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

Hey everyone. So this problem is dealing with the atomic structure. Let's see what it's asking us in an experiment. The energies of some unknown element in the five S five P and five D states are measured to be negative 3.43 electron volts, negative two point to one electron volts and negative 0.54 electron volts respectively determine the value of the effective or the screened atomic number given by the effective for each state. And then we're asked to observe a pattern uh in the effective based off of our calculations. Our multiple choice answers are given here to solve this problem. We can first recall the energy of an atom is given by the equation E sub N is equal to negative Z effective squared divided by N squared multiplied by 13.6 E V. Where N is our principal quantum number. When we rearrange this equation to solve for Z effective isolating that on the left hand side of the equation, we get Z effective is equal to the square root of negative E sub N multiplied by N squared divided by 13.6 E V. And so now all we need to do for each of these elemental states is plug in the energies and the N or the principal quantum number for each. So because we have the five S five P and five D for each of these problems and is equal to five. And so for five S will have Z effective is equal to the square root of negative negative 3.43 E V. So those negatives cancel multiplied by N squared, which is five squared all divided by 13.6 E V. We plug that into our calculator and we get 2.5. Similarly, for the five P state, we have Z effective is equal to negative negative 2.21 E V multiplied by five squared all divided by 13. E V. And that comes out to 2.0. Lastly 45 D, we have Z effective is equal to negative negative 0.54 E B multiplied by five squared all divided by 13. E V. And that comes out to 1.0. So the pattern here is that as our um orbital quantum number L increases our Z effective decreases, which is indicative that the screening is more complete, which makes sense as we get closer to the nucleus of the atom. And so our multiple choice answers align here with answer choice A that's all we have for this one. We'll see you in the next video.

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Problem 32

Textbook Question

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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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?

Key Concepts

Effective Nuclear Charge (Zeff)

Electron Shielding

Energy Levels and Orbital Types

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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?