## Introduction: My experiment

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In calculus, we learned about Newton's Law of Cooling- that is, the rate at which liquids cool depends on the difference between their temperature and the temperature of the surroundings, or ambient temperature. Being a cook, I thought this was pretty interesting- if I left cookies out on the counter, for example, they would cool faster the first minute, and the closer they got to room temperature, the slower they would cool.

Newton's equation includes a constant K. This k is different for different substances based on their material properties, and in calc, we had always solved for it using data. I wanted to experimentally determine the different k values for a couple of liquids, then compare them.

Capstone Project: Newton’s Law of Cooling

Purpose: To determine the positive constant “K” for different liquids in the Newton’s Law of Cooling equation:

T-Ta = (T0-Ta)ekt

• T: Current temperature of liquid at time t
• Ta: Ambient temperature (room temperature)
• T0: Initial temperature of liquid
• k: A positive constant
• t: the time that has passed

Materials:

• 2 temperature sensors, cables
• Vernier data-collection interface
• Logger Pro software
• Liquids (1/2 cup each, heated to about 50 degrees Celsius, in identical bowls)
• Water
• Salt water (1:4 ratio of water to salt)
• Oil

Procedure:

• Connect the Vernier data-collection interface to the laptop, turn on the box, and turn on the computer. Connect the temperature sensors to Channel A and Channel B on the interface. Open the Logger Pro “Capstone Lab” document. Place one of the temperature sensors in the empty beaker. This sensor will be measuring the ambient (room) temperature for the experiment.
• Place the water on the insulating pad. Data recording is set to last for 5 minutes. Data will be recorded twice every minute (once every thirty seconds).
• Click on the “start” button to begin data acquisition.
• Print the tables and graphs of temperature vs. time.
• Repeat the previous steps for salt water and oil.

Data Analysis:

• Set up Excel spreadsheets of the data points for each liquid. Then, determine “k” for each value of time. You can do this by solving for k in the equation:
1. T-Ta = (T0-Ta)ekt
2. (T-Ta) / (T0-Ta) = ekt
3. Ln [(T-Ta) / (T0-Ta)] = kt
4. Ln [(T-Ta) / (T0-Ta)] / t = k
• Once you do this, you can enter these values for each data point, then average them for each liquid to obtain an average k for each.
• What do these values tell you about each liquid?

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