Question

Which sct of quantum numbers represent the clectron of the lowest encrgy? (1) \(n=2, l=0, m=0, s=-\frac{1}{2}\) (2) \(n=2, l=1, m=0, s=+\frac{1}{2}\) (3) \(n=4, l=1, m=0, s=+\frac{1}{2}\) (4) \(n=4, l=0, m=0, s=-\frac{1}{2}\)

Short answer

Option (1): n=2, l=0, m=0, s=-1/2.

Step-by-step solution

Understanding Quantum Numbers

Quantum numbers describe the properties of atomic orbitals and the properties of electrons in these orbitals. The four quantum numbers are: Principal Quantum Number (n), Azimuthal Quantum Number (l), Magnetic Quantum Number (m), and Spin Quantum Number (s). The principal quantum number, n, determines the energy level and size of the orbital. The azimuthal quantum number, l, determines the shape of the orbital. The magnetic quantum number, m, determines the orientation of the orbital, and the spin quantum number, s, determines the spin orientation of the electron.

Determine the Energy Level

The principal quantum number (n) primarily determines the energy of the electron. The lowest energy state has the smallest value of n. In the given options, the smallest value of n is 2.

Compare within the Same Principal Quantum Number

For the same value of n, the energy increases with the azimuthal quantum number (l). Therefore, we need to compare the options with the smallest n value (which is 2) and then look at the value of l. The options are: (1) n=2, l=0, and (2) n=2, l=1. Since l=0 is lower than l=1, (1) will have the lower energy.

Conclusion

Option (1) with quantum numbers n=2, l=0, m=0, s=-1/2 represents the electron with the lowest energy.

Key concepts

Principal Quantum Number

The principal quantum number, denoted as n, is one of the four quantum numbers assigned to each electron in an atom. It primarily dictates the electron's energy level and the average distance from the nucleus. Electrons with the same principal quantum number are in the same electron shell. The values of n are positive integers (1, 2, 3,...).

The principal quantum number affects:

• Electron energy levels: Higher n means higher energy and a farther distance from the nucleus.
• Size of orbitals: Orbitals grow larger with increasing n.

For example, in the exercise option (1) has n=2, which places this electron in the second energy level, indicating relatively higher energy compared to a hypothetical n=1 but lower than n=3 and beyond.

Azimuthal Quantum Number

The azimuthal quantum number, represented as l, determines the shape of an electron's orbital. It can take any integer value from 0 to n-1. This means for a given principal quantum number n, if n=2, l can be 0 or 1. The values correspond to the subshell types:

• l = 0: s-orbital (spherical shape)
• l = 1: p-orbital (dumbbell shape)
• l = 2: d-orbital
• l = 3: f-orbital

The energy of the subshells increases with higher l values for the same principal quantum number. In our exercise, the option (1) corresponds to l=0 (s-orbital) making it lower in energy compared to l=1 (p-orbital) in option (2) for n=2 electrons.

Electron Energy Levels

Electron energy levels are determined by the principal quantum number and the azimuthal quantum number. The combination of these quantum numbers specifies the energy of an electron within an atom. The general hierarchy for energy levels is:

• Electrons in lower principal quantum numbers (n) have lower energy.
• Within the same principal quantum number, subshells with lower azimuthal quantum numbers (l) have lower energy.

This can be illustrated using the options in our exercise. The electron in option (1) with n=2 and l=0 has a lower energy than those in options with either higher n or higher l values. Thus, it is in the lowest energy state.

Spin Quantum Number

The spin quantum number, represented as s, specifies the intrinsic spin or angular momentum of an electron. It can have values of +1/2 or -1/2, which corresponds to 'spin-up' or 'spin-down' respectively. This quantum number does not directly affect the energy of an electron in isolated atoms but is crucial for understanding electron pairing inside orbitals. In any given orbital, two electrons can coexist provided they have opposite spins, adhering to Pauli's exclusion principle.

Looking at our exercise choices, the spin quantum numbers do not affect the overall energy comparison among the options. However, they are essential for pairing electrons within the same orbital. For example, in option (1), the electron has a spin of -1/2, which will pair with an electron of spin +1/2 in the same orbital.