An electron in a singly ionized helium atom (He ^+ ) has a potential energy of -6.8 eV. The radial distance of the electron from the nucleus in this state is approximately : (Take 1 eV = 1.6 10⁻¹⁹ J, e = 1.6 10⁻¹⁹ C and 1 4 ₀ = 9 10⁹ N m ^2 /C ^2 )

Step-by-step solution

The potential energy U of an electron in a hydrogen-like ion is given by: U = - 1 4 ₀ Z e^2 r For a singly ionized helium atom (He ^+ ), the atomic number Z = 2 . The given potential energy is U = -6.8 eV. Converting this to Joules: U = -6.8 1.6 10⁻¹⁹ J Equating the two expressions for potential energy: -6.8 1.6 10⁻¹⁹ = - 9 10^9 2 (1.6 10⁻¹⁹)^2 r Dividing both sides by -1.6 10⁻¹⁹ : 6.8 = 9 10^9 2 1.6 10⁻¹⁹ r 6.8 = 28.8 10⁻¹⁰ r r = 28.8 10⁻¹⁰ 6.8 r 4.235 10⁻¹⁰ m This is approximately 4.2 10⁻¹⁰ m. Answer: 4.2 10⁻¹⁰ m