Effects of time in the decrease of temperature on different substances
IB Diploma Programme
Internal Assessment
Mathematics Analysis and Approaches SL
Student personal code: kmf095
Page count: 21 pages, appendices attached in additional pages.
1. Introduction
Why does water freeze faster than other liquids? That is the question that pushed me to do
this experiment. Some years ago I made popsicles with my grandparent to sell, water-based
and milk-based. Milk-based popsicles always took longer to freeze and I never knew why. If
both are liquids and are suffering the same impact of temperature, why does the time vary?
The freezing point depression tells that with dissolved substances in the water the freezing
point will be lower. “ By dissolving a solute into the liquid solvent [...] more energy must be
removed from the solution in order to freeze it, and the freezing point of the solution is lower
than that of the pure solvent.” (CK-12 Foundation, 2018.) Milk can be watered down
naturally or by the producer to be more efficient. If we see the ingredients at the back of the
milk, in the majority of cases they will have water. This is the basis of this experiment.
Figure 1. Ingredients of Milk
The aim of this work is to measure how the temperature (dependent variable) decreases
depending on the time (independent variable) on different substances such as water and milk.
1
The temperature will be measured with a thermometer and the time with a chronometer. The
control variable is the amount of ice each liquid will have and the time the experiment will
last. I will pass the data obtained to a table and graphicate and model it mathematically. The
goal of this experiment is to find the equation to find the temperature with the time,
where
is the variable in minutes (time) and
is the variable in Celsius
degrees (time). In addition, the results of this experiment will prove the freezing point
depression, and water and milk times of cooling would be contrasted and compared. The
hypothesis is that water will cool faster than milk.
2. Description of the experimental setup and variables
Variable
Units
Description
Symbol
Type
Time
Minutes
Time lapse
t
Independent
Temperature
Celsius degrees
How much it
c
Dependent
decreases
Figure 2. Table of the definition of variables, notation, and symbols.
The first step for this experiment was to pour one cup of water into a glass at room
temperature. Secondly, two ice cubes were placed in the glass, to serve as the factor which
would make the temperature (°C) decrease. The next step was putting a special culinary
thermometer into the glass, and starting running the chronometer on my phone to take the
time (min). Finally, write down the data it is being collected.
2
Figure 3. The glass of water
Figure 3. The glass of water
The same steps were repeated with the milk.
Figure 4. The glass of Milk
3
3. Experimental results
Temperature (C°)
Time in minutes (t)
Water
Milk
0
30.4
35.2
1
29.6
35
2
28.3
34.9
3
26.8
33.6
4
24.2
33.2
5
23.1
32.8
6
21.5
32.3
7
20.6
31.7
8
20.3
31.4
9
20.1
29.8
10
19.9
29.3
11
19.8
29
12
19.6
28.6
13
19.5
27.3
14
19.4
26
15
19.5
25.8
16
19.4
25.6
17
19.2
25.3
18
19.3
25
19
19.4
24.9
20
19.6
25.1
21
19.7
25.3
22
19.8
25.4
23
20
25.5
24
20.1
25.5
25
20.1
25.6
26
20.2
25.7
27
20.4
25.9
28
20.5
26
29
20.6
26
30
20.8
26.1
4
Figure 5. Table of raw data results from the experiment
Figure 6. Graph with raw data coordinates
4. Functions to model the experiment
The functions were obtained using the software GeoGebra. In this experiment polynomial,
logistic, and rational functions were used because they were the ones that best modeled and
fitted the coordinates of the data obtained. Principally, the polynomial one for its concavity
since it is a function of third degree. That is known because of the coefficient of
determination, which according to the Corporate Finance Institute (2022):
The coefficient of determination (R² or r-squared) is a statistical measure in a
regression model that determines the proportion of variance in the dependent variable
that can be explained by the independent variable. In other words, the coefficient of
determination tells one how well the data fits the model (the goodness of fit).
5
The coefficient of determination goes between 0 and 1, the closer it is to 1, the greater the
adjustment of the model to the variable. All of the results in both, milk and water, were
higher than 0.9, with one exception of 0.8, which means the models fit the data very well.
Which can be seen in the following figures.
Figure 7. Graph of water temperature over time functions
6
Figure 8. Graph of milk temperature over time functions
In both cases (milk and water), the polynomial function is appropriate to model the
experiment results because it is the one that best adjusts the data, with
the water, and
in
in the milk being really high in both. However, it does not
have continuity because at the moment the data ends, it falls or raises radically and it does not
show how it would continue.
In contrast, the logistic function shows good and logical continuity that demonstrates how
the experiment could continue. In addition, it has a great fit of
in water and
in milk, which is very appropriate.
Finally, the rational function is the one that least adjusts the data, but it is still not bad at all.
In water
and in milk
which is still very stable. Its
continuity is the best one because it makes more sense that temperature stabilizes to ambient
temperature after 30 minutes of having just two ice cubes rather than increasing or
decreasing, as the other two functions imply.
4.1.2 Residual Analysis of functions
The residuals are as its name says, the residue of the subtraction of the original values, and
the predicted values that the functions made. The lowest this number means the
correspondent functions model the experiment in a better way, or in other words, it predicts
better how the temperature is affected by time in both cases, water and milk.. “Analysis of the
7
residuals is frequently helpful in checking the assumption that the errors are approximately
normally distributed with constant variance, and in determining whether additional terms in
the model would be useful.” (Montgomery, D. C., & Runger, G. C., 2010)
Residuals can vary to the top or bottom of the axis, if the predicted value is higher than the
original value, they are positive residuals, and if the predicted value is lower than the original
value, they are called negative residuals. This does not affect the validation of the models.
(Zach, 2020)
Figure 9. Graph of the residual temperature of the water
8
Figure 10. Graph of the residual temperature of the milk
In this case, it can be observed that the polynomial functions have less margin of error or
lower residual values, matching their high coefficient of determination. All the values are
below 1 and greater than -1, which is a good sign for the adaptability of the model, because as
the Corporate Finance Institute (2022) says: “the coefficient of determination can take any
values between 0 to 1”.
Following, the logistic functions take second place in the deviation of the data. There is a
higher variation between the original and the predicted values, very different from the first
functions, since some of the values even reach 2 in the case of water , and 1.5 in milk,
moving further away from 0 and therefore having a greater margin of error. Meaning, it is
not that properly adapted to the original values.
9
At last, the rational functions have the most radical changes. Its behavior is similar to the
logistic functions and very different from the polynomial functions. The predicted values
differ a lot from the original values despite their good coefficient of determination,
considering some of the values exceed 2 in the case of water, and 1.5 in milk 1.5.
(Consult annex 1 and 2 on the appendix to see complete tables of residuals)
4.2 Functions analysis
Coordinates for the Inflection point
The first test to see the accuracy of the functions to the original data was to obtain the
coordinates for the inflection point. It is important to mention this only applies to the
polynomial functions. The temperature decreases until a point where the temperature starts
stabilizing to ambient temperature. The point where the temperature changes from decreasing
to increasing is an inflection point.
Water:
To calculate the inflection point, the first step is to get the first and second derivative.
Having the second derivative it is equal to 0 so the variable
can be cleared, and that would
be the x-value.
10
Next, the value of variable
,that is now known, is substituted in the original function to
get the y-value.
Lastly, the x and y-value are expressed as a coordinate, and that would be the inflection point.
For the water, the coordinate is (20.91, 19.77), which is similar to the original one (17, 19.2),
with a margin of error of 3.91 minutes and 0.57 degrees.
The same steps were made with the
function.
Milk:
1. Get first and second derivatives.
2. Equal to 0 and clear variable
3. Substitute variable
.
in the original function.
4. Write as coordinate.
In contrast, milk has a bigger margin of error of 15.79 minutes and 8.81 degrees (3.21, 33.71)
being the inflection point coordinate and (19, 24.9) the original coordinate. Meaning the
11
mathematical models predict better the behavior of water rather than milk, and due to the
inflection point it is proved that water cools faster than milk.
The results are shown in the next figure (10).
Function
Inflection Point
w (t)
(20.91, 19.77)
v (t)
None
l (t)
None
m (t)
(3.21, 33.71)
n (t)
None
p (t)
None
Figure 10. Table of water and milk inflection points
Value of x for a specific y value of interest
The second test made was the value of x for a specific y value of interest. This test applied
with all except with the logarithmic function in the water scenario
. It helps to know the
time in which the substance, either water or milk, reached its minimum temperature. It is
important and relevant for the aim of this project since it makes it possible to contrast the
lowest temperature in each substance so it can be identified which one cools faster and more.
Water:
12
Since the y-value of interest is 19.2 because it is the lowest temperature the substance arrived
at, the first step is to equal the function to that so the x-value can be found.
The second step is to subtract that quantity (19.2) to both sides so the function is equal to 0.
Next, a solution is given by the Newton-Raphson method, since it is a third degree
polynomial.
1. Divide.
2. Equal to 0.
3. Apply the general formula to clear the variable
The same steps were made with the
function, and the rest of the functions.
Milk:
Equal the function to the y-value of interest.
Subtract y-value to both sides of the equation.
13
Use Newton-Raphson method.
1. Divide.
2. Equal to 0.
3. Apply general formula.
The polynomial function had the best approximation to the real value, calculating 17.64 when
the real one is 17 in water, and 23.53 when the real one is 19. Then it follows the rational
function which differs by 12.05 minutes in water and 8.14 minutes in milk. Lastly, the
rational function has a margin of error of 9.34 minutes in milk. This occurs because
polynomial functions have the greater coefficient of determination in both substances, which
is 0.97, and because of the concavity.
The results are shown in the next figure (11).
Function
Value of t
w (t)
17.64
v (t)
29.05
l (t)
None
m (t)
23.53
n (t)
27.14
p (t)
28.34
Figure 11. Table of water and milk values of t in the lowest temperature
14
4.3 Instantaneous rate and shape analysis
This graph (figure 12) shows the first derivative from the polynomial, logistic and rational
functions of water. It represents the instantaneous rate of change from the original function.
Figure 12. First derivative graph from water
In figure 7 we can see that the original polynomial function
is predicting that
temperature is decreasing until the inflection point where the temperature starts to increase
slowly, this is because the temperature is adapting and balancing to ambient temperature after
the ice has melted. Since it is a third degree polynomial it has two turning points, but we are
interested in just one due to the domain. Similarly, there is a change of direction in the first
derivative. It seems to be increasing until the minute 20 and 0.2 degrees (20.9, 0.2), where it
is the inflection point, and then it starts decreasing in small quantities due temperature values
are in decimals. The model predicts the temperature will continue decreasing very slowly.
This makes sense with Newton's law of cooling, which says: "The rate at which the
15
temperature of an object changes is proportional to the difference between the temperature of
the object and that of the surrounding medium." (Connor, 2020) Which means that as time
passes, the temperature of the substance (water or milk) decreases more slowly. In contrast,
the logistic
and rational
functions are very similar, both are negative and they
are increasing until their acceleration is practically 0, reaching a constant speed. This tells us
the temperature is stopping to rise and is starting to remain constant, since they are adapting
to ambient temperature.
Now, this graph (figure 13) shows the rate of change from the original functions of milk.
Figure 13. First derivative graph from milk
The original function in figure 8, shows something similar to water, but in this case, the
temperature decreases more slowly. In the first derivative logistic
functions
and rational
are very similar to the case of water. Both are negative and increasingly
approaching zero. The difference is in this model they are a bit further away from zero, and it
16
seems they will continue increasing really slowly and in little quantities until they remain
constant. With this we can infer milk takes a bit more time on remaining constant than water.
In the case of the polynomial function
the temperature is decreasing until the
inflection point in minute 3 and degree -0.7 (3.2, -0.7) that it starts increasing. The model
predicts it will continue rising.
This graph (figure 14) shows the second derivatives in both cases, water and milk, it is the
rate of rate of change. That means it explains the behavior of the first derivative and the shape
of the original function.
Figure 14. Second derivative graph of water
17
Figure 14. Second derivative graph of milk
In this case, the second derivative shows the function is slowing down, given that the speed
oscillates very close to 0. Almost all the functions are combined on the x-axis for both water
and milk, meaning the acceleration is 0 and the velocity is constant. Differently, the
polynomials show other things. In the case of water, the model predicts after intercepting the
x-axis the temperature will continue decreasing. And in the case of milk, the model predicts it
will start to rise, meaning it is concave up. The signs on the first derivative and the second
derivative are opposite, which indicates the speed of the functions is going slower, which
makes complete sense because the temperature is adapting to ambient temperature and that
means it will not decrease or increase much more.
(Consult annex 3 and 4 on the appendix to see the process to get first and second
derivatives)
18
5. Conclusions and recommendations
The conclusion I arrived at the end of this experiment and mathematical analysis is that the
hypothesis was correct, water cools faster than milk being congruent with the freezing point
depression. What was surprising is how both substances reacted when the ice melted (around
minute 14 in water, and 19 in milk). What I expected was that the temperature would
continue to decrease over time or in which case it would remain at the coldest temperature it
reached, which would be the temperature of ice. But surprisingly, the temperature started to
increase after that inflection point, where the temperature was the lowest, until stabilizing at
ambient temperature. I say surprisingly because I did not expect that, but it is really totally
logical because since the substances no longer have a factor that cools them, what would
continue to lower the temperature? And since the substances were not in the fridge or freezer,
why would they stay cold?
The models that best explained this were the polynomial functions. They showed, in both the
water and milk scenario, the decreasing of temperature until the inflection point where they
started increasing, making a concave up model with
average which fits
really well, and they also had the lowest margin of error on residuals.. The weakness or
limitation is that they only serve to predict until minute 30, which was the duration of the
experiment, and after that they do not have continuity, or rather, not a logical one. Because in
the case of water it predicts to continue increasing, and differently, in the case of milk it
predicts to continue decreasing, which does not make sense. But comparing them to different
mathematical approaches, such as the logistics and rationals functions, polynomials fit better
because the logistics and rationals do not show the inflection point where the temperature
starts increasing to stabilize to ambient temperature. Instead, they just drop to room
19
temperature without any change of direction and start to stabilize. Their strength is that they
show better and more logical continuity.
20
Bibliography
CK-12 Foundation. (September 12, 2018). 16.14 Freezing Point Depression.
https://flexbooks.ck12.org/cbook/ck-12-chemistry-flexbook-2.0/section/16.14/primary
/lesson/freezing-point-depression-chem/
Connor, N. (2020, 7 enero). ¿Cuál es la Ley de Enfriamiento de Newton? Definición.
Thermal Engineering.
https://www.thermal-engineering.org/es/cual-es-la-ley-de-enfriamiento-de-newton-def
inicion/
Corporate Finance Institute. (May 5, 2022). Coefficient of Determination.
https://corporatefinanceinstitute.com/resources/knowledge/other/coefficient-of-determ
ination/
Grainger Engineering Office of Marketing and Communications. (October 22, 2007).
Freezing Point of Milk | Physics Van | UIUC.
https://van.physics.illinois.edu/ask/listing/1606
Montgomery, D. C., & Runger, G. C. (2010). Applied Statistics and Probability for
Engineers. Wiley.
https://spada.uns.ac.id/pluginfile.php/196559/mod_resource/content/1/Douglas%20C.
%20Montgomery%20Applied%20Statistics%20and%20Probability%20for%20Engin
eers%203ed.pdf
Zach. (December 7, 2020). What Are Residuals in Statistics? Statology.
https://www.statology.org/residuals/
21
Appendix
Annex 1. Table of residuals of water
Water
Time in
Temperatur
minutes (t) e (C°)
w(t)
y-w(t)
v(t)
y-v(t)
l(t)
y-l(t)
0
30.4
31.00
-0.60
31.89
-32.49
32.01
-64.50
1
29.6
29.10
0.50
28.27
-27.78
28.75
-56.52
2
28.3
27.41
0.89
26.15
-25.26
26.52
-51.78
3
26.8
25.91
0.89
24.75
-23.85
24.94
-48.79
4
24.2
24.58
-0.38
23.76
-24.14
23.77
-47.91
5
23.1
23.42
-0.32
23.02
-23.34
22.89
-46.23
6
21.5
22.43
-0.93
22.44
-23.37
22.22
-45.59
7
20.6
21.58
-0.98
21.99
-22.97
21.69
-44.66
8
20.3
20.87
-0.57
21.62
-22.19
21.28
-43.47
9
20.1
20.29
-0.19
21.31
-21.50
20.95
-42.45
10
19.9
19.83
0.07
21.05
-20.98
20.69
-41.67
11
19.8
19.48
0.32
20.82
-20.51
20.48
-40.98
12
19.6
19.23
0.37
20.63
-20.26
20.31
-40.57
13
19.5
19.07
0.43
20.46
-20.03
20.17
-40.20
14
19.4
18.99
0.41
20.31
-19.90
20.06
-39.96
15
19.5
18.98
0.52
20.18
-19.66
19.97
-39.62
16
19.4
19.02
0.38
20.06
-19.69
19.89
-39.58
17
19.2
19.12
0.08
19.96
-19.88
19.83
-39.71
18
19.3
19.25
0.05
19.86
-19.81
19.78
-39.60
19
19.4
19.42
-0.02
19.77
-19.79
19.74
-39.53
20
19.6
19.60
0.00
19.69
-19.69
19.71
-39.40
21
19.7
19.79
-0.09
19.62
-19.71
19.68
-39.39
22
19.8
19.98
-0.18
19.55
-19.74
19.66
-39.40
23
20
20.16
-0.16
19.49
-19.65
19.64
-39.30
24
20.1
20.32
-0.22
19.43
-19.66
19.63
-39.28
25
20.1
20.45
-0.35
19.38
-19.73
19.61
-39.35
26
20.2
20.54
-0.34
19.33
-19.67
19.60
-39.27
27
20.4
20.57
-0.17
19.29
-19.46
19.60
-39.06
28
20.5
20.55
-0.05
19.24
-19.29
19.59
-38.88
29
20.6
20.45
0.15
19.20
-19.06
19.58
-38.64
30
20.8
20.28
0.52
19.16
-18.64
19.58
-38.22
22
Annex 2. Table of residuals of milk
Milk
Time in
Temperatur
minutes (t) e (C°)
m(t)
y-m(t)
n(t)
y-n(t)
p(t)
y-p(t)
0
35.2
35.87
-0.67
36.64
-37.31
36.74
-74.05
1
35
35.21
-0.21
35.39
-35.60
35.50
-71.10
2
34.9
34.53
0.37
34.32
-33.95
34.41
-68.35
3
33.6
33.85
-0.25
33.37
-33.62
33.44
-67.06
4
33.2
33.17
0.03
32.54
-32.51
32.58
-65.09
5
32.8
32.49
0.31
31.80
-31.50
31.81
-63.31
6
32.3
31.82
0.48
31.14
-30.66
31.13
-61.79
7
31.7
31.16
0.54
30.55
-30.01
30.51
-60.52
8
31.4
30.51
0.89
30.01
-29.12
29.95
-59.07
9
29.8
29.88
-0.08
29.52
-29.60
29.44
-59.05
10
29.3
29.27
0.03
29.08
-29.05
28.98
-58.03
11
29
28.68
0.32
28.67
-28.35
28.57
-56.92
12
28.6
28.13
0.47
28.30
-27.82
28.18
-56.00
13
27.3
27.60
-0.30
27.95
-28.25
27.83
-56.08
14
26
27.11
-1.11
27.63
-28.74
27.51
-56.25
15
25.8
26.66
-0.86
27.33
-28.19
27.22
-55.41
16
25.6
26.25
-0.65
27.05
-27.70
26.95
-54.65
17
25.3
25.88
-0.58
26.79
-27.38
26.70
-54.08
18
25
25.57
-0.57
26.55
-27.12
26.47
-53.59
19
24.9
25.31
-0.41
26.33
-26.73
26.26
-52.99
20
25.1
25.10
0.00
26.11
-26.11
26.07
-52.18
21
25.3
24.96
0.34
25.91
-25.57
25.88
-51.45
22
25.4
24.87
0.53
25.72
-25.20
25.72
-50.91
23
25.5
24.86
0.64
25.54
-24.91
25.56
-50.47
24
25.5
24.92
0.58
25.38
-24.79
25.42
-50.21
25
25.6
25.05
0.55
25.22
-24.67
25.28
-49.95
26
25.7
25.26
0.44
25.06
-24.63
25.16
-49.79
27
25.9
25.55
0.35
24.92
-24.57
25.04
-49.62
28
26
25.93
0.07
24.78
-24.72
24.94
-49.65
29
26
26.40
-0.40
24.65
-25.05
24.84
-49.89
30
26.1
26.96
-0.86
24.53
-25.39
24.74
-50.13
Annex 3. Process to get derivative of
23
Annex 4. Process to get second derivative of
24