What is the electronic configuration of the ground state of the carbon atom? Write it in the following ways: (a) using spectroscopic notation \(\left(1 s^{2}, \ldots\right) ;\) (b) listing the four quantum numbers for each of the electrons. Note that there may be more than one possibility in (b).

To write the electronic configuration of carbon, fill the electrons in the atomic orbitals according to the Aufbau principle (1s, 2s, 2p, 3s, 3p, …). Carbon has 6 electrons, and they will be filled like this: 1s, 2s, 2px, 2py, 2pz. In spectroscopic notation, the notation will look like this: \(1s^{2}, 2s^{2}, 2p^{2}\)

We'll use the following notation for the quantum numbers: (n, l, m) Where n is the principal quantum number, l is the angular momentum quantum number, and m is the magnetic quantum number. 1s: (1, 0, 0) 2s: (2, 0, 0) 2px: (2, 1, -1) 2py: (2, 1, 0) 2pz: (2, 1, 1)

To satisfy the Pauli Exclusion Principle, each orbital can have a maximum of two electrons with opposite spins (+1/2 and -1/2). For each orbital, we'll list both electrons if applicable, and remember that there can be alternative possibilities. Electron 1: (1, 0, 0, +1/2) Electron 2: (1, 0, 0, -1/2) Electron 3: (2, 0, 0, +1/2) Electron 4: (2, 0, 0, -1/2) Electron 5: (2, 1, -1, +1/2) Electron 6: (2, 1, 0, +1/2) or (2, 1, 1, +1/2) In conclusion, the electronic configuration of the ground state of the carbon atom in the spectroscopic notation is \(1 s^{2}, 2 s^{2}, 2 p^{2}\), and the possible quantum numbers for each electron are Electron 1: (1, 0, 0, +1/2), Electron 2: (1, 0, 0, -1/2), Electron 3: (2, 0, 0, +1/2), Electron 4: (2, 0, 0, -1/2), Electron 5: (2, 1, -1, +1/2), Electron 6: (2, 1, 0, +1/2) or (2, 1, 1, +1/2).