Determine the effective nuclear charge

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SUMMARY

The effective nuclear charge (Zeffective) for the 3s state of sodium (Na) can be calculated using the formula Zeffective = Z - S, where Z is the actual nuclear charge and S is the shielding constant. Given that the energy of the 3s state is -5.14 eV, the binding energy of an electron in a hydrogen atom serves as a reference point for this calculation. The core electrons in Na shield some of the nuclear charge, which must be accounted for to determine the effective nuclear charge experienced by the outer electron.

PREREQUISITES
  • Understanding of atomic structure and electron configurations
  • Familiarity with the concept of effective nuclear charge
  • Knowledge of binding energy in atomic physics
  • Basic algebra for manipulating equations
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  • Research the concept of shielding constants in multi-electron atoms
  • Learn about the calculation of binding energy for different atomic states
  • Explore the relationship between effective nuclear charge and atomic radii
  • Study the differences in effective nuclear charge across the periodic table
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Students studying chemistry, particularly those focusing on atomic theory and quantum mechanics, as well as educators teaching concepts related to effective nuclear charge and electron behavior in atoms.

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1. Homework Statement

The 3s state of Na has an energy of -5.14 eV. Determine the effective nuclear charge.

2. Homework Equations

Zeffective = Z - S (not listed in textbook, found this online when trying to figure out the problem)

3. The Attempt at a Solution

Nothing in the textbook in this section talk about determining nuclear charge. I know that Na is in the 2s^2 2p^6 group
 
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What is the binding energy for an electron in a hydrogen atom? Then you can ask what is the binding energy for a single electron with a nuclear charge of +Z.

Now you can pretend that you have something similar here, where the core electrons are shielding some of the nuclear charge of Na. You have know idea how much effective nuclear charge the single electron feels outside, so knowing the binding energy you can work backwards to get this effective nuclear charge.
 

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