Question

A uranium and iron atom reside a distance r = 42.30 nm apart. the uranium atom is singly ionized; the iron atom is doubly ionized. calculate the distance r from the uranium atom necessary for an electron to reside in equilibrium. ignore the insignificant gravitational attraction between the particles

Answer 1

Using this equation:

E(U) = E(Fe) kq / r² = k2q / (R-r)² 1/r² = 2/(R-r)²

(R-r)² = 2r² 

R² - 2Rr + r² = 2r² 

r² + 2Rr - R² = 0 is a quadratic equation to solve 
r² + 84.6r - 1789 = 0 ► solving for the quadratic equation

r² x 10 + 84.6r x 10 - 1789 x 10 = 0 x 10

refining: 10r² + 846r - 17890 = 0
so a = 10, b = 846, and c = -17890

r = -846 + √846² - 4 x 10 (-17890) / 2 x 10

r = 17.52 nm