Experimentally it is found that 12.8 eV energy is required to separate a hydrogen atom into a proton and an electron. So the orbital radius of the electron in a hydrogen atom is 9/x × 10⁻¹⁰ m. The value of the x is __________.
(1 eV=1.6 × 10⁻¹⁹ J, 1/(4 π ε₀)=9 × 10⁹ Nm² / C² and electronic charge =1.6 × 10⁻¹⁹ C)

Solution

Physics sir · 4 steps

Energy needed to free the electron is the binding energy, and that is exactly ke²/2r.

1. Binding energy. In a Bohr orbit, kinetic energy is ke²/2r and potential energy is -ke²/r, so the total energy is -ke²/2r. To pull the electron out to rest at infinity you must supply +ke²/2r, here 12.8 eV.

   r = ke²/2E = (9 × 10⁹)(1.6 × 10⁻¹⁹)²/(2 × 12.8 × 1.6 × 10⁻¹⁹)

2. Cancel one charge. One factor of 1.6 × 10⁻¹⁹ cancels, leaving r = (9 × 10⁹ × 1.6 × 10⁻¹⁹)/25.6 = (14.4 × 10⁻¹⁰)/25.6 m.
3. Simplify. 14.4/25.6 = 9/16, so r = 9/16 × 10⁻¹⁰ m.
4. So. Comparing with 9/x × 10⁻¹⁰ m gives x = 16.

The common slip: 8. Setting the ionisation energy equal to ke²/r and dropping the factor of 2 gives r = 9/8 × 10⁻¹⁰ m, so x = 8.

The idea to keep. For an electron in a circular Coulomb orbit, kinetic energy is half the size of potential energy, so the ionisation energy is ke²/2r, not ke²/r.

Answer from the NTA final answer key. Solution by PrepWiser.