Quantum Numbers: Spin Quantum Number - Online Tutor, Practice Problems & Exam Prep

The spin quantum number is a critical concept in quantum mechanics, describing the intrinsic angular momentum of electrons within an atomic orbital. According to the Pauli exclusion principle, no two electrons in the same orbital can have identical sets of quantum numbers. Therefore, they must have opposite spins, denoted by the spin quantum number ($m_s$). Electrons with upward spin are assigned a value of $\frac{1}{2}$, while those with downward spin have a value of $-\frac{1}{2}$. This distinction ensures that each electron in a shared orbital has a unique quantum state, which is pivotal in determining the arrangement of electrons around an atom and the resulting chemical behavior of elements.

Quantum Numbers: Spin Quantum Number

With the spin quantum number we actually look at the individual spins of the electrons within a given orbital. We're going to say an orbital can hold a maximum of two electrons that have opposite spins according to the Pauli exclusion principle. Now the Pauli exclusion principle says that no two electrons within the same orbital can have the same four quantum numbers by having opposite spins. That affects their spin quantum number, which is M_s.

Now it deals with the rotational spin of an electron inside an atomic orbital. We're going to say here that the way we do it is we start out filling an orbital with an electron that points up followed by the next one pointing down. This indicates that each electron has an opposite spin. We're going to say an electron that points up has an M_s value of 12, and one that points down has an M_s value of -12.

So here we have our illustration of an orbital in red, and we have those two electrons within that orbital. One is spinning up, the first one, and then one is spinning down. So just remember when it comes to the spin of an electron within an orbital, we look at the M_s quantum number.

Quantum Numbers: Spin Quantum Number Example 1

Here we need to provide the NLM_LM_S quantum numbers for the two highlighted electrons in a 3D principle level. All right, so because they told us it's a 3D principle level, that means n=3.

Next, we're going to see that our M_L remember that's the range is -1 to +1. M_L is the range of L so that would mean that L=1. We're looking specifically at electrons in the orbital marked 0, so that M_L for both electrons is 0.

Now though, one electron is pointing up and one is pointing down. That would give us two different M_S quantum numbers. So let's call this electron one. For this electron here, since it's spinning up, it's M_S would be +12.

And then for electron 2, the one that's pointing down, it would have the same NL and M_L, but because it's pointing down it would be -12. So these would be the four quantum numbers for the two electrons within this given orbital.

Select the correct quantum numbers for the highlighted electrons in a set of 5d orbitals.

• A. n = 5, l = 3, m_l = – 4, m_s = +1/2
• B. n = 4, l = 4, m_l = +1, m_s = +1/2
• C. n = 5, l = 2, m_l = +1, m_s = +1/2
• D. n = 5, l = 5, m_l = – 2, m_s = +1/2
• E. n = 5, l = 2, m_l = +5, m_s = +1/2

Which of the following set of quantum numbers is possible?

• A. n = 8, l = 3 , ml_l = 0, m_s = 0
• B. n = 7, l = 2, m_l = 1, m_s = –1/2
• C. n = 9, l = 1, m_l = – 2, m_s = +1/2
• D. n = 3, l = 0, m_l = +3, m_s = +1/2
• E. n = 4, l = 2, m_l = –2, m_s = +1

Which of the following set of quantum numbers is possible for an electron in a set of 6f orbitals?

• A. n = 6, l = 3 , ml_l = 0, m_s = 0
• B. n = 6, l = 2, m_l = 1, m_s = –1/2
• C. n = 9, l = 1, m_l = – 2, m_s = +1/2
• D. n = 6, l = 3, m_l = 0, m_s = +1/2
• E. n = 4, l = 2, m_l = –2, m_s = +1

Which of the following statements is false?

• A. If an electron has n = 2, it possesses only s and p orbitals.
• B. Each orbital within a given atom can hold up to 2 electrons.
• C. The second shell of an atom possesses d orbitals.
• D. A set of f orbitals can hold a maximum of 14 electrons.
• E. The first energy level contains only s orbitals.

Here’s what students ask on this topic:

What is the significance of the spin quantum number?

The spin quantum number is a fundamental property of electrons that describes their intrinsic angular momentum, or "spin." Unlike the other quantum numbers that describe an electron's position in an atom (like the principal, azimuthal, and magnetic quantum numbers), the spin quantum number indicates the direction of an electron's spin around its own axis. It can take on one of two possible values, $\frac{1}{2}$ or $-\frac{1}{2}$, often referred to as "spin-up" or "spin-down."

This quantum number is significant because it explains the behavior of electrons in a magnetic field, which is crucial for understanding phenomena such as electron pairing in atomic orbitals due to the Pauli exclusion principle. This principle states that no two electrons in an atom can have the same set of four quantum numbers, which includes the spin quantum number. Consequently, the spin quantum number is essential for determining the electronic structure of atoms and the chemical properties of elements. It also plays a key role in the field of quantum mechanics and is pivotal in technologies like magnetic resonance imaging (MRI) and electron spin resonance spectroscopy.

What information is given by the spin quantum number?

The spin quantum number, denoted by the symbol $m_s$, provides information about the orientation of the intrinsic angular momentum, or 'spin', of an electron within an atom. This quantum number can have only two possible values: $\frac{1}{2}$ or $\frac{-1}{2}$. These values indicate the two allowed spin states of an electron, often referred to as 'spin-up' and 'spin-down'. The concept of spin is a fundamental property of particles, akin to their charge or mass, and it plays a crucial role in the Pauli exclusion principle, which states that no two electrons in an atom can have the same set of four quantum numbers. The spin quantum number is essential in determining the magnetic properties of an atom and is also a key factor in the fine structure of atomic spectra.