Question

As is done for helium in Table 8.3 , determine for a carton atom the various states allowed according to \(\angle S\) coupling. The coupling is between carbon's two \(2 p\) electrons (its filled 2 s subshell not participating). one of which always remains in the \(2 p\) state. Consider cases in which the other is as high as the \(3 d\) level. (Note: When both electrons are in the \(2 p\), the exclusion principle Iestricts the number of states. The only allowed states are those in which \(s_{r}\) and \(\ell_{T}\) are both even or both odd.)

Short answer

The total number of states for a Carbon atom's two 2p electrons according to \(\angle S\) coupling, considering the other electron being as high as the 3d level is 16.

Step-by-step solution

Understanding the Quantum Numbers

Every electron in an atom is described by four quantum numbers. The principal quantum number (n) indicates the main energy level occupied by the electron. The azimuthal quantum number (\(l\)) describes the subshell and takes the values 0 to (n -1). The magnetic quantum number (\(m_l\)) indicates the numbers of possible states for each subshell. The spin quantum number (s) describes the direction of the spin of the electron.

Calculating the Number of States for Carbon Atom

Since we are considering the electron in the \(2p\) state and up to the \(3d\) level, we need to take into consideration the quantum numbers for these levels. For the \(2p\) state (n=2, \(l=1\)), there are three magnetic quantum states (\(m_l = -1,0,+1\)). For each magnetic state, there are two possible spin states (\(s = +1/2, -1/2\)) making a total of six states. According to the exclusion principle, the allowed states must have either even or odd values of (\(s_r, \ell_T\)) so the only allowed states are \(0, 0\) and \(2, 2\). For the \(3d\) level (n=3, \(l=2\)), there are five magnetic quantum states (\(m_l = -2,-1, 0, +1, +2\)). For each magnetic state, there are two possible spin states (\(s = +1/2, -1/2\)) making a total of ten states.

Summing Up the States

The total number of states from the \(2p\) level and up to \(3d\) level is then 6 from the \(2p\) level plus 10 from the \(3d\) level to give 16 states.

Key concepts

Spin Quantum Number

Every electron has an innate property known as spin, which gives rise to its magnetic moment. This spin is represented by the spin quantum number, denoted as 's'. The spin quantum number can take one of two possible values:

• \(+\frac{1}{2}\)
• \(-\frac{1}{2}\)

These values indicate the orientation of the electron's spin—either "up" or "down". This is crucial because each electron in an atom can exist in a spin-up or spin-down state. Therefore, for any orbital, there are always two possible spin states.
In our analysis of the carbon atom, for every magnetic quantum state considered, there are two spin states available. This makes it possible to have different states even if the principal and azimuthal quantum numbers are the same. The inclusion of spin therefore doubles the number of possible electron configurations for each magnetic quantum state.

Principal Quantum Number

The principal quantum number, symbolized by 'n', indicates which energy level an electron occupies in an atom. It's essentially the shell number in which the electron resides. The principal quantum number is always a positive integer (n = 1, 2, 3, ...), and it determines the overall size and energy of the electron's orbit.
Larger values of 'n' mean electrons are found in higher energy levels, further from the nucleus. Thus, as 'n' increases, the electron's potential energy increases due to its separation from the nucleus. In our carbon atom scenario, the electrons reside in the 2p and 3d states. Here, 'n=2' refers to the second shell, while 'n=3' denotes the third shell, corresponding to a higher energy state.
Understanding these energy levels is critical when considering electronic transitions and the electron configuration of atoms, allowing for the calculation of possible states when electrons are in differing shell levels.

Exclusion Principle

The Pauli Exclusion Principle is a fundamental rule in quantum mechanics. In simple terms, it states that no two electrons in the same atom can have the same set of all four quantum numbers. What this means for multi-electron atoms like carbon is that each electron must occupy a unique quantum state defined by its quantum numbers \(n\), \(l\), \(m_l\), \(s\).In our exercise, this principle becomes crucial. It restricts the number of viable electron configurations in these atoms. When applying this principle to the carbon atom's 2p and 3d orbitals, the allowed states are limited, especially when both electrons are in the 2p state.
The principle necessitates that the states must have both total spin (\(s_r\)) and total azimuthal quantum number (\(\ell_T\)) as either both even or both odd (e.g., 0, 0 or 2, 2). This restriction is necessary to ensure that no electrons share the same quantum state, maintaining the individual uniqueness required by the exclusion principle.