Final answer:
Using the energy levels of a hydrogen atom, we find that a 12.5 eV electron beam can excite a hydrogen atom up to the n=3 energy level because the transition from n=1 to n=3 requires 12.09 eV, which is within the provided energy limit.
Step-by-step explanation:
The question involves determining up to which energy level a 12.5 eV electron beam can excite a hydrogen atom. To solve this, we need to understand the energy levels of a hydrogen atom and how these correspond to different principal quantum numbers (n). The energy of an electron in a hydrogen atom is given by the formula En = -13.6 eV / n2, where n is the principal quantum number of the electron's orbit.
To find out up to which energy level (n) the hydrogen atom can be excited with 12.5 eV, we can calculate the energy required to move an electron from n=1 to higher levels until the required energy surpasses 12.5 eV. Here are the energies required for the first four transitions (from n=1):
- From n=1 to n=2: E2 - E1 = -3.4 eV - (-13.6 eV) = 10.2 eV
- From n=1 to n=3: E3 - E1 = -1.51 eV - (-13.6 eV) = 12.09 eV
Here we see that the transition from n=1 to n=2 requires 10.2 eV, and from n=1 to n=3 requires 12.09 eV, which is close to our 12.5 eV limit. Any transition above n=3 would exceed the 12.5 eV provided by the electron beam. Hence, the hydrogen atom can be excited up to the n=3 energy level with a 12.5 eV electron beam.