Which of the following sets of quantum numbers represents the highest energy of an atom? (a) n = 3, l = 0, m = 0, s = +1/2 (b) n = 3, l = 1, m = 1, s = +1/2 (c) n = 3, l = 2, m = 1, s = +1/2 (d) n = 4, l = 0, m = 0, s = +1/2 ?.

To determine which set of quantum numbers represents the highest energy of an atom, we need to analyze the significance of each quantum number. The four quantum numbers are: the principal quantum number (n), the azimuthal quantum number (l), the magnetic quantum number (m), and the spin quantum number (s). Each of these plays a role in defining the energy levels and sublevels of electrons in an atom.

Understanding Quantum Numbers

Let’s break down what each quantum number signifies:

• Principal Quantum Number (n): This number indicates the main energy level of an electron. The higher the value of n, the higher the energy level and the farther the electron is from the nucleus.
• Azimuthal Quantum Number (l): This number defines the shape of the orbital and can take values from 0 to (n-1). Higher values of l correspond to higher energy sublevels (e.g., s, p, d, f).
• Magnetic Quantum Number (m): This number describes the orientation of the orbital in space and does not directly affect energy levels but is dependent on l.
• Spin Quantum Number (s): This indicates the spin of the electron, which can be +1/2 or -1/2. It does not influence the energy level significantly.

Evaluating the Options

Now, let’s evaluate each option based on the values of n and l, as these are the most critical for determining energy:

• (a) n = 3, l = 0: This corresponds to a 3s orbital.
• (b) n = 3, l = 1: This corresponds to a 3p orbital, which has higher energy than 3s.
• (c) n = 3, l = 2: This corresponds to a 3d orbital, which has even higher energy than both 3s and 3p.
• (d) n = 4, l = 0: This corresponds to a 4s orbital, which has higher energy than any 3rd level orbital.

Comparing Energy Levels

In terms of energy levels, the order generally follows:

• 4s < 3d < 3p < 3s

Thus, while the 3d orbital (option c) is higher than the 3s and 3p orbitals, the 4s orbital (option d) is actually at a higher energy level than the 3d orbital, even though it is filled before the 3d in the Aufbau principle.

Final Decision

Considering all of this, the set of quantum numbers that represents the highest energy of an atom is:

(d) n = 4, l = 0, m = 0, s = +1/2

This set corresponds to the 4s orbital, which is at a higher energy level than the 3rd energy level orbitals. Therefore, option (d) is the correct answer.