The energies of the 4s4s, 4p4p, and 4d4d states of potassium are given in Example 41.1041.10. Calculate ZeffZ_{eff} for each state. What trend do your results show? How can you explain this trend?

Ch 41: Quantum Mechanics II: Atomic Structure

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 41: Quantum Mechanics II: Atomic Structure

Problem 32

Chapter 41, Problem 32

The energies of the $4 s$ , $4 p$ , and $4 d$ states of potassium are given in Example $41.10$ . Calculate $Z_{e f f}$ for each state. What trend do your results show? How can you explain this trend?

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Identify the formula for the effective nuclear charge (Zeff). Zeff can be calculated using the equation:

Z eff = Z - S

, where Z is the atomic number of the element (potassium has Z = 19), and S is the shielding constant, which accounts for the screening effect of inner electrons.

Relate the energy of each state (4s, 4p, 4d) to Zeff. The energy of an electron in a given state is influenced by Zeff, and the relationship can be expressed as:

E = - R Z_{eff}^2 / n^2

, where R is the Rydberg constant, n is the principal quantum number, and E is the energy of the state. Rearrange this equation to solve for Zeff:

Z_{eff} = n^2 \sqrt{(- E / R)}

Substitute the given energy values for the 4s, 4p, and 4d states into the formula for Zeff. Use the principal quantum number n = 4 for all three states. Ensure that the energy values are converted to the appropriate units (e.g., joules or electron volts) if necessary.

Calculate Zeff for each state (4s, 4p, and 4d) using the formula derived in step 2. Note that the shielding effect (S) is different for each state due to the varying penetration and spatial distribution of the orbitals. The 4s orbital experiences the least shielding, while the 4d orbital experiences the most.

Analyze the trend in Zeff values for the 4s, 4p, and 4d states. Typically, Zeff decreases as the orbital angular momentum quantum number (l) increases because orbitals with higher l values (e.g., d orbitals) are less penetrating and experience greater shielding. Explain this trend in terms of the spatial distribution of the orbitals and the effectiveness of shielding by inner electrons.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Effective Nuclear Charge (Zeff)

Effective Nuclear Charge (Zeff) is the net positive charge experienced by an electron in a multi-electron atom. It accounts for the actual nuclear charge (the number of protons) and the shielding effect caused by other electrons. The formula Zeff = Z - S is often used, where Z is the atomic number and S is the shielding constant. Understanding Zeff is crucial for analyzing electron energies and their distribution in atomic orbitals.

Electron Shielding

Electron shielding refers to the phenomenon where inner-shell electrons partially block the attractive force of the nucleus on outer-shell electrons. This effect reduces the effective nuclear charge felt by the outer electrons, influencing their energy levels. As the number of inner electrons increases, the shielding effect becomes more pronounced, leading to variations in energy states among different orbitals, such as 4s, 4p, and 4d.

Energy Levels and Orbital Types

Energy levels in an atom correspond to the different shells and subshells where electrons reside, such as s, p, and d orbitals. Each type of orbital has a distinct shape and energy, with s orbitals generally being lower in energy than p orbitals, which in turn are lower than d orbitals within the same principal energy level. The arrangement and energy of these orbitals are influenced by both the effective nuclear charge and electron shielding, leading to observable trends in atomic properties.

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