Lecture on Logarithms, Antilogarithms, and pH Calculation

Introduction to the Topic

• Today's focus:
  1. Usage of logarithm and antilogarithm tables
  2. Application of these concepts in pH calculation
• Key Terms:
  • Logarithm (log X): Logarithm of X to the base 10 (log X = log₁₀ X)
  • Natural Logarithm (ln X): Logarithm of X to the base e (ln X = logₑ X)
• Euler's constant e ≈ 2.71

Logarithm and Antilogarithm Tables

Structure of Log Table:

1. Segment A: Numbers from 10 to 99
2. Segment B: Entries from 0 to 9
3. Segment C: Entries from 1 to 9, referred to as the 'mean difference' column

Structure of Antilogarithm Table:

• Segment A: Numbers from 0.00 to 0.99
• Segment B and C: Same as the log table

Example Calculation Using Log Table:

• Find log of 1234
  • Characteristic (number of decimals) and Mantissa (from table)
  • Calculation yields: log(1234) ≈ 3.0913

Introduction to Logarithmic Functions

• Functions map one variable to another: f(x) = y
• Examples:
  • Algebraic Functions: y = x²
  • Trigonometric Functions: y = sin(x)
• Logarithmic Function: y = log₁₀(x)
  • Simplifies multiplicative scales to additive ones
  • Example: log of 1, 10, 1000 simplifies to 0, 1, 3 respectively

pH Calculation Basics

• Definition: pH = -log [H+]
• Water pH: 7, neutral solutions
• Mathematical relations: pH + pOH = 14
• Example Calculation: HCl solution
  1. HCl dissociates to H⁺ and Cl⁻ ions
  2. Calculate [H⁺] → Find pH through -log [H⁺]
  3. Example for 2 x 10⁻³ M HCl yields: pH ≈ 2.69

Handling Dilute Acid Solutions:

• Example: 10⁻⁸ M HCl
• Contributions from both the acid and water
• Calculate total [H⁺]
  • [H⁺] from H₂O: 10⁻⁷ M
  • Summed concentration: 1.1 x 10⁻⁷ M
  • pH = 6.9586, consistent with mild acidity

Common Questions Addressed

• Accuracy of Log/Antilog Calculations: Verify by back-calculating (inverse verification)
• Handling Values Greater than 1 in Antilog: Only the decimal part is considered after applying 10^('characteristic+1')

Conclusion

• Application of logarithms and antilogarithms simplifies complex calculations
• Essential for precise pH computations in chemistry
• Practice with tables ensures accuracy in solving real-life problems
• Engage in exercises to reinforce these concepts