A uranium and iron atom reside a distance R = 28.70 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. What is the magnitude of the force on the electron from the uranium ion?

To calculate the distance r from the uranium atom necessary for an electron to reside in equilibrium, we need to consider the balance between the attractive Coulomb force exerted by the uranium ion and the repulsive Coulomb force exerted by the iron ion on the electron.

The attractive force between the uranium ion and the electron can be calculated using Coulomb's law:

F_u = (k * q_u * q_e) / r_u

Where:
F_u = Force exerted by the uranium ion on the electron
k = Coulomb's constant (k = 8.99 × 10^9 N m^2/C^2)
q_u = Charge of the uranium ion (in Coulombs)
q_e = Charge of the electron (in Coulombs)
r_u = Distance between the uranium ion and the electron (given as 28.70 nm)

Since the uranium atom is singly ionized, it has a charge of +1e, where e is the elementary charge (e = 1.6 × 10^-19 C). Therefore, q_u = +1e.

The repulsive force between the iron ion and the electron can be calculated in a similar way:

F_i = (k * q_i * q_e) / r_i

Where:
F_i = Force exerted by the iron ion on the electron
q_i = Charge of the iron ion (in Coulombs)
r_i = Distance between the iron ion and the electron (which is equal to r)

Since the iron atom is doubly ionized, it has a charge of +2e. Therefore, q_i = +2e.

In equilibrium, the forces F_u and F_i must be equal in magnitude and opposite in direction. So, we can equate them:

F_u = F_i

Substituting the known values, we get:

(k * q_u * q_e) / r_u = (k * q_i * q_e) / r

Cancelling the common terms, we can solve for r:

r = (q_i * r_u) / q_u

Substituting the values:

r = (2e * 28.70 * 10^-9 m) / (1e)

r = 57.40 nm

So, the distance r from the uranium atom necessary for an electron to reside in equilibrium is 57.40 nm.

To calculate the magnitude of the force on the electron from the uranium ion, we can substitute the value of r into the equation for F_u:

F_u = (k * q_u * q_e) / r_u

Substituting known values:

F_u = (8.99 × 10^9 N m^2/C^2) * (1e) * (1.6 × 10^-19 C) / (28.70 × 10^-9 m)

Calculating the magnitude of the force F_u, we get:

F_u ≈ 4.69 × 10^-8 N

So, the magnitude of the force on the electron from the uranium ion is approximately 4.69 × 10^-8 Newtons.