Problem 6.146 — Pauli Principle: Electron Configuration of Oxygen

Problem Statement

Problem 6.146 — Pauli Principle: Electron Configuration of Oxygen

Given Information

• All quantities, constants, and constraints stated in the problem above
• Physical constants used as needed (see Concepts section)

Physical Concepts & Formulas

This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.

• Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
• State the mathematical form of that law as it applies here
• Check dimensions at every step: both sides of an equation must have the same units

Step-by-Step Solution

Problem Statement

Problem 6.146 — Pauli Principle: Electron Configuration of Oxygen

Given Information

• All quantities, constants, and constraints stated in the problem above
• Physical constants used as needed (see Concepts section)

Physical Concepts & Formulas

This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.

• Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
• State the mathematical form of that law as it applies here
• Check dimensions at every step: both sides of an equation must have the same units

Step-by-Step Solution

Problem Statement

Problem 6.146 — Pauli Principle: Electron Configuration of Oxygen

Given Information

• See problem statement for all given quantities.

Physical Concepts & Formulas

This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.

• See the step-by-step solution for the specific equations applied.
• All quantities are in SI units unless otherwise stated.

Step-by-Step Solution

Step 1 — Identify given quantities and set up the problem: Configuration: $1s^2 2s^2 2p^4$

Step 2 — Apply the relevant physical law or equation: The four $2p$ electrons fill three degenerate $2p$ orbitals. By Hund’s rule:

Step 3 — Solve algebraically for the unknown:

• Maximize total spin $S$ (parallel spins preferred — lower electron repulsion)
• Then maximize $L$ (within the spin constraint)

Step 4 — Substitute numerical values with units: $2p^4$ assignment: $(m_l=+1,\uparrow)$, $(m_l=0,\uparrow)$, $(m_l=-1,\uparrow)$, $(m_l=+1,\downarrow)$

Step 5 — Compute and check the result: This gives: $S = 1$ (2 unpaired electrons), $L = 1$ ($M_L = +1+0-1+1 = 1$), $J = |L-S| = 0$ to $L+S = 2$, ground state $J=2$.

Step 6: Ground state term: $^3P_2$

Worked Calculation

Full substitution shown in the steps above.

Configuration: $1s^2 2s^2 2p^4$

The four $2p$ electrons fill three degenerate $2p$ orbitals. By Hund’s rule:

• Maximize total spin $S$ (parallel spins preferred — lower electron repulsion)
• Then maximize $L$ (within the spin constraint)

$2p^4$ assignment: $(m_l=+1,\uparrow)$, $(m_l=0,\uparrow)$, $(m_l=-1,\uparrow)$, $(m_l=+1,\downarrow)$

This gives: $S = 1$ (2 unpaired electrons), $L = 1$ ($M_L = +1+0-1+1 = 1$), $J = |L-S| = 0$ to $L+S = 2$, ground state $J=2$.

Ground state term: $^3P_2$

Oxygen is paramagnetic with magnetic moment $\approx 2\mu_B$, consistent with 2 unpaired electrons.

Answer

See final result in the worked calculation above.

Physical Interpretation

The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.

Worked Calculation

Substituting all given numerical values with their units into the derived formula:

$$\text{Numerical result} = \text{given expression substituted with values}$$

Answer

$$\boxed{\approx 2\mu_B}$$

Physical Interpretation

The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.

Worked Calculation

Substituting all given numerical values with their units into the derived formula:

$$\text{Numerical result} = \text{given expression substituted with values}$$

Answer

$$\boxed{\boxed{\approx 2\mu_B}}$$

Physical Interpretation

The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.