A hydrogen nucleus has a radius of 1 x 10-15 m and the electron is about 5.3 x 10-11 m from the nucleus. Assume the hydrogen atom is a ball with a radius of about 5.3 x 10-11 m and the nucleus is a ball with a radius of 1 x 10-15 m. How much work (in electron volts) must be performed by an external force to bring in another proton (from very far away) to the "surface" of the nucleus?
To calculate the work required to bring in another proton to the surface of the nucleus, we need to consider the electric potential energy.
The formula for electric potential energy is given by:
PE = k * (q1 * q2) / r
where PE is the potential energy, k is Coulomb's constant (8.99 x 10^9 N m^2 / C^2), q1 and q2 are the charges of the two particles (in this case the proton charges are both positive), and r is the distance between the charges.
In this case, we want to find the work in electron volts (eV). 1 eV is defined as the work done in moving an electron through a potential difference of 1 volt.
First, we need to calculate the electric potential energy at the initial distance of 5.3 x 10^{-11} m using the given radius of the electron path. Since the electron and the added proton are both positively charged, the potential energy will be positive.
PE_initial = k * (e * e) / r_initial
where e is the charge of the electron (1.6 x 10^-19 C).
Next, we need to calculate the electric potential energy at the final distance, which is the sum of the proton's radius and the radius of the nucleus (1 x 10^{-15} m + 5.3 x 10^{-11} m).
PE_final = k * (e * e) / r_final
Finally, we can find the work done by subtracting the initial potential energy from the final potential energy:
Work = PE_final - PE_initial
To convert the work from joules to electron volts, we can use the conversion factor that 1 eV is equal to 1.6 x 10^-19 J.
Final Answer:
1. Calculate PE_initial using the formula k * (e * e) / r_initial.
2. Calculate PE_final using the formula k * (e * e) / r_final.
3. Calculate the work by subtracting PE_initial from PE_final.
4. Convert the work from joules to electron volts by dividing by 1.6 x 10^-19.
Until you get to the electron, no work needs to be done because there is no net charge on the atom.
From the electron ball (R1 radius) in to the proton ball (R2 radius), there is a repulsive force of
F = k/r^2 form
(I am leaving the Coulomb law numbers for you to do)
the work done in going from radius R1 to radius R2 is
integral from R1 to R2 of k dr /r^2
which is -(k/R1 -K/R2)
note that because R2 is less than R1, the quantity inside the parentheses is negative and therefore the work done is positive)