A 12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. What series of wavelengths will be emitted?
The hydrogen atoms will be excited to n=3 state, and both Lyman series and Balmer series wavelengths will be emitted.
Explanation
The textbook context states that at room temperature, most hydrogen atoms are in the ground state. The energy required to excite from ground state (n=1) to n=2 is 10.2 eV, and to n=3 is 12.09 eV. Since the electron beam has 12.5 eV, it can excite atoms to n=3 (12.09 eV < 12.5 eV) but not to n=4 (would require more energy). From n=3, electrons can transition to n=1 directly (Lyman series) or cascade via n=2 (Balmer series then Lyman series).
Solution Steps
Step 1: At room temperature, hydrogen atoms are in ground state (n=1).
Step 2: From context: Energy to excite to n=2 = 10.2 eV; Energy to excite to n=3 = 12.09 eV.
Step 3: Since 12.5 eV > 12.09 eV, atoms can be excited to n=3. But 12.5 eV < energy needed for n=4, so maximum excitation is n=3.
Step 4: From n=3, possible transitions are: n=3→n=1 (Lyman series), n=3→n=2 (Balmer series), and n=2→n=1 (Lyman series).
Step 5: Therefore, both Lyman series and Balmer series wavelengths will be emitted.