Calculating N for Stokes Law Half Life

Stokes' Law describes the settling velocity of spherical particles in a fluid, while half-life measures the time it takes for a quantity to reduce by half. Calculating n for Stokes' Law half-life involves determining the exponent in the sedimentation equation that relates particle size to settling time.

Introduction

When analyzing particle sedimentation, understanding how particle size affects settling time is crucial. The exponent n in Stokes' Law half-life equations helps quantify this relationship. This guide explains how to calculate n, its significance, and practical applications in fields like environmental science and material engineering.

Stokes' Law Basics

Stokes' Law describes the settling velocity (v) of a small spherical particle in a viscous fluid:

v = (2gr²Δρ)/(9η)

Where:

g = acceleration due to gravity (9.81 m/s²)
r = particle radius
Δρ = density difference between particle and fluid
η = dynamic viscosity of the fluid

For non-spherical particles, the equation becomes more complex, often involving an exponent n to account for shape effects.

Half-Life Concept

In sedimentation studies, half-life refers to the time it takes for half of the particles to settle out of suspension. The relationship between particle size and half-life is often described by power-law equations:

t₁/₂ = kd⁻ⁿ

Where:

t₁/₂ = half-life
d = particle diameter
k = constant
n = exponent to be calculated

The exponent n provides insight into the sedimentation mechanism and helps predict settling behavior for different particle sizes.

Calculating n

To calculate n, you need experimental data showing how half-life changes with particle size. The calculation involves:

Collecting sedimentation data for multiple particle sizes
Plotting the data on log-log coordinates
Determining the slope of the resulting line
The slope equals -n

For spherical particles, n typically ranges between 1 and 2. Values less than 1 indicate diffusion-dominated settling, while values greater than 2 suggest aggregation effects.

Worked Example

Consider the following sedimentation data:

Particle Diameter (μm)	Half-Life (hours)
10	5
20	2
30	1

Plotting this data on log-log coordinates and calculating the slope gives n = 1. This indicates simple Stokes' Law behavior where half-life is inversely proportional to particle diameter.

FAQ

What does the exponent n represent in Stokes' Law half-life?

The exponent n quantifies how particle size affects settling time. It accounts for shape effects and sedimentation mechanisms beyond simple spherical particles.

How accurate is the n calculation method?

The method provides a good approximation when using well-characterized particles and accurate sedimentation data. For complex systems, additional factors may need consideration.

Can n be negative?

No, n is always positive. Negative values would imply an inverse relationship that doesn't physically make sense in sedimentation contexts.