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 determine the distance r from the uranium atom necessary for an electron to reside in equilibrium, we can use Coulomb's law.
Coulomb's law states that the force between two charged particles is given by the equation:
F = (k * q1 * q2) / r^2
Where:
- F is the force between the particles
- k is the electrostatic constant (8.99 x 10^9 Nm^2/C^2)
- q1 and q2 are the charges of the respective particles
- r is the distance between the particles
In this case, the uranium atom is singly ionized, so it has a charge of +1e, and the iron atom is doubly ionized, so it has a charge of +2e. The charge of an electron is -e.
Plugging in the values:
F = (8.99 x 10^9 Nm^2/C^2) * (1e) * (2e) / (28.70 nm)^2
To calculate the magnitude of the force, we take the absolute value of F:
|F| = (8.99 x 10^9 Nm^2/C^2) * (1e) * (2e) / (28.70 nm)^2
Next, we need to convert the distance from nm to meters to match the units of the electrostatic constant.
1 nm = 1 x 10^-9 m
Substituting the values:
|F| = (8.99 x 10^9 Nm^2/C^2) * (1.60 x 10^-19 C) * (3.20 x 10^-19 C) / (28.70 x 10^-9 m)^2
Calculating |F| gives us the magnitude of the force on the electron from the uranium ion.
To determine the distance (r) from the uranium atom necessary for an electron to reside in equilibrium, we need to find the point where the electric force from the uranium ion cancels out the electric force from the iron ion.
The electric force between two charged particles can be calculated using Coulomb's law: F = k * (q1 * q2) / r^2, where F is the force, k is the electrostatic constant (8.988 × 10^9 N m^2/C^2), q1 and q2 are the charges, and r is the separation distance.
In this case, the uranium atom is singly ionized with a charge of +1 (q1 = +1e, where e is the elementary charge 1.6 × 10^-19 C), and the iron atom is doubly ionized with a charge of +2 (q2 = +2e).
Since we want to find the distance r where the forces cancel out, we set the magnitudes of the forces equal to each other and solve for r: F_uranium = F_iron.
Using Coulomb's law, we have:
k * (q_uranium * q_electron) / r^2 = k * (q_iron * q_electron) / (R - r)^2
Simplifying and plugging in the given values:
((8.988 × 10^9 N m^2/C^2) * (1e) * (1.6 × 10^-19 C)) / r^2 = ((8.988 × 10^9 N m^2/C^2) * (2e) * (1.6 × 10^-19 C)) / (28.70 × 10^-9 m - r)^2
Now, we can solve for r. Rearranging the equation and canceling out constants:
r^2 = ((2e) / (e)) * (28.70 × 10^-9 m - r)^2
r^2 = 2 * (28.70 × 10^-9 m - r)^2
Expanding:
r^2 = 2 * (28.70 × 10^-9 m)^2 - 2 * r * 2 * (28.70 × 10^-9 m) + 2 * r^2
Canceling out r^2 terms:
0 = 2 * (28.70 × 10^-9 m)^2 - 4 * r * (28.70 × 10^-9 m)
Simplifying:
r = (2 * (28.70 × 10^-9 m)^2) / (4 * (28.70 × 10^-9 m))
r = 14.35 × 10^-9 m
Therefore, the distance from the uranium atom necessary for an electron to reside in equilibrium is approximately 14.35 nm.
To find the magnitude of the force on the electron from the uranium ion, we can use Coulomb's law again. Plugging in the values:
F_uranium = (8.988 × 10^9 N m^2/C^2) * (1e) * (1.6 × 10^-19 C) / (14.35 × 10^-9 m)^2
Simplifying:
F_uranium ≈ 1.81 × 10^-8 N
Thus, the magnitude of the force on the electron from the uranium ion is approximately 1.81 × 10^-8 Newtons.