Mixing of Water of Different Temperatures

  1. Introduction

    In this session we will mix water of different temperatures and devise a hypothesis concerning the temperature of the mixture. Then we will test the hypothesis with further experiments.

  2. Objectives

    Device and test a hypothesis about the final temperature of water when water of two different temperatures are mixed. Identify how all of our actions fit into the scientific method. Devise effective ways of record keeping and graphing. Evaluate how errors in technique and measurement affect our results.

  3. Materials
  4. Vocabulary
  5. Presentation

    As the students enter, offer them various drinks. For example, cold water, hot tea, hot coffee, fruit juice. The instructor also drinks a hot drink. The instructor declares the hot drink too hot and pours some cold water into it to cool it.

    With a little showmanship, convert the cooling of the drink into a physics lesson.

    Referring to the chart showing the steps of the scientific method, note that we have made some observations concerning temperatures of liquids when they are mixed, specifically, the final temperature of a liquid when it is comprised of two liquids of different initial temperature. We now need to make measurements and devise a mathematically based hypothesis.

    Guide the students to the hot and cold water, show them the basic mixing technique with an emphasis on safety. The tell them that we need to make measurements and devise a hypothesis on mixing water of different temperature. We also need experiments to test the hypothesis. Then let them proceed as they see fit. Wander among the students offering help and suggestions as appropriate.

    One of the important principals in experiment is to eliminate as many variable components of the experiment as possible. Each variable component can introduce unexpected change to the results. We strive to have exactly one part the experimental procedure change (input) and we hope to measure exactly one result (output). In this case, the students may be tempted to use different volumes of water prior to mixing. No, No No. Only the temperature should change. The volume (or mass) of both of the initial cups of water should be very nearly the same.

    In this same vein, encourage the use of graphs to help with the scientific method. Graphing will be a powerful visual aid in assisting the students to define a single control variable and a single result variable.

    Demonstrate that several different functions can be drawn on the same graph. Each function could be on its own graph but plotting several functions on the same graph can save time and hint at other possible relationships.

    The graphs should help students discover that a constant temperature for one of the cups will be a requirement for graphing. The second cup of water changes the equilibrium temperature for the mixture. If the students have several different temperatures for the first cup, they will have several different functions plotted on their graph. See the graph below.

    Graphing several different starting
temperatures.

    Near the end of class time, ask about the accuracy of measurements, and their experimental technique with special emphasis on sources of error.

    If time remains and a student has a good mathematical conjecture about mixing water, show a simple example of error analysis. (If there is no time remaining or there is no student who has devised a mathematical expression like the first one below, then omit this discussion.) The following equations, while not strictly correct, show the general thrust of the error analysis we will use in this course.

    Error equations for equlibribum 
temperature of water

    Tf is the final equilibrium temperature of the mixture. T1 and T2 are the initial temperatures of the water in beaker 1 and 2 respectively. The δTs are the errors associated with each temperature. What is missing is the error in each mass of water. We have not yet discussed the role of mass in this problem so there is no need in adding confusion at this point.

  6. Evaluation

    Devise a short homework set that reviews the basic ideas of mixing water of different temperatures. Due next meeting. Suggested framework for homework questions: