Problem 93 - Chemistry: The Central Science

Question

Determine whether each of the following sets of quantum numbers for the hydrogen atom are valid. If a set is not valid, indicate which of the quantum numbers has a value that is not valid: (a) \(n=4, l=1, m_{l}=2, m_{s}=-\frac{1}{2}\) (b) \(n=4, l=3, m_{l}=-3, m_{s}=+\frac{1}{2}\) (c) \(n=3, l=2, m_{l}=-1, m_{s}=+\frac{1}{2}\) (d) \(n=5, l=0, m_{l}=0, m_{s}=0\) (e) \(n=2, l=2, m_{l}=1, m_{s}=+\frac{1}{2}\)

Step-by-Step Solution

Answer

(a) The set of quantum numbers is not valid because \(m_l\) is not valid. (b) The set of quantum numbers is valid. (c) The set of quantum numbers is valid. (d) The set of quantum numbers is not valid because \(m_s\) is not valid. (e) The set of quantum numbers is not valid because \(l\) is not valid.

Step 1: Check Principal Quantum Number (n)

n must be a positive integer. In this case, n = 4, which is valid.

Step 2: Check Angular Momentum Quantum Number (l)

l must be an integer such that 0 ≤ l < n. In this case, l = 1, which is valid.

Step 3: Check Magnetic Quantum Number (m_l)

m_l must be an integer such that -l ≤ m_l ≤ l. In this case, m_l = 2, which is NOT valid because -1 ≤ 2 ≤ 1 is not true.

Step 4: Conclusion for (a)

The set of quantum numbers (a) is not valid because m_l is not valid. (b) n=4, l=3, m_l=-3, m_s=+1/2

Step 5: Check Principal Quantum Number (n)

n must be a positive integer. In this case, n = 4, which is valid.

Step 6: Check Angular Momentum Quantum Number (l)

l must be an integer such that 0 ≤ l < n. In this case, l = 3, which is valid.

Step 7: Check Magnetic Quantum Number (m_l)

m_l must be an integer such that -l ≤ m_l ≤ l. In this case, m_l = -3, which is valid because -3 ≤ -3 ≤ 3 is true.

Step 8: Check Spin Quantum Number (m_s)

m_s can be either +1/2 or -1/2. In this case, m_s = +1/2, which is valid.

Step 9: Conclusion for (b)

The set of quantum numbers (b) is valid. (c) n=3, l=2, m_l=-1, m_s=+1/2

Step 10: Check Principal Quantum Number (n)

n must be a positive integer. In this case, n = 3, which is valid.

Step 11: Check Angular Momentum Quantum Number (l)

l must be an integer such that 0 ≤ l < n. In this case, l = 2, which is valid.

Step 12: Check Magnetic Quantum Number (m_l)

m_l must be an integer such that -l ≤ m_l ≤ l. In this case, m_l = -1, which is valid because -2 ≤ -1 ≤ 2 is true.

Step 13: Check Spin Quantum Number (m_s)

m_s can be either +1/2 or -1/2. In this case, m_s = +1/2, which is valid.

Step 14: Conclusion for (c)

The set of quantum numbers (c) is valid. (d) n=5, l=0, m_l=0, m_s=0

Step 15: Check Principal Quantum Number (n)

n must be a positive integer. In this case, n = 5, which is valid.

Step 16: Check Angular Momentum Quantum Number (l)

l must be an integer such that 0 ≤ l < n. In this case, l = 0, which is valid.

Step 17: Check Magnetic Quantum Number (m_l)

m_l must be an integer such that -l ≤ m_l ≤ l. In this case, m_l = 0, which is valid because 0 ≤ 0 ≤ 0 is true.

Step 18: Check Spin Quantum Number (m_s)

m_s can be either +1/2 or -1/2. In this case, m_s = 0, which is NOT valid.

Step 19: Conclusion for (d)

The set of quantum numbers (d) is not valid because m_s is not valid. (e) n=2, l=2, m_l=1, m_s=+1/2

Step 20: Check Principal Quantum Number (n)

n must be a positive integer. In this case, n = 2, which is valid.

Step 21: Check Angular Momentum Quantum Number (l)

l must be an integer such that 0 ≤ l < n. In this case, l = 2, which is NOT valid because 0 ≤ 2 < 2 is not true.

Step 22: Conclusion for (e)

The set of quantum numbers (e) is not valid because l is not valid.

Key Concepts

Principal Quantum Number

The principal quantum number, symbolized as 'n', plays a crucial role in determining the size and energy level of an electron's orbit in an atom. It is a positive integer (1, 2, 3,...) that indicates the main energy level occupied by the electron. The larger the value of 'n', the larger the orbit and the higher the energy level of the electron.

In the sets of quantum numbers given in the exercise, we initially check whether each 'n' is a valid positive integer. For instance, in cases (a), (b), (c), and (d), 'n' is 4, 3, and 5 respectively, which are valid principal quantum numbers. However, understanding 'n' goes beyond just the integer value; it provides insights into the electronic structure of atoms, which directly correlates with an atom’s chemical properties.

Angular Momentum Quantum Number

The angular momentum quantum number, denoted as 'l', is indicative of the shape of the electron's orbital and contributes to the atom’s subshell structure. The value of 'l' ranges from 0 to 'n-1', where 'n' is the principal quantum number. This arrangement results in each energy level having a set of shaped orbitals: spherically shaped 's' orbitals for 'l = 0', dumbbell-shaped 'p' orbitals for 'l = 1', and more complex shapes for higher values of 'l'.

For example, in case (e) of the exercise, 'l' is given as 2 for 'n' equal to 2, which is not possible because 'l' must be less than 'n'. Understanding these restrictions helps ensure that students assign correct subshells when depicting an electron configuration.

Magnetic Quantum Number

The magnetic quantum number, represented as 'm_l', concerns the orientation of the orbital in three-dimensional space. It depends on the angular momentum quantum number 'l' and can take on integer values that range from '-l' to '+l', including zero. This range encompasses all possible orientations an orbital can have around a nucleus.

When assessing whether a particular value of 'm_l' is valid, as seen in case (a) where 'm_l' equals 2 for 'l' equal to 1, it is crucial to remember that 'm_l' must lie within the range established by 'l'. Its value here is not valid since it does not fall within the '-1 ≤ m_l ≤ 1' range for 'l = 1'. By considering the allowed orientation values of orbitals, one can better understand magnetic quantum numbers in the context of atomic structure.

Spin Quantum Number

The spin quantum number, labeled 'm_s', characterizes the orientation of the intrinsic angular momentum or spin of an electron within an orbital. The only two possible values for 'm_s' are +1/2 or -1/2, which describe the spin direction of the electron as either 'up' or 'down'. The concept of electron spin is crucial for explaining the magnetic properties of materials and the fine structure of atomic spectra.

Occurrences such as in case (d), where 'm_s' equals 0, highlight a misunderstanding. Since 'm_s' can only be +1/2 or -1/2, a value of 0 is not valid, emphasizing the necessity for students to accurately apply the concept of electron spin when considering atomic and molecular structures.

Quantum Mechanics Validation

Understanding Quantum Numbers in Context

Quantum mechanics validation entails checking each of the quantum numbers against the rules governing their possible values. The validation process exemplified in the exercise clarifies how each quantum number must align with these fundamental principles to describe a physically plausible electron state.

Implications in Quantum Chemistry

Accuracy in assigning quantum numbers is not just a matter of rote memorization; it reflects understanding of the principles that govern atomic behavior on the quantum scale. This understanding is critical when studying chemical bonding, electron configurations, and transition rules that underpin quantum chemistry and the prediction of chemical properties.