Wet Math
Deep lak es, shallow lak es – an d – Ice in my drink
Introduc tion
See what influences how quickly water cools
Materials
Jug of hot water (as hot as is safe)
Ice
Stopwatch
Thermometer (2)
Beakers (80 ml works well) or cups (2), narrow graduated cylinder
Optional: Styrofoam or plastic cups (2)
To Do and Notice
Optional: Make stable holders for your thermometers. Stick the bulb of the
thermometer through the bottom of the Styrofoam cups. This keeps the
thermometer bulb from hitting the sides of the beaker.
A. Deep lakes, shallow lakes
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Put the thermometers in the jug of hot water. Record this
temperature. (Is there a difference in the thermometer readings? If
so, make note).
Put one thermometer in the beaker, one in the graduated cylinder
Fill beaker and graduated cylinder with the same volume of hot water.
Quickly take the temperature on both thermometers. Then start the
stopwatch. This is the temperature at t = 0.
Record the temperature on both thermometers at the following times:
30 sec, 1 min, 1 min 30 sec, and then at 2, 3, 4, 5, 10, 15, 20, 30, 40, 50,
60 minutes. The exact times are not important, just write down the
time on the stopwatch when you measure.
Calculate the change in temperature from the start: ie., delta T =
measured temperature minus initial temperature (eg., 30 C – 35 C).
This will be negative because it is cooling!
Graph change in temperature versus time.
B. Ice in my drink
© 2006 Exploratorium, all rights reserved
Stephanie Chasteen
Exploratorium Teacher Institute
http://www.exo.net/~drsteph
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Put the thermometers in the jug of hot water. Record this
temperature. (Is there a difference in the thermometer readings? If
so, make note).
Put one thermometer in each beaker.
Fill both beakers with water, but leave enough room to add an ice cube
without overflowing.
Quickly take the temperature on both thermometers. Then start the
stopwatch. This is the temperature at t = 0.
Record the temperature on both thermometers at the following times:
30 sec, 1 min, 1 min 30 sec, and then at 2, 3, 4, 5, 10, 15, 20, 30, 40, 50,
60 minutes. The exact times are not important, just write down the
time on the stopwatch when you measure.
At some point during that hour, add the ice cube to the second beaker.
Record the temperature every minute for the first five minutes after
you add the ice.
Calculate the change in temperature from the start: ie., delta T =
measured temperature minus initial temperature (eg., 30 C – 35 C).
This will be negative because it is cooling!
Graph change in temperature versus time.
What’s go ing o n?
Cooling of full beaker vs. half beaker of water
5
Change in temperature (C)
A sample graph is to the right. The
curves are clearly not straight lines.
The slope becomes smaller as time
goes on. These cooling curves you see
are exponentials. That’s because the
change in the water temperature
during a change in time (t) is
proportional to the difference
between the room temperature (T0)
and the temperature of the liquid
"T
right now (T) or
# T0 $ T . The
"t
change in temperature over change is
time is the slope of our graph! So as
the liquid’s temperature decreases, so
!
does the slope.
full beaker
half beaker
0
-5
-10
-15
-20
-25
-30
-35
0
10
20 30 40 50
Time (minutes)
60
Stephanie Chasteen
Exploratorium Teacher Institute
http://www.exo.net/~drsteph
© 2006 Exploratorium, all rights reserved
A. Deep lakes, shallow lakes
The graduated cylinder cools more rapidly.
The water cools by transferring energy at the surface. The volume in both
cases is the same, but the surface area of the water in cylinder is less than
that in the beaker.
That is also why deeper lakes don’t get cool enough to freeze during the
winter, but shallow ones do.
Beaker with water versus beaker with water and ice
B. Ice in my drink
When the ice is added, heat energy
flows from the water to the ice.
This accomplishes three things.
First, the ice is heated to the
melting point (0 C).
5
Change in temperature (C)
A sample graph is at right. The
temperature of the water drops
abruptly when the ice is added, and
then resumes its gradual descent.
Water only
Water and ice
0
-5
-10
-15
-20
-25
-30
-35
0
10
20
30
40
50
60
Then, the ice melts, undergoing a
Time (minutes)
phase change from solid to liquid.
The temperature of the ice doesn’t rise while it melts, but this process takes a
lot of energy! It takes a lot more energy to melt a gram of ice than to raise a
gram of water 1 0C. All the energy goes into changing the phase.
Third, the liquid water from the melted ice is warmed. All of these things
require heat to flow from the warm water, cooling it rapidly.
Notice that, after a few minutes, the rate of cooling is slower in the beaker
with the ice than in the one without the ice. Will the curves all eventually
meet? (Yes, when the water equilibrates with room temperature)
How do you get the coldest water at, say, 10 minutes? Should we add the ice
cube immediately? At 5 minutes? At 9 minutes? This would answer the ageold question of – when do I add the milk to my coffee in order to have it cool
quickest? After I first pour the coffee, or a little later?
Stephanie Chasteen
Exploratorium Teacher Institute
http://www.exo.net/~drsteph
© 2006 Exploratorium, all rights reserved
Etc.
Specific heat and latent heat
The amount of heat that a material requires to raise (or lower) its
temperature is the specific heat. The specific heat of water is very high – it
takes longer to warm the water in the pan than to warm the pan itself,
because it requires more heat (calories). You can measure the heat lost by
the water using the formula Heat = mass * specific heat * change in
temperature or Q = mC • "T . The specific heat of water is 4.18 J/(g*0C) or 1
calorie/(g*0C). The latent heat of fusion is the heat required to melt the ice,
which is 334 J/g. Thus, it takes much more heat to melt a gram of ice than to
raise the ice 1 0C.
!
The high specific heat of water (and thus its slow cooling and heating) is
important for life on earth. Water stores huge amounts of heat energy from
the sun during the day. At night, it gradually cools and warms the air. This
gives the coasts their mild climates. Since we’re mostly made of water, we’re
also more easily able to maintain a stable internal temperature.
Cooling factor
"T
# T0 $ T is known as a differential
"t
equation. You need to use calculus to solve this kind of equation to find out
what the temperature would be at some time t.
The equation for the cooling rate,
!
That gives the equation T " e#Ct , where “C” is the “cooling factor”. That’s a
special number that tells you how the liquid cools. It depends on the liquid
and the container. The e is a special number in mathematics. It is called the
exponential. Its
! approximate value is 2.71828. Like π, it actually has an
infinite number of decimal digits.
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.
Can hot water freeze faster than cold water?
Sometimes, yes! This is called the Mpemba effect. It is believed to be caused
by greater evaporation from the warmer water compared to the cooler water,
and convection. Read more here:
http://math.ucr.edu/home/baez/physics/General/hot_water.html
Stephanie Chasteen
Exploratorium Teacher Institute
http://www.exo.net/~drsteph
© 2006 Exploratorium, all rights reserved