Question

enter the correct ground - state (or lowest energy) configuration based on the number of electrons: $1s^3 2s^3 2p^9$. express your answer in complete form in the order of orbital filling as a string without blank space between orbitals. for example, $1s^2 2s^2$ should be entered as 1s^22s^2.

Explanation:

Step1: Recall orbital capacity

The \(1s\) orbital can hold a maximum of 2 electrons, \(2s\) can hold 2 electrons, and \(2p\) can hold a maximum of 6 electrons.

Step2: Correct the electron counts

- For \(1s\): The given \(1s^3\) is incorrect. It should be \(1s^2\) (since \(1s\) can hold at most 2 electrons).
- For \(2s\): The given \(2s^3\) is incorrect. It should be \(2s^2\) (since \(2s\) can hold at most 2 electrons).
- For \(2p\): The given \(2p^9\) is incorrect. It should be \(2p^6\) (since \(2p\) can hold at most 6 electrons). But wait, let's check the total electrons. Wait, maybe the original problem has a typo, but assuming we correct each orbital to its maximum capacity first, but actually, the correct ground - state electron configuration for a neutral atom (or ion, but let's assume neutral) should follow the Aufbau principle. Wait, maybe the intended is to correct each orbital to its valid maximum. So \(1s\) max 2, \(2s\) max 2, \(2p\) max 6. But if we sum the electrons: original is \(3 + 3+9=15\) electrons? Wait, no, maybe the problem is to correct the wrong exponents. The correct way is: \(1s\) can have at most 2, so \(1s^2\); \(2s\) at most 2, so \(2s^2\); \(2p\) at most 6, so \(2p^6\). But wait, maybe the problem is that the user made a mistake in the exponents, and the correct configuration (after correcting the over - filled orbitals) would be \(1s^22s^22p^6\) plus the remaining electrons? Wait, no, the original is \(1s^32s^32p^9\). Let's calculate the total electrons: \(3 + 3+9 = 15\) electrons. The electron configuration for 15 electrons (which is phosphorus, but wait, no, 15 electrons: \(1s^22s^22p^63s^23p^3\)). Wait, maybe the problem is that the initial configuration has wrong exponents (since \(1s\) can't have 3, \(2s\) can't have 3, \(2p\) can't have 9). So first, correct each orbital to its maximum: \(1s^2\), \(2s^2\), \(2p^6\) (total 10 electrons), then the remaining \(15 - 10=5\) electrons would go to \(3s^23p^3\). But maybe the problem is just to correct the exponents of \(1s\), \(2s\), and \(2p\) to their maximum possible (since they are over - filled). Wait, the problem says "Enter the correct ground - state (or lowest energy) configuration based on the number of electrons: \(1s^32s^32p^9\)". So first, let's find the total number of electrons: \(3 + 3+9 = 15\) electrons. Now, the correct electron configuration for 15 electrons (atomic number 15, phosphorus) is \(1s^22s^22p^63s^23p^3\). But let's check the steps again.
1. \(1s\) orbital: maximum 2 electrons, so change \(1s^3\) to \(1s^2\).
2. \(2s\) orbital: maximum 2 electrons, so change \(2s^3\) to \(2s^2\).
3. \(2p\) orbital: maximum 6 electrons, so change \(2p^9\) to \(2p^6\). Now, the number of electrons accounted for is \(2 + 2+6 = 10\) electrons. The remaining electrons: \(15-10 = 5\) electrons. These go to the next orbitals, \(3s\) (which can hold 2) and \(3p\) (which can hold 3). So \(3s^23p^3\). So putting it all together: \(1s^22s^22p^63s^23p^3\). But in the format required (like the example \(1s^22s^2\) is entered as \(1s^22s^2\)), so we need to write it as \(1s^22s^22p^63s^23p^3\) (but let's check the original problem's example: the example is \(1s^22s^2\) entered as \(1s^22s^2\), so no spaces. So the correct configuration, after correcting the over - filled orbitals and following the Aufbau principle for 15 electrons, is \(1s^22s^22p^63s^23p^3\). Wait, but maybe the problem is simpler: the user just wants to correct the exponents of \(1s\), \(2s\), and \(2p\) to their maximum (even if the total electrons don't match, but that doesn't make sense). Alternatively, maybe the original problem has a typo, and the intended configuration is for a different number of electrons. But given the problem statement, we need to correct the exponents of each orbital to their maximum allowed (since \(1s\) can't have 3, \(2s\) can't have 3, \(2p\) can't have 9). So:

- \(1s\): max 2, so \(1s^2\)
- \(2s\): max 2, so \(2s^2\)
- \(2p\): max 6, so \(2p^6\)

But then the total electrons would be \(2 + 2+6 = 10\) (neon). But the original has \(3 + 3+9 = 15\) electrons. There is a contradiction. Maybe the problem is that the user made a mistake in writing the exponents, and the correct approach is to recognize that \(1s\) can hold 2, \(2s\) can hold 2, \(2p\) can hold 6. So the correct configuration, assuming that we first correct each orbital to its maximum (even if the total electrons are wrong, but that's the only way to correct the given exponents), would be \(1s^22s^22p^6\). But that's for 10 electrons. Alternatively, maybe the problem is to write the correct configuration by following the Pauli exclusion principle (each orbital can hold at most 2 electrons with opposite spins, and sub - shells have maximum electrons: \(s = 2\), \(p=6\), \(d = 10\), \(f = 14\)). So, to correct \(1s^3\) (wrong, should be \(1s^2\)), \(2s^3\) (wrong, should be \(2s^2\)), \(2p^9\) (wrong, should be \(2p^6\)). Then, if we have remaining electrons, we move to the next orbitals. But the total electrons from the original are \(3+3 + 9=15\). After correcting \(1s\), \(2s\), \(2p\) to \(2\), \(2\), \(6\) (total 10), we have \(15 - 10 = 5\) electrons left. These go to \(3s\) (2 electrons) and \(3p\) (3 electrons). So the correct configuration is \(1s^22s^22p^63s^23p^3\).

Answer:

\(1s^22s^22p^63s^23p^3\)