Question

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).

Short answer

Answer: 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).

Step-by-step solution

Finding electronic configuration using spectroscopic notation

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: \(1s^{2}, 2s^{2}, 2p^{2}\)

Write the first three quantum numbers (n, l, m) for each orbital

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)

Assign the fourth quantum number (ms) to each electron and list all 4 quantum numbers

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).