Dimensional Analysis & Its Calculations | Chemistry for EmSAT Achieve PDF Download

Introduction

Dimensional analysis addresses intriguing questions, such as comparing the magnitudes of different units like meters and seconds, or the feasibility of adding disparate units like kilograms and eggs. By delving into this section, you'll gain insights into how dimensional analysis tackles such inquiries effectively. 

Principles of Dimensional Analysis

• Use of Dimensions and Formulas:
  • Dimensional analysis utilizes dimensions and dimensional formulas of physical quantities to establish interrelations among them.
  • Based on established facts in physics, including the International System of Units, measurement of length, mass, time, significant figures, accuracy, precision of instruments, and errors in measurement.
• Principle of Homogeneity:
  • Physical laws remain invariant regardless of the units used for measurement.
  • Expressions representing physical situations must possess consistent dimensions throughout.

Applications of Dimensional Analysis

• Conversion of Units: 
  • Physical quantities' dimensions remain constant irrespective of the system of units used.
  • Equations derived from different unit systems must equate based on the principle of homogeneity, facilitating unit conversions.
• Consistency Checking:
  • Ensuring consistency in physical equations by verifying that the dimensions on both sides match.
• Deriving Relations in Physical Phenomena:
  • Utilizing the principle of homogeneity to establish relationships between various physical quantities observed in phenomena.

Example

Example: The Time period (T) of a simple pendulum is observed to depend on the  following factors:

• length of the pendulum (L),
• mass of the Bob (m)
• acceleration due to gravity (g)

Sol: Let T ∼  L^αm^βg^γ or …… (5) 
[T] = [L]^α[M]^β[L]^γ[T]^-2γ or 
Solving the above for α, β and γ we have: 
β = 0; α + γ = 0; 
-2γ = 1 or γ= -1/2 
Using these in equation (5), we have: