Vernier Software & Technology

# Titration Curves: An Application of the Logistic Function

## Introduction

Think about how cold germs spread through a school. One person comes to class with a cold and infects other students. At first, the disease spreads slowly, but as more students catch cold and spread it to other classmates, the disease spreads more rapidly. The rate of infection slows down again when most students are infected and there is no one left at school to infect. The maximum number of students in the school who can contract the disease is the number of students in the school.

A logistic function is often used to model this type of situation. The logistic function is an exponential function, but it contains a ratio and offset which make its behavior interesting. The formula for a logistic function is:

$y = \frac{A} {{1 + {B^{x - C}}}} + D$

In this activity, you will add base to an acid and use a logistic function to model the data and locate the equivalence point.

## Objectives

• Record pH versus base volume data for an acid-base titration.
• Manually model the titration curve using a logistic function.
• Describe the role of each parameter in the logistic function.

## Sensors and Equipment

This activity features the following Vernier sensors and equipment.

You may also need an interface and software for data collection. What do I need for data collection?

## Real-World Math with Vernier

See other experiments from the lab book.

 1 Walk the Line - Straight Line Distance Graphs 2 Making Cents of Math: Linear Relationship between Weight and Quantity 3 Pool Plunge - Linear Relationship between Water Depth and Pressure 4 Funnel Volumes - Volume and Weight 5 Keep It Bottled Up - Rates of Pressure Increase 6 Mix It Up - Mixing Liquids of Different Temperatures 7 Spring Thing - Newton's Second Law 8 Stretch It to the Limit - The Linear Force Relation for a Rubber Band 9 What Goes Up - Position and Time for a Cart on a Ramp 10 That's the Way the Ball Bounces - Height and Time for a Bouncing Ball 11 Walk This Way - Definition of Rate 12 Velocity Test - Interpreting Graphs 13 From Here to There - Applications of the Distance Formula 14 Under Pressure - The Inverse Relationship between Pressure and Volume 15 Light at A Distance - Distance and Intensity 16 Chill Out: How Hot Objects Cool 17 Charging Up, Charging Down - Charging a Capacitor 18 Bounce Back - The Pattern of Rebound Heights 19 Sour Chemistry - The Exponential pH Change 20 Swinging Ellipses - Plotting an Ellipse 21 Lights Out! - Periodic Phenomena 22 Tic, Toc: Pendulum Motion 23 Stay Tuned: Sound Waveform Models 24 Up And Down: Damped Harmonic Motion 25 How Tall? Describing Data with Statistical Plots 26 And Now, the Weather - Describing Data with Statistics 27 Meet You at the Intersection: Solving a System of Linear Equations 28 Titration Curves: An Application of the Logistic Function 29 Clock Design: Period and Length of a Simple Pendulum 30 Graph It in Pieces: Piecewise Defined Functions 31 Stepping to the Greatest Integer: The Greatest Integer Function 32 Crawling Around: Parametric Plots

### Activity 28 from Real-World Math with Vernier Lab Book

#### Included in the Lab Book

Vernier lab books include word-processing files of the student instructions, essential teacher information, suggested answers, sample data and graphs, and more.