Final answer:
The highest occupied state for 6 electrons in a box with a length of 10 nm is n=2. This is because the n=1 state can hold up to 2 electrons, and the n=2 state can hold up to 8 electrons, thus accommodating all 6 electrons comfortably without the need to occupy a higher state.
Step-by-step explanation:
The student is asking about the highest occupied quantum state in a box with a given number of electrons. The principal quantum number, n, describes the energy level in which electrons can reside. The electrons fill from the lowest energy state up, with each energy level (or shell) accommodating a maximum number of electrons based on the formula 2n².
For example, when n = 2, there are four orbitals and according to the formula, this level can hold up to 2(2²) = 8 electrons. Moving forward, if there are 6 electrons, they would fill up the n = 1 state completely (which can hold 2 electrons) and the remaining 4 electrons would go into the n = 2 state. Since the n = 2 state can accommodate 8 electrons and we are only adding 4, there is still room and no need to go to a higher state.
Therefore, the highest occupied state for 6 electrons in a box is n = 2.