Newton’s Law of Cooling

A container of hot water at temperature, *T*, placed in a room of lower temperature *T _{room}*, will result in an exchange of heat from the hot water to the room. The water will eventually cool to the same temperature as the room. You observe this cooling process every time you wait for a hot drink to cool. In this experiment, you will examine the cooling of hot water, with the goal of creating a model that describes the process. You can also predict the time it takes for the hot water to cool to room temperature.

Isaac Newton modeled the cooling process by assuming that the rate at which thermal energy moved from one body to another is proportional (by a constant, *k)* to the difference in temperature between the two bodies, *T _{diff}*. In the case of a sample of water cooling in room temperature air:

*cooling rate = -kT _{diff}*

From this simple assumption, he showed that the temperature change is exponential in time and can be predicted by:

*T _{diff} = T_{0} e^{-kt}*

where *T _{0}* is the initial temperature difference. Exponential changes are common in science. Systems in which a rate of change is proportional to the changing quantity show exponential behavior.

To complete this experiment in a short time, you will use a small quantity of hot water, at a temperature about 30°C above room temperature. A Temperature Probe will record the water’s temperature as it cools.

**Objectives**

- Use a Temperature Probe to record the cooling process of hot water.
- Test Newton’s law of cooling using your collected water temperature data.
- Use Newton’s law of cooling to predict the temperature of cooling water at any time.

**Materials**

- Laptop
- Vernier Logger Pro
- Vernier Stainless Steel Temperature Probe
- Cup
- Hot water (50–60°C)

**Directions**

Work in your assigned groups to complete the following tasks:

- Answer the
**Preliminary Questions**in complete sentences. - Follow the
**Procedure**to collect data using Vernier equipment.

Individually write up a lab report answering the **Analysis **questions. Write your answers in complete sentences.

Create an electronic version of this completed assignment including the cover page, and turn in on Canvas by the due date. You may scan a hardcopy of this worksheet to turn in, or you can create a PDF directly from Google Docs. This is an individual assignment.

**Preliminary Questions **

A coffee drinker is faced with the following dilemma. She is not going to drink her hot coffee with cream for ten minutes but wants it to still be as hot as possible.

- Is it better to immediately add the room-temperature cream, stir the coffee, and let it sit for ten minutes, or is it better to let the coffee sit for ten minutes and then add and stir in the cream?

- Which results in a higher temperature after ten minutes?

**Procedure **

- Connect a Temperature Probe to the Vernier LabQuest handheld computer, and connect the LabQuest to a laptop and start Logger Lite software.
- Determine the room temperature. To do this, hold the sensor in the air away from heat sources and sunlight. Monitor the temperature and record the value in the data table in the Analysis section.
- Obtain approximately 1 cup of hot water in a cup.
- Place the Temperature Probe into the water. Position the tip of the Temperature Probe in the middle of the water; it should not touch the sides or the bottom of the cup.
- Wait about 10s for the Temperature Probe to reach the temperature of the water. Change the time duration to 20 minutes by clicking on the
*Duration*button on the right side of the LabQuest screen. Click to begin data collection. Data will be collected for 20 minutes.**Note**: If the temperature of the water cools to within 5° of room temperature, you can stop data collection early.

**Analysis **

Analysis Data Table

Room Temperature (^{o}C) |
24.7 ^{o}C |

A | 75.709 |

B | 24.7 |

C | 0.011 |

- Fit an exponential function to the data. To do this, export the data you collected from Logger Lite to a CSV file. Open this CSV file in Excel. If you don’t have Excel on your laptop, you may use it through Citrix. Highlight all of the data in the .csv file, and
*Insert*a*Scatter Chart.*Click on the data points in the chart so they are highlighted, and then right click and press*Add Trendline*. In the*Format Trendline*dialogue box, click on*Exponential*and click the box to*Display Equation on Chart*. You should see an equation of the form y=A*exp(–Ct)+B on the chart. Record the parameters A, B, and C in the data table above.

- Newton’s law of cooling was given above as:

*T _{diff} = T_{0} e^{-kt}*

where *T _{diff}*is the difference between the temperature of the water sample,

*T*, and room temperature,

*T*.

_{room}*T*is the initial temperature difference; in other words, the value of

_{0}*T*at

_{diff}*t*= 0.

*T *–* T _{room} = T_{0} e^{-kt}*

Rearranging these variables, we have:

*T = T _{0} e^{-kt}*+

*T*

_{room}

Match the variables x, y, A, B, and C in the fitted equation to the terms *T*, *T _{room}*,

*k*, and

*t*in the expression above. What are the units of A, B, and C? Compare your value for B to the room temperature you recorded earlier. During data collection, was the sensor ever at room temperature?

- When
*t*= 0, what is the value of*e*?^{–kt}

- When
*t*is very large, what is the value of the temperature difference? What is the temperature of the water at this time?

- What could you do to your experimental apparatus to decrease the value of
*k*in another run? What quantity does*k*measure?

- Use your equation to calculate the temperature after 800 seconds. Compare your calculated value with the actual data value.

- Use your equation to predict the time it takes the water to reach a temperature 1°C above room temperature.

- If the starting temperature difference is cut in half, does it take half as long to get to 1°C above room temperature? Explain.