Question

Which of the following quantum number sets describes a possible element? A. \(n=2 ; l=2 ; m_l=1 ; m_{\mathrm{s}}=+1 / 2\) B. \(n=2 ; l=1 ; m_l=-1 ; m_{\mathrm{s}}=+1 / 2\) C. \(n=2 ; l=0 ; m_l=-1 ; m_{\mathrm{s}}=-1 / 2\) D. \(n=2 ; l=0 ; m_l=1 ; m_{\mathrm{s}}=-1 / 2\)

Short answer

Set B is the correct choice.

Step-by-step solution

- Understand Quantum Numbers

Quantum numbers describe the properties of atomic orbitals and the properties of electrons in orbitals. There are four quantum numbers: 1) Principal quantum number (), 2) Angular momentum quantum number (), 3) Magnetic quantum number (), and 4) Spin quantum number ().

- Verify Principal Quantum Number (n)

For all options, the principal quantum number is given as = 2. This is valid as is an integer starting from 1.

- Verify Angular Momentum Quantum Number (l)

The value of must satisfy 0 <= Where: . For = 2:A. = 2 (This is incorrect because must be less than B. = 1 (This is correct)C. = 0 (This is correct)D. = 0 (This is correct)

- Verify Magnetic Quantum Number (ml)

The value of can range between - and + . For valid options from Step 3:B. = 1: should be within the range (-1 to +1). -1 is correct.C. = 0: should be within the range (-0 to 0). -1 is incorrect.D. = 0: should be within the range (-0 to 0). 1 is incorrect.

- Verify Spin Quantum Number (ms)

The value of must be +1/2 or -1/2. For valid option from Step 4:B. = +1/2 (This is correct)

Conclusion

Only the set B = 2; = 1; = -1; = + 1/2 is a possible element.

Key concepts

Principal Quantum Number (n)

The Principal Quantum Number, symbolized as n, indicates the main energy level or shell that an electron occupies in an atom. It is a positive integer starting from 1. The larger the value of n, the higher the energy level and the farther it is from the nucleus. For example, an electron with n = 2 is in the second energy level. This number also indirectly tells us the size of the electron's orbital. Lower n means smaller orbitals and higher n means larger ones.

Angular Momentum Quantum Number ( l )

The Angular Momentum Quantum Number, represented by l, describes the shape of the electron's orbital. It is also called the azimuthal quantum number. l can take on integer values from 0 to < (n-1) >. Each value of l corresponds to a different type of orbital shape:

• =0 : s-orbital (spherical shape)
• =1 : p-orbital (dumbbell shape)
• =2 : d-orbital (complex shape)
• =3 : f-orbital (even more complex shape)

For example, with n = 2, l can be 0 or 1. If l = 0, you get an s-orbital; if l = 1, you get a p-orbital.

Magnetic Quantum Number ( m_l )

The Magnetic Quantum Number, denoted as m_l, describes the orientation of the orbital in space. Given a particular value of l, m_l can take on integer values from < -l > to < +l >. For instance, if l = 1, the possible values of m_l are -1, 0, and 1. This quantum number gives us an idea of the number of orbitals present in a subshell and their orientations. For l = 1 (p-orbitals), there are three orientations, corresponding to m_l values of -1, 0, and 1.

Spin Quantum Number ( m_s )

The Spin Quantum Number, represented as m_s, describes the intrinsic spin of the electron within an orbital. It can only take two possible values: +1/2 or -1/2. This quantum number accounts for the magnetic properties of electrons based on their spin, which can be thought of as either 'up' (+1/2) or 'down' (-1/2). In any given orbital, two electrons must have opposite spins (one with m_s = +1/2 and one with m_s = -1/2) to comply with the Pauli Exclusion Principle which states no two electrons can have the same set of all four quantum numbers.