Equipment:
- A thermometer. We will loan you one. We want them all back
when you are done.
- A way of heating water to around 50°C (=112°F).
A saucepan on a stove works well.
- At least two different containers to place the liquid in to
cool. The saucepan itself works well for one of them. It is
fun if one of them is well insulated, like using a Styrofoam
cup or wrapping towels around the saucepan.
- Notebook paper on which to write down the temperatures of
the liquid every minute for an hour.
- Excel, CricketGraph, and word processing (either ClarisWorks
or Word) software.
Data Collection Procedure:
- Record the temperature in whatever environment is surrounding
the container. This is called the ambient temperature, or TA.
Run experiments that use at least two different ambient temperatures.
You could possibly allow the liquid to cool outdoors or in a
refrigerator instead of indoors.
- Heat the liquid in the saucepan to 50°C. Heat it slowly
so that you don't overshoot 50°C. Stir it constantly to
be sure that the liquid is all the same temperature. Be
careful! While 50°C is well short of boiling, it is still
hot enough to burn you.
- Pour the liquid into whatever container you are using for
this experimental run. Place them thermometer into the liquid.
Record the starting temperature. This is called the original
temperature, or To.
- Every minute, observe the thermometer and record the temperature
of the liquid as it cools. Do this for one hour. Stir the liquid
gently to be sure that it is all the same temperature. Even
though the °C marks are 2° apart on the thermometer,
try to record the temperature to
the nearest degree.
- When you are done with the experiment, enter the time (in
minutes from the start of the experiment) in the first column
of an Excel spreadsheet. Enter the temperature (in ûC)
in the second column.
- Copy and paste the time and temperature spreadsheet columns
into CricketGraph.
- Create a graph of the temperature (y) versus time (x). Experiment
with different curve fits. (But, be sure to check out the comment
at the end of this write-up before taking any curve fits too
seriously.)
- Compute the cooling factor C for each experiment. To determine
C for an experiment, divide 0.693 by the number of minutes it
takes for the liquid to cool to a temperature halfway between
the original temperature and the ambient temperature. The cooling
factor will be
explained in more detail below.
After the Lab Turn in:
- The thermometer.
- A report describing what you did, what containers and ambient
temperatures you used, what numbers you collected, and the graphs
that resulted. Copy and paste the numbers from the Excel spreadsheets
and the graphs from CricketGraph right into your word processor
report.
How it Really Works – Calculus
meets Physics:
In most of the math we learn in high school, things have equations
for what they are. For example, the equation for a straight line
on a graph is y = mx + b. If we know x, then we know what y is.
In the world of physics, a lot of things do not have equations
for what they are, but for how they are changing. This is because
nature likes to put things in equilibrium. In your temperature
experiment, the forces of nature will work towards making the
temperature of the liquid the same as the temperature of the surrounding
air. We call this surrounding air the ambient temperature. When
you take calculus, yoiu will learn to write mathematical equations
that use the triangle symbol and the Capital letter T.
One of the equations that you will learn to write describes the
the change of the liquid's Temperature during a change in time
as proportional to the difference between the ambient temperature
and the liquid's current temperature. In other words: The bigger
the difference, the faster the rate at which nature will try to
cool it off. (This is one of the nice things about calculus. The
math may be harder, but understanding what is going on doesn't
have to be.)
So, we start out with an equation for how the temperature changes
and then we use calculus to
solve for what the temperature will be as the liquid cools. When
you do this, you get:
T = TA + ( T o -TA ) e - Ct
In this equation:
- t is the time in minutes that the liquid has been cooling
- T is the temperature of the liquid in °C at time t
- TA is the ambient temperature in °C
- To is the original temperature of the liquid in °C (this
should be approximately 50°C)
e is a special number in mathematics. It is called the exponential.
Its approximate value is 2.71828. Like p, it actually has an infinite
number of decimal digits. On scientific calculators, you can take
e to the x power with a key that usually looks like ex. In spreadsheets,
you take e to the x power with the expression exp(x).
C is called the cooling factor. It is a special number that tells
you how fast the liquid cools. C depends on what liquid you are
using and on how well the container is insulated. Heat transfer
scientists spend lots of time determining the different C values
for different materials and containers so that they have an equation
to predict heat transfer behaviors in the future. (For example,
will a truckload of frozen strawberries reach the market before
they thaw? What if the driver drives at night instead of the hot
day?)
To determine C for your experiments, divide 0.693 by the number
of minutes it takes for the liquid to cool to a temperature halfway
between the original temperature and the ambient temperature.
This now gives you an equation to predict how the liquid in this
container will cool.
You can put your predictions right into your spreadsheet. For
example, suppose your original temperature is 50°C and the
ambient temperature is 22°C. If you determine that C for this
experiment is 0.028 and you have the times in column A starting
with cell A1 and the recorded temperatures in column B starting
with cell B1, you can place the theoretical temperature in cell
C1 with the following equation:
= 22 + (50-22)*exp(-0.028*A1)
You can also check how close your measurements were to the theoretical
temperatures by placing in cell D1 the equation:
= B1 - C1
You can then copy and paste these formulas into the other rows
in columns C and D.
Things to Notice:
- Some of the curve fits in CricketGraph may seem to fit your
data quite well, but in fact none of the curve fitting equations
available in CricketGraph matches the form that we know from
physics that the equation must take. Thus, if you were to blindly
use one of these curve fitting equations to predict how the
liquid will continue cooling, you would get the wrong answer.
This is why the blind use of computers does not replace a science
and math education! You must understand the science and math
to properly direct the computer to a correct solution. The computer
is a tool for scientists, not a replacement for them.
- Nature tries to bring things into equilibrium faster when
they are more out of balance. This is why your liquid cools
quickly at first and then cools slower. This is also why liquids
cool faster when the difference between the original and ambient
temperatures is greater.
- Notice how the cooling factor C varies between different
containers. Larger cooling factors represent faster cooling,
smaller ones represent slower cooling.
Example Graph:
When we ran this experiment at home, we got a graph like this
one.
For this experiment, the ambient temperature, TA, was 22°C.
The original temperature, To, was 50°C.
Halfway between TA and To is 36°. We found that it took 25
minutes to reach 36°, so the cooling factor for this experiment
was 0.693/25 = 0.028.
So the temperature of this cooling liquid has an equation of:
T = 22 + 28 e - 0.028t
Or, in Excel:
= 22 + 28*exp(-0.028*A1)
This experiment was only run for 40 minutes. Be sure to run your
experiments for the full hour to get a better idea of the shape
of the curve.
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