Question

Given the following sets of electron quantum numbers, indicate those that could not occur, and explain your answer. (a) \(3,0,0,-\frac{1}{2}\) (b) \(2,2,1,-\frac{1}{2}\) (c) \(3,2,1,+\frac{1}{2}\) (d) \(3,1,1,+\frac{1}{2}\) (e) \(4,2,-2,0\)

Short answer

Answer: (b) \(2,2,1,-\frac{1}{2}\) and (e) \(4,2,-2,0\) could not occur due to invalid quantum numbers. In set (b), the azimuthal quantum number (l) is not valid, as it should be less than the principal quantum number (n). In set (e), the electron spin quantum number (m_s) is not valid, as it should be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

Step-by-step solution

Understand Quantum Number Rules

There are four quantum numbers: 1. Principal quantum number (n) - An integer greater than zero (n = 1, 2, 3, ...). 2. Azimuthal quantum number (l) - An integer ranging from 0 to n-1. 3. Magnetic quantum number (m_l) - An integer ranging from -l to +l, including 0. 4. Electron spin quantum number (m_s) - Only two values are possible: \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

Analyze Each Set of Electron Quantum Numbers

We will analyze each of the given electron quantum number sets, checking if they follow the rules: (a) \(3,0,0,-\frac{1}{2}\) Principal quantum number (n) = 3, which is valid. Azimuthal quantum number (l) = 0, which is valid (0 ≤ l ≤ n-1). Magnetic quantum number (m_l) = 0, which is valid (-l ≤ m_l ≤ l). Electron spin quantum number (m_s) = \(-\frac{1}{2}\), which is valid. (b) \(2,2,1,-\frac{1}{2}\) Principal quantum number (n) = 2, which is valid. Azimuthal quantum number (l) = 2, which is NOT valid (0 ≤ l ≤ n-1). The maximum value for l in this case should be 1. (c) \(3,2,1,+\frac{1}{2}\) Principal quantum number (n) = 3, which is valid. Azimuthal quantum number (l) = 2, which is valid (0 ≤ l ≤ n-1). Magnetic quantum number (m_l) = 1, which is valid (-l ≤ m_l ≤ l). Electron spin quantum number (m_s) = \(+\frac{1}{2}\), which is valid. (d) \(3,1,1,+\frac{1}{2}\) Principal quantum number (n) = 3, which is valid. Azimuthal quantum number (l) = 1, which is valid (0 ≤ l ≤ n-1). Magnetic quantum number (m_l) = 1, which is valid (-l ≤ m_l ≤ l). Electron spin quantum number (m_s) = \(+\frac{1}{2}\), which is valid. (e) \(4,2,-2,0\) Principal quantum number (n) = 4, which is valid. Azimuthal quantum number (l) = 2, which is valid (0 ≤ l ≤ n-1). Magnetic quantum number (m_l) = -2, which is valid (-l ≤ m_l ≤ l). Electron spin quantum number (m_s) = 0, which is NOT valid (m_s can only be \(+\frac{1}{2}\) or \(-\frac{1}{2}\)).

Conclude Invalid Quantum Number Sets

Among the given sets of electron quantum numbers, the following could not occur: (b) \(2,2,1,-\frac{1}{2}\): The azimuthal quantum number (l) is not valid, as it should be less than the principal quantum number (n). The maximum value for l in this case should be 1. (e) \(4,2,-2,0\): The electron spin quantum number (m_s) is not valid, as it should be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).

Key concepts

Principal Quantum Number

In the quantum world of the atom, the principal quantum number, denoted as 'n', is akin to address for the electrons, denoting which shell they reside in around the nucleus. It is always a positive integer such as 1, 2, 3, and so on. Think of it like the floors in a hotel; the ground floor is n=1, the first floor is n=2, and it continues upward. As 'n' increases, the electron's energy and its probable distance from the nucleus also increase.

For example, in the step-by-step solution, we saw that 'n' must be greater than zero. An electron with a principal quantum number of 'n' = 3 is further out from the nucleus compared to an electron with 'n' = 2. This concept is paramount as it lays the groundwork for understanding where an electron can possibly be, providing the first piece of our quantum puzzle.

Azimuthal Quantum Number

Diving deeper into the atom's architecture, the azimuthal quantum number, usually represented as 'l', tells us about the subshells or shape of the electron cloud within the principal shell. It ranges from 0 up to (n-1), where each value of 'l' correlates to a particular subshell. For instance, if 'l' is 0, we're talking about an s subshell, which is spherical; if 'l' is 1, it points to a p subshell with a dumbbell shape, and so on.

The given solution clearly demonstrates that the value of 'l' is crucial in determining the correct subshell. A principal quantum number 'n' of 2, for example, does not allow for an 'l' value of 2 as it goes beyond the range of 0 to n-1. This would be like trying to put a puzzle piece in a space that's too small. The azimuthal quantum number is an integral part to understand atomic orbitals and electron configurations, acting as the 'shape identifier' for electron positions.

Magnetic Quantum Number

Zooming into the electron's address, we come to the magnetic quantum number, symbolized as 'm_l'. This number is integral in laying out the electron's orientation within its subshell. It can take on integer values from -l to +l, which includes zero. These numbers correspond to the different orientations an orbital can have in a three-dimensional space.

For example, if the azimuthal quantum number 'l' is 1 (which means we are looking at a p subshell), then 'm_l' can be -1, 0, or +1, representing the three orientations of the p orbitals. In our solution set, each magnetic quantum number falls within the correct range, except for one instance where the electron spin quantum number does not comply. Understanding the magnetic quantum number is much like knowing which way a room faces in a house; is it north, east, south, or west? Each orientation is a possible home for an electron.

Electron Spin Quantum Number

Last but not least is the electron spin quantum number, given the symbol 'm_s'. This number is a fundamental property of electrons and denotes the direction of the electron's intrinsic spin. There are only two possible values: +1/2 or -1/2, often referred to as 'spin-up' or 'spin-down'. The concept of spin is akin to the earth rotating on its axis; it can go one of two ways.

Electron spin is vital in many quantum phenomena, including magnetic fields and the Pauli Exclusion Principle, which states that no two electrons can have the same set of all four quantum numbers. The solution given pointed out an incorrect instance where the spin was denoted as '0', a value that is not allowed, as electrons cannot be 'spinless'. Grasping this quantum number is essential for understanding how electrons pair up within orbitals and contribute to the magnetic properties of atoms.