Periodic Table: A Comprehensive Guide

The periodic table organizes the 118 known elements by atomic number, revealing patterns in their chemical and physical properties. Developed by Dmitri Mendeleev in 1869, it predicts element behavior and bonding. This MathMultiverse guide explores periodic trends, group characteristics, electron configurations, and applications, with examples and visualizations.

Mendeleev’s table, with gaps for undiscovered elements like gallium, showcases its predictive power. Periods (rows) and groups (columns) reflect electron shell filling, driving properties like reactivity and atomic size.

Periodic Trends

Trends arise from nuclear charge and electron shielding.

Atomic Radius

Decreases across periods, increases down groups:

\[ Z_{\text{eff}} = Z - S \]

Examples: \( \ce{Li} \) (152 pm), \( \ce{F} \) (72 pm).

Ionization Energy

Increases across, decreases down:

\[ IE \propto \frac{Z_{\text{eff}}}{r} \]

\( \ce{He} \): 2372 kJ/mol, \( \ce{K} \): 419 kJ/mol.

Electronegativity

Increases across, decreases down:

\[ \Delta EN = |EN_1 - EN_2| \]

\( \ce{HCl} \): \( 3.16 - 2.20 = 0.96 \).

Periodic Trends (Period 2)

Atomic radius and ionization energy for Period 2 elements.

Major Groups

Alkali Metals (Group 1)

1 valence electron (\( ns^1 \)):

\[ \ce{Na -> Na+ + e-} \]

Alkaline Earth Metals (Group 2)

2 valence electrons (\( ns^2 \)):

\[ \ce{Mg -> Mg^2+ + 2e-} \]

Halogens (Group 17)

7 valence electrons (\( ns^2 np^5 \)):

\[ \ce{Cl + e- -> Cl-} \]

Noble Gases (Group 18)

Full valence shells (\( ns^2 np^6 \)).

Electron Configurations

s-Block

\( ns^1 \) or \( ns^2 \):

\[ \ce{K} : [Ar] 4s^1 \]

p-Block

\( ns^2 np^{1-6} \):

\[ \ce{P} : [Ne] 3s^2 3p^3 \]

d-Block

\( (n-1)d^{1-10} ns^{0-2} \):

\[ \ce{Ni} : [Ar] 4s^2 3d^8 \]

f-Block

\( (n-2)f^{1-14} \):

\[ \ce{Ce} : [Xe] 6s^2 4f^2 \]

Examples

Effective Nuclear Charge

For \( \ce{Na} \):

\[ Z_{\text{eff}} = 11 - 8.8 \approx 2.2 \]

Ionization Energy

For \( \ce{Mg} \):

\[ IE_1 = 738 \, \text{kJ/mol} \]

Electronegativity Difference

For \( \ce{HCl} \):

\[ \Delta EN = 3.16 - 2.20 = 0.96 \]

Electron Configuration

For \( \ce{Br} \):

\[ [Ar] 4s^2 3d^{10} 4p^5 \]

Applications

Reaction Prediction

Sodium and chlorine:

\[ 2\ce{Na} + \ce{Cl2} -> 2\ce{NaCl} \]

Alloys

Stainless steel:

\[ \approx 74\% \ce{Fe}, 18\% \ce{Cr}, 8\% \ce{Ni} \]

Medicine

Iodine-131 decay:

\[ \ce{^{131}I -> ^{131}Xe + e-} \]

Energy

Uranium-235 fission:

\[ \ce{^{235}U + n -> ^{141}Ba + ^{92}Kr + 3n} \]