Gideon Robert University Assignment Notes

GIDEON ROBERT UNIVERSITY

SCHOOL OF MATHEMATICS AND NATURAL SCIENCES

Assignment Two

Due Date: 15/04/2026
Time: 13 hours
Lecturer: Mambwe W.

Instructions

• Write your full name and ID number on the cover page.
• Note that no other assignment will be given.

Problems to Solve:

1. Calculate the radius of Bohr’s orbit in the ground state.

   • To find the radius of Bohr's orbit (specifically the first orbit or ground state), the formula is given by: $r_n = n^2 imes \frac{h^2}{4 ext{π}^2 m e^2}$ Where:
     • $n$ = principal quantum number (for ground state, $n = 1$)
     • $h$ = Planck's constant ($h ext{ = } 6.626 imes 10^{-34} ext{ J·s}$)
     • $m$ = mass of electron (approximately $9.11 imes 10^{-31} ext{ kg}$)
     • $e$ = elementary charge (approximately $1.602 imes 10^{-19} ext{ C}$)

2. Show that the energy of an electron in its orbit is:

   • The energy of the electron in a hydrogen atom can be expressed as:
     $E_n = -\frac{m e^4}{2 ext{ħ}^2 n^2}$
     Where:
   • $E_n$ = energy of the electron in orbit
   • $m$ = mass of the electron
   • $e$ = elementary charge
   • $ ext{ħ}$ = reduced Planck's constant ($ ext{ħ} = rac{h}{2 ext{π}}$)
   • $n$ = principal quantum number

3. Calculate the longest wavelength in the Lyman series.

   • The longest wavelength in the Lyman series corresponds to the transition from $n = 2$ to $n = 1$.
   • Using the Rydberg formula for hydrogen:
     $\frac{1}{ ext{λ}} = R_H imes \bigg(\frac{1}{n_1^2} - \frac{1}{n_2^2}\bigg)$
     Where:
   • $R_H = 1.097 imes 10^7 ext{ m}^{-1}$ (Rydberg constant)
   • Solve for $ ext{λ}$ as follows:
     $ext{λ}_{ ext{longest}} = \frac{1}{R_H imes \bigg(\frac{1}{1^2} - \frac{1}{2^2}\bigg)}$

4. The longest wavelength emitted in the Paschen series.

   • The longest wavelength in the Paschen series corresponds to the transition from $n = 4$ to $n = 3$.
   • Applying the same Rydberg formula as in part 3:
     $ext{λ}_{ ext{Paschen}} = \frac{1}{R_H imes \bigg(\frac{1}{3^2} - \frac{1}{4^2}\bigg)}$

5. The shortest wavelength emitted in the Balmer series.

   • The shortest wavelength in the Balmer series occurs when the transition is from $n = 3$ to $n = 2$.
   • Use the Rydberg formula:
     $ext{λ}_{ ext{shortest}} = \frac{1}{R_H imes \bigg(\frac{1}{2^2} - \frac{1}{3^2}\bigg)}$

6. A metal surface is receiving light of wavelength $325 imes 10^{-9} ext{ m}$. If the work function for the metal is $2.46 ext{ eV}$, calculate the maximum kinetic energy of the photoelectrons emitted from the metal surface.

   • The energy of the incident photon can be calculated using:
     $E = \frac{hc}{ ext{λ}}$
   • Where:
     • $h$ = Planck's constant
     • $c$ = speed of light ($c = 3.00 imes 10^8 ext{ m/s}$)
   • Kinetic energy can be found using:
     $K.E. = E - ext{Work Function}$
   • Thus:
     $K.E. = \frac{hc}{ ext{λ}} - 2.46 ext{ eV}$

7. Light of wavelength $5.00 imes 10^{-7} ext{ m}$ falls on a material that has a photoelectric work function of $2.0 ext{ eV}$. Find:

   • (a) The energy of each individual atom.
     • The energy is given by:
       $E = \frac{hc}{ ext{λ}}$
   • (b) The kinetic energy of the most energetic photoelectron.
     • Use:
       $K.E. = E - ext{Work Function}$
   • (c) The stopping potential.
     • The stopping potential can be calculated using:
       $V = \frac{K.E.}{e}$
       Where $e$ is the charge of an electron ($1.602 imes 10^{-19} ext{ C}$).

8. Rest mass and speed of electrons and protons.

   • The rest mass of a proton is approximately $2000$ times the rest mass of an electron.
   • To find the speed at which the electron should move so that its mass is equal to the rest mass of the proton, we use the relativistic mass increase formula:
     $m = \frac{m_0}{ ext{sqrt}(1 - \frac{v^2}{c^2})}$
   • Set $m = m_p$ (rest mass proton) and solve for $v$:
     v = c imes ext{sqrt}igg{(}1 - igg{(}rac{m_e}{m_p}igg{)}^2igg{)}
   • Where:
     • $m_e$ is rest mass of the electron
     • $m_p$ is rest mass of the proton
     • $c$ is the speed of light ($3.00 imes 10^8 ext{ m/s}$)

9. Radioactive Decay Process:

   • A radioactive nucleus $X$ undergoes $\beta^-$ decay to nucleus $Y$.
     • (i) What could be the process giving rise to the positrons?
     • This could be a process involving $\beta^+$ decay (positron emission).
     • (ii) What are the expected end-point energies of the two positron groups?
     • These energies can be categorized as the energy levels involved in the decay process, which correspond to distinct energy values seen in the provided figure:
       • $5.53 ext{ MeV}$
       • $4.14 ext{ MeV}$
       • $1.38 ext{ MeV}$
       • $0 ext{ MeV}$

Conclusion

• Ensure to follow both the provided instructions and calculations thoroughly, as accuracy is crucial for the completion of your assignment.
• Follow up on any remaining questions or clarifications needed from your lecturer, Mambwe W.