and PE = -e^2/(4 pi e0 r). Notice PE is exactly -2 times KE, so PE = -2 KE. Total E = KE + PE = KE - 2KE = -KE, so E = -KE. Substituting KE = -E into PE = -2KE gives PE = -2(-E) = 2E. This is a result of the inverse-square Coulomb force and is called the virial theorem. It is always true for a 1/r attractive force.
Kinetic energy is ALWAYS positive because KE = (1/2)mv^2 and both mass and v^2 are positive. In numbers KE = +13.6 Z^2/n^2 eV. Only the potential energy and the total energy are negative. A common exam mistake is to write KE as negative - never do that.
Use two shortcuts. KE = -E (same size as total energy but positive). PE = 2E (twice the total energy, stays negative). Example: if E = -3.4 eV, then KE = +3.4 eV and PE = 2 x (-3.4) = -6.8 eV. This exact case was asked in NEET 2019.
KE : PE : E = 1 : -2 : -1. So KE and total energy are equal in magnitude (ratio 1 : -1), and PE is double the total energy. Learn the trio -2 : +1 : -1 for PE : KE : Total and you can answer any sign or ratio question in seconds.
The ratio of kinetic energy to the total energy of an electron in a Bohr orbit of the hydrogen atom is:
The total energy of an electron in an atom in an orbit is -3.4 eV. Its kinetic and potential energies are, respectively:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
KE = e^2/(8 pi e0 r) = +13.6 Z^2/n^2 eV, PE = -e^2/(4 pi e0 r) = -27.2 Z^2/n^2 eV, and Total E = -e^2/(8 pi e0 r) = -13.6 Z^2/n^2 eV. In magnitude PE is twice KE, and total energy equals KE in size but is negative.
KE = -E (total energy). They have the same magnitude but opposite signs. For hydrogen ground state, E = -13.6 eV and KE = +13.6 eV.
PE = 2E. The potential energy is exactly twice the total energy and is always negative. For hydrogen ground state, E = -13.6 eV so PE = -27.2 eV.
E = -13.6 Z^2/n^2 eV. Energy is most negative (deepest) at n = 1 and rises toward 0 as n increases. For hydrogen-like ions it scales with Z^2, so He+ (Z=2) has E = -54.4 eV in its ground state.
A free electron at infinity has E = 0. A bound electron has E = -13.6 Z^2/n^2 eV, which is below 0. To lift it from a negative level up to 0 you must supply energy - that supplied amount is the ionisation energy, equal to +13.6 Z^2/n^2 eV.
