Chemistry: Degeneracy of states caused by multiplicity of spin

In summary, for a carbon atom with 2 electrons in the 2p orbital, there are 36 possible states, but only 3 independent states due to the Pauli exclusion principle and Hund's rule.
  • #1
filippo
12
0

Homework Statement



For C atom (2 electrons in 2p orbital) I get L=0,1 or 2 and S=0 or 1: so spin multiplicity/terms (in braket generacy of states) are 1S(1), 3S(3), 1P(3), 3P(9), 1D(5), 3D(15)...for a total of 36 states. I am sure I am overestimating because in reality those should be microstates: microstates are not states while states are mixtures of microstates so no. of indipendent states are no. of indipendents microstates.

Homework Equations


spin multiplicity: 2S+1 so S=0 => multip 1 (singlet); S=1 => 3 (triplet).
When L=0 term is S; L=1 term is P; L=0 term is D.

The Attempt at a Solution


Pauli principle + Hund's rule are the only things that come into my mind but I am not really sure if they can help or there is something else I am missing out.
 
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  • #2




Hi there! You are correct in your understanding of spin multiplicity and the relationship between states and microstates. In this case, for a carbon atom with 2 electrons in the 2p orbital, there are indeed 36 possible states. However, as you mentioned, not all of these states are independent and some may be degenerate.

To determine the number of independent states, we can use the Pauli exclusion principle and Hund's rule. The Pauli exclusion principle states that no two electrons in an atom can have the same set of quantum numbers, meaning that no two electrons can occupy the same state. Hund's rule states that for a given set of quantum numbers, the state with the highest total spin is the most stable.

Applying these principles to the 2p orbital of a carbon atom, we can see that there are 3 possible values of L (0, 1, or 2) and 2 possible values of S (0 or 1). This gives us a total of 6 possible states (3 x 2 = 6). However, since the Pauli exclusion principle tells us that no two electrons can occupy the same state, we must divide this number by 2 (since there are 2 electrons in the 2p orbital), giving us a total of 3 independent states.

Therefore, the correct number of independent states for a carbon atom with 2 electrons in the 2p orbital is 3, not 36. I hope this helps clarify your understanding!
 
  • #3


I would like to clarify and expand on the concept of degeneracy of states caused by the multiplicity of spin. In quantum mechanics, degeneracy refers to the phenomenon where multiple states of a system have the same energy. In the case of atoms, this can occur due to different combinations of orbital and spin angular momentum.

In the homework statement, the example given is a carbon atom with two electrons in the 2p orbital. In this case, the possible values for orbital angular momentum (L) are 0, 1, or 2, and for spin (S) are 0 or 1. This results in a total of 36 possible states, which are a combination of different L and S values. However, as correctly pointed out, this is an overestimation as it does not take into account the Pauli exclusion principle and Hund's rule.

The Pauli exclusion principle states that no two electrons in an atom can have the same set of quantum numbers (n, l, m, s). This means that for a given set of L and S values, only one electron can occupy that state. Therefore, the actual number of independent states is equal to the number of unique combinations of L and S, which is much lower than 36.

Hund's rule, on the other hand, states that the lowest energy state for a given electron configuration is the one with the maximum possible spin. This results in a higher number of states with higher spin values, as seen in the example given. However, these states are not independent as they are mixtures of different microstates, which are different arrangements of electrons within the same energy level.

In conclusion, the concept of degeneracy of states caused by multiplicity of spin is an important aspect of quantum mechanics in understanding the electronic structure of atoms. However, it is important to consider the Pauli exclusion principle and Hund's rule to accurately determine the number of independent states and their corresponding energies.
 

1. What is degeneracy of states?

Degeneracy of states refers to the phenomenon in which multiple quantum states have the same energy level. This occurs when the energy of a system is determined by more than one quantum number, resulting in different combinations of quantum numbers that can produce the same energy level.

2. How is degeneracy of states caused by multiplicity of spin?

Multiplicity of spin refers to the number of possible orientations of the spin of particles in a system. When there are multiple particles with different spin values in a system, the total spin and therefore the energy of the system can vary, resulting in degeneracy of states.

3. How does degeneracy of states affect chemical reactions?

Degeneracy of states can affect the kinetics and thermodynamics of chemical reactions. In systems with degeneracy of states, there are more possible ways for the reaction to occur, leading to a higher reaction rate. It can also affect the stability and reactivity of molecules, as molecules with degenerate energy levels may have different electronic configurations and therefore exhibit different properties.

4. Can degeneracy of states be observed in everyday life?

Yes, degeneracy of states can be observed in everyday life. For example, in the case of nuclear magnetic resonance (NMR) spectroscopy, the spin of atomic nuclei can have different orientations, resulting in degenerate energy levels. This allows scientists to study the structure and properties of molecules in various fields such as medicine, chemistry, and materials science.

5. How is degeneracy of states important in quantum mechanics?

Degeneracy of states plays a crucial role in quantum mechanics, as it allows for a more accurate description of the behavior of particles and systems. It also has practical applications in fields such as quantum computing and quantum information processing, where the manipulation of degenerate states is necessary for performing calculations and storing information.

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