Problem 76 - Chemistry: An Atoms-Focused Approach

Question

How many quantum numbers are needed to identify an electron in an atom?

Step-by-Step Solution

Answer

Answer: A total of four quantum numbers (n, l, m_l, m_s) are needed to uniquely identify an electron in an atom. They provide information about the electron's energy level, orbital shape, orbital orientation, and spin.

Step 1: Understanding Quantum Numbers

Quantum numbers are a set of numerical values that help describe the characteristics and position of an electron within an atom. They give information about the energy level, shape, orientation, and spin of an electron's orbital. In total, there are four quantum numbers that are needed to define an electron's position in an atom.

Step 2: Describing Principal Quantum Number (n)

The first quantum number is called the principal quantum number and is denoted by "n". It is a positive integer (n = 1, 2, 3, ...) and represents the energy level (or shell) in which the electron resides. As the value of n increases, the distance of the electron from the nucleus also increases, meaning the electron is less tightly bound to the nucleus.

Step 3: Describing Azimuthal Quantum Number (l)

The second quantum number is the azimuthal quantum number, represented by "l". This number describes the shape and type of an electron's orbital, and it has integer values ranging from 0 to (n-1). The different types of orbitals are denoted by letters s, p, d, f, where l=0 corresponds to s, l=1 is p, l=2 is d, and l=3 is f.

Step 4: Describing Magnetic Quantum Number (m_l)

The third quantum number is the magnetic quantum number, denoted as "m_l". It describes the orientation of the orbital in space. The value of m_l ranges from -l to +l, including zero. This means that an orbital with an azimuthal quantum number l will have (2*l + 1) different orientations in space.

Step 5: Describing Spin Quantum Number (m_s)

The fourth and final quantum number is the spin quantum number, represented by "m_s". It describes the intrinsic angular momentum or spin of the electron. Electrons, being fermions, have spin values of ±1/2. This means that for any given electron, m_s can either be +1/2 (spin-up) or -1/2 (spin-down).

Step 6: Conclusion

In conclusion, a total of four quantum numbers (n, l, m_l, m_s) are needed to uniquely identify an electron in an atom. These quantum numbers provide information about the electron's energy level, orbital shape, orbital orientation, and spin.