Treasure Hunts
Calculus and Chemistry
Introduction
Calculus can be used to predict the melting point and boiling point of a chemical. Your task is to use calculus to develop a model that will predict the melting point and boiling point of various chemicals, then compare your results with actual melting points and boiling points of these chemicals.
Guiding Question
How accurately can you predict the melting point and boiling point of naphthalene and other chemicals?
Questions
1. The vapor pressure P, in millimeters of mercury (mm Hg), of a liquid or a solid increases as the temperature increases. The rate of change of the vapor pressure, dP/dT, is directly proportional to P and inversely proportional to the square of the Kelvin temperature, T. In physical chemistry, you will learn that this relationship is called the Clausius-Clapeyron equation. Write a differential equation that expresses dP/dT in terms of P and T. Integrate the equation, then solve for P in terms of T.
2. Do research to look up the vapor pressure (in mm Hg) for naphthalene (mothballs, C10H8) at various temperatures. What is the melting point? What is the boiling point? Use the data to find the two constants in the equation you wrote in part (1). You may solve the system of simultaneous equations either in their logarithmic form or in their exponential form, whichever is more convenient. Don't be afraid of large numbers! And don’t round them off!
3. How well does your function fit the actual data? Does the same equation fit well above the melting point? If so, give information to support your conclusion. If not, find an equation that fits better above the melting point. Do any other types of functions available on your graphing calculator seem to fit the data better than the function from the Clausius-Clapeyron equation?
4. How well does your equation predict the boiling point of naphthalene? Does the same equation fit well above the boiling point? If so, give information to support your conclusion. If not, find an equation that fits better above the boiling point. Do any other types of functions available on your graphing calculator seem to fit the data better than the function from the Clausius-Clapeyron equation?
5. Repeat this project for another substance besides naphthalene.
Internet Resources
Physical Chemistry
Physical Properties of Naphthalene
Vapor Pressure
Vapor Pressures at Different Temperatures
Definition of Boiling Points
Definition of Melting Points
The Clausius-Clapeyron Equation
Physical Properties of Benzene
Physical Properties of Coronene
Separable Differential Equations
Simultaneous Linear Equations
Graphing Data on Graphing Calculator
Exponential and Logarithmic Equations
Temperature Conversions
Activities
Your write-up must include:
1. Objective: The main question you are trying to answer with this project.
2. Bibliography: List of all resources used.
3. Data: Table of values of vapor pressure at various temperatures.
4. Procedures: Description of what you did in this project and methods used to answer the questions posed.
5. Visuals: Graphs of the functions developed in this project, complete with labels and captions.
6. Calculations: Calculations and written explanation of the calculations.
7. Conclusions: Your conclusions.
Jean Nanjo, Teacher, and Adriana Miu, AP student, collaborated to develop this web hunt.