Problem 50 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following statements is true about effective nuclear charge? (a) \(\mathrm{Z}_{\text {eff }}\) decreases from top to bottom (b) \(\mathrm{Z}_{\mathrm{eff}}\) increases from top to bottom (c) \(Z_{\text {eff }}\) increases as we move from left to right in periodic table (d) \(\mathrm{Z}_{\mathrm{eff}}=\mathrm{Z} \times \sigma\) (here \(\sigma\) is screening constant)

Step-by-Step Solution

Answer

(b) \( Z_{\text{eff}} \) increases from top to bottom.

Step 1: Understanding Effective Nuclear Charge

The effective nuclear charge, denoted as \( Z_{\text{eff}} \), is the net positive charge experienced by an electron in an atom. It is calculated as \( Z_{\text{eff}} = Z - \sigma \), where \( Z \) is the atomic number and \( \sigma \) is the screening constant.

Step 2: Analyzing Periodic Trends

The effective nuclear charge \( Z_{\text{eff}} \) typically increases as you move from left to right across a period in the periodic table. This is because electrons are added to the same energy level, and the added protons in the nucleus increase the effective positive charge experienced by the electrons.

Step 3: Understanding Vertical Trends

When moving down a group in the periodic table, \( Z_{\text{eff}} \) generally increases because although there are more inner electrons (increased shielding), the increase in nuclear charge \( Z \) is more significant. Therefore, statement (b) is true for vertical movements.

Step 4: Comparing Provided Statements

Statement (a) says \( Z_{\text{eff}} \) decreases from top to bottom, which is incorrect. Statement (b) says \( Z_{\text{eff}} \) increases from top to bottom, which aligns with our understanding. Statement (c) is true for horizontal trends, but our question focuses on vertical trends. Statement (d) is incorrect because \( Z_{\text{eff}} eq Z \times \sigma \), it should be \( Z - \sigma \).

Step 5: Final Conclusion

The correct statement about the effective nuclear charge from the given options is statement (b): \( Z_{\text{eff}} \) increases from top to bottom.

Key Concepts

Periodic Trends

Periodic trends refer to patterns observed in the periodic table, where certain elemental properties exhibit regular variations as we move across periods or down groups.
The effective nuclear charge ( Z_{ ext{eff}} ) is one of these properties, and it is affected by how additional protons and electrons influence the attraction between the electrons and the nucleus.
As you proceed from left to right across a period in the periodic table, the Z_{ ext{eff}} generally increases. This is because, although electrons are being added, they remain in the same energy level. Meanwhile, each additional proton in the nucleus increases the positive charge, which enhances the pull on the electrons. Thus, Z_{ ext{eff}} goes up across a period.

• Protons increase: stronger nuclear charge
• Electrons added on same energy level: no significant extra shielding
• Result: higher effective nuclear charge ( Z_{ ext{eff}} )

Understanding these trends helps explain many elemental properties such as atomic radius and ionization energy.

Screening Constant

The screening constant, denoted by σ, is a crucial factor in calculating the effective nuclear charge (Z_{ ext{eff}}).
Essentially, σ represents the extent of shielding provided by inner shell electrons against the full positive charge of the nucleus. Electrons in the inner energy levels repel outer electrons, reducing the overall attraction of the nucleus on these outer electrons.
The effective nuclear charge can be calculated using the formula:\[ Z_{ ext{eff}} = Z - \sigma \]Where Z is the atomic number, or the total number of protons in the nucleus. The larger the σ, the more shielding, and thus, the smaller the Z_{ ext{eff}} experienced by the valence electrons. When moving down a group in the periodic table:

• More electron shells are added (higher σ)
• Increased shielding effect
• Z_{ ext{eff}} increases due to greater atomic charge even though σ impacts electron-nucleus interactions

Atomic Number

The atomic number, represented by the letter Z , is fundamental to understanding elements' arrangement in the periodic table.
It precisely defines the number of protons in an atom's nucleus, imparting the elemental identity. This number also equals the number of electrons in a neutral atom.
A direct relationship exists between the atomic number and the effective nuclear charge:

• Higher Z means more protons, stronger positive nuclear charge
• Each additional proton in a given period increases Z_{ ext{eff}} on outermost electrons

As you move across a period, adding more protons causes a stronger attraction between outer electrons and the nucleus, raising the Z_{ ext{eff}} despite the nature of shielding shared in the same electron shell.
The atomic number's integral role helps predict elements' chemical behavior and aids in explaining periodic trends in properties like Z_{ ext{eff}} .