Physics · Atoms · NEET
Just three, each with one base number. Radius: r_n = 0.53 (n^2/Z) angstrom (base 0.53 A at n=1, Z=1). Velocity: v_n = 2.19 x 10^6 (Z/n) m/s. Energy: E_n = -13.6 (Z^2/n^2) eV. Almost every numerical is one of these scaled by n and Z. Memorise the three base numbers 0.53, 2.19 x 10^6 and 13.6, and you can rebuild any other quantity.
They come from two conditions solved together. The Coulomb force equals the centripetal force (ke^2/r^2 = mv^2/r), and Bohr quantisation says mvr = nh/2pi. Solving these two gives r proportional to n^2 (radius grows fast) and v proportional to 1/n (electron slows in higher orbits). So outer orbits are much bigger AND the electron there moves slower.
It is negative: E_n = -13.6 (Z^2/n^2) eV. Negative means the electron is bound (trapped). The relations are KE = -E_n (positive) and PE = 2E_n (negative). So total energy = KE + PE = (-E_n) + (2E_n) = E_n, which is negative. If E were positive the electron would be free, not in an atom.
For hydrogen-like atoms: PE = 2 x (total energy) and KE = -(total energy). So if total energy E = -3.4 eV, then KE = +3.4 eV and PE = -6.8 eV. Quick ratio to memorise: KE : PE : E = 1 : -2 : -1. This exact ratio is a repeated NEET question.
Z is the nuclear charge number (H = 1, He+ = 2, Li2+ = 3). Radius has Z in the bottom: r_n = 0.53 (n^2/Z), so higher Z means a smaller orbit. Velocity and energy have Z on top: v proportional to Z/n, and E = -13.6 (Z^2/n^2) eV, so higher Z means faster electron and more tightly bound (more negative energy). For He+ ground state, E = -13.6 x 4 = -54.4 eV.
Radius = size of the orbit (in angstrom or metres), scales as n^2/Z. Energy = how bound the electron is (in eV), scales as -Z^2/n^2. Trap check: radius has n^2 on TOP and Z on the BOTTOM; energy has Z^2 on TOP and n^2 on the BOTTOM. They are almost mirror images, which is why students swap them.
The radius of the innermost orbit of a hydrogen atom is 5.3 x 10^-11 m. What is the radius of the third allowed orbit of the hydrogen atom?
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.
The first Bohr radius (n = 1, Z = 1) is a_0 = 0.53 angstrom = 0.53 x 10^-10 m = 5.3 x 10^-11 m. It is the smallest orbit of hydrogen and the base for r_n = 0.53 (n^2/Z) A.
v_1 = 2.19 x 10^6 m/s (about 1/137 of the speed of light). For any orbit v_n = 2.19 x 10^6 (Z/n) m/s, so it drops as 1/n and rises with Z.
Putting n = 1 and Z = 1 into E_n = -13.6 (Z^2/n^2) eV gives E_1 = -13.6 eV. The minus sign shows the electron is bound; 13.6 eV is also the ionisation energy of hydrogen (energy to free the electron).
The revolution frequency f = v_n / (2 pi r_n). Since v is proportional to 1/n and r is proportional to n^2, f is proportional to 1/n^3. So the electron revolves much more slowly in higher orbits.
For a hydrogen-like atom, KE : PE : E = 1 : -2 : -1. So KE = -E (positive), PE = 2E (negative), and total energy E is negative. This single ratio answers many NEET numericals directly.
