Investigatory Project

Modern physics-Cooling Curve of a Liquid – Measuring

Temperature vs. Time

NAME:ARK RATURI ROLL NO.:

CLASS:

CERTIFICATE
 THE CAMBRIDGE INTERNATIONAL SCHOOL, BANGALORE

This is to certify that

Has satisfactorily completed the investigatory project

in______________________________ as prescribed by The Central Board of
Secondary Education for the year 2025-2026.

____________________ _______________________
Internal Examiner External Examiner

Date: _____________ ________________________

Principal

Name of Candidate: __________________

Examination Centre: __________________
Date of examination: __________________

Index
SNO CONTENT PAGE NO.
1 Acknowledgement 1
2 Introduction 2
3 Content 3
4 Aim 4
5 Apparatus 5
6 Diagram 6
7 Theory 7-8
8 Procedure 9-11
9 Observation 12-14
10 Results and Precautions 15
11 Error Analysis and Sources of 16
Uncertainty
12 Conclusion 17
13 Bibliography 18

Acknowledgement
Apart from the efforts of me, the success of any project depends largely on the
encouragement and guidance of many others.
I express a deep sense of gratitude and sincere thanks to almighty God for
giving me the strength for the successful completion of this project. I wish to
express my deep gratitude and sincere thanks to the Principal Ms. Roshni
Vijayan for her encouragement and for all the facilities that she provided for this
project work. I sincere appreciate her generosity by taking me into her fold for
which I shall remain indebted to. I extend my hearty thanks to my physics
teacher Ms. Manju who guided me to the successful completion of this project. I
take this opportunity to express my deep sense of gratitude for her invaluable
guidance, constant encouragement, immense motivation, which has sustained
my efforts at all stages of this project.
I would like to offer my sincere thanks to my parents and also my classmates
who helped me to carry out this project successfully and for their valuable
advice and support.

Introduction
When a liquid at a high temperature is left to cool in an environment at a lower
temperature, it loses heat to the surroundings until it reaches thermal
equilibrium. This process occurs due to the transfer of heat energy through
mechanisms such as conduction, convection, and radiation. The study of how
temperature changes with time during cooling helps us understand the laws
governing heat transfer, particularly Newton’s law of cooling. By plotting the
cooling curve of a liquid, we can observe the rate at which it cools and the
factors that influence this rate.
The cooling curve is a graphical representation of temperature versus time,
showing the gradual decrease in temperature as the liquid approaches the
ambient temperature. This curve often follows an exponential decay pattern,
where the rate of cooling is proportional to the temperature difference between
the liquid and its surroundings. Understanding this relationship not only
deepens our knowledge of thermal physics but also has practical applications in
industries such as food processing, metallurgy, and climate studies. In this
project, we aim to experimentally determine the cooling curve of a liquid and
analyse the data to evaluate its conformity to Newton’s law of cooling.

Content
This project investigates how a liquid cools over time by measuring and plotting
its temperature at regular intervals. Using Newton’s law of cooling as the
theoretical foundation, it examines the relationship between the cooling rate and
the temperature difference between the liquid and its surroundings. By heating a
liquid to a set starting temperature and recording its temperature as it cools to
room temperature, we obtain a cooling curve, which is then analysed to
determine the cooling constant and understand the heat transfer process.
The work involves setting up the experiment with basic lab apparatus, recording
accurate observations, and plotting temperature versus time to observe the
curve’s shape. Analysis includes linearising the data by plotting the natural log
of the temperature difference against time, allowing us to confirm the
exponential decay predicted by theory. The project not only reinforces concepts
of thermal physics but also highlights the influence of experimental conditions,
errors, and precautions on the accuracy of results.

Aim
The aim of this project is to study and analyse the cooling behaviour of a liquid
by measuring its temperature at fixed intervals as it cools from an elevated
temperature to the surrounding room temperature. By recording and plotting the
cooling curve, the experiment seeks to observe the nature of temperature change
over time and verify the predictions made by Newton’s law of cooling.
Another objective is to determine the cooling constant of the liquid under given
experimental conditions. Through data collection, graph plotting, and
mathematical analysis, the project aims to connect theoretical principles of heat
transfer with practical measurements. This will help in understanding the
influence of factors such as the nature of the liquid, environmental conditions,
and initial temperature on the rate of cooling.

Apparatus
Beaker (100–250 mL)
Liquid sample (distilled water / cooking oil / glycerin)
Bunsen burner / hot plate or electric kettle
Thermometer (digital or mercury) — range 0–110 °C, least count 0.1 °C (or a
temperature probe/thermocouple connected to a data logger)
Stop watch or digital timer
Tripod stand and wire gauze ( Bunsen burner)
Insulating mat / cotton wool
Stirrer (glass rod)
Retort stand and clamp (for holding thermometer)
Paper and pen for observations; graph paper

Diagram
Theory
When a body or liquid at a temperature higher than its surroundings is left
exposed, it loses heat to the environment. This occurs through conduction
(direct transfer of energy through molecular contact), convection (movement of
fluid layers carrying heat away), and radiation (emission of infrared energy).
The rate of heat loss is influenced by the temperature difference between the
body and its surroundings, the surface area exposed, the properties of the
medium, and environmental conditions such as air flow.
Newton’s law of cooling provides a mathematical framework for this process. It
states that the rate of decrease of temperature of a body is directly proportional
to the temperature difference between the body and the surroundings, provided
the temperature difference is not too large. Mathematically:

where:
T is the temperature of the liquid at time ,
TENV is the ambient temperature,
k is the cooling constant (depends on material, medium, and conditions).
Solving this differential equation gives:

Where TO is the initial temperature of the liquid. This expression predicts that
the cooling curve is an exponential decay approaching Tenv asymptotically.
To experimentally verify this law, we record the temperature of the liquid at
fixed intervals. By plotting T versus t, we get the cooling curve. To determine
the cooling constant k, we can rearrange the equation:

This suggests that a plot of ln(T-Tenv) versus t will yield a straight line with
slope-k . This linearisation helps in analysing data and extracting k with
minimal influence from small measurement errors.
In practice, factors such as stirring, heat loss to the container, and fluctuations in
room temperature can cause deviations from the ideal behaviour predicted by
Newton’s law. Liquids with different viscosities, specific heats, and thermal
conductivities will cool at different rates. Environmental conditions such as
airflow and humidity also affect the rate of cooling. Understanding these effects
and accounting for them in the analysis is essential for accurate results.
The study of cooling curves has wide applications: in food science for cooling
cooked materials, in metallurgy for understanding solidification, in climatology
for predicting temperature drops, and in forensic science for estimating time of
death from body cooling. Thus, this experiment not only reinforces theoretical
concepts of thermal physics but also connects them with real-world phenomena.

Procedure
A. Preparation
 1. Assemble apparatus: place the retort stand, clamp, beaker, thermometer
(or temperature probe), insulating mat, and stopwatch on a steady bench.
Ensure the thermometer is calibrated and the digital probe (if used) is
connected to the data logger or multimeter.
2. Clean and dry the beaker to remove contaminants. Measure and note the
volume of liquid to be used (e.g., 150 mL) so repeated trials use the same
quantity.
3. Measure and record the ambient room temperature (T_env). Keep the
experimental area draft-free (close windows or turn off fans) to reduce
convection currents.
4. If using a digital probe, set the sampling interval (for example, 5–10 s) on
the data logger. If recording manually, decide on a consistent time
interval for readings (e.g., every 30 s for the first 5 minutes, then every 60
s).
B. Heating and Starting the Run
5. Pour the measured quantity of liquid into the beaker and place it on the
hot plate or over a Bunsen burner. Heat gently to the chosen initial
temperature (T₀) — for water, 70–90 °C is a practical range. Avoid
vigorous boiling unless part of the experiment design.
6. Monitor temperature closely while heating. When the liquid approaches
the desired T₀, remove the heat source and allow the temperature to
stabilise for a few seconds. Record the exact starting temperature and the
time as t = 0.
7. Immediately clamp the thermometer/probe so that its sensing element is
immersed well below the liquid surface but not touching the beaker walls
or bottom. Start the stopwatch or the data-logger recording at the same
instant you note T₀.
C. Recording Data
8. Take temperature readings at the pre-decided regular intervals. For
manual recording, call out the time and the temperature and write them in
the data table. For an electronic probe, ensure timestamps are recorded
automatically.
9. If manual readings are used, keep the same observer or practice consistent
reaction timing to reduce human error. Record at least 10–15 points
 spanning from T₀ until the temperature is within about 1–2 °C of
ambient.
10.Gently stir the liquid at fixed, pre-decided times if necessary to maintain
uniform temperature throughout the beaker. If stirring is used, note the
exact times and method (e.g., slow circular stirring for 2 s every 60 s)
because stirring affects convection and cooling.
D. Ending the Run and Repeats
11.Stop data collection when the temperature difference (T − T_env)
becomes very small (≈1–2 °C) or when readings plateau near room
temperature.
12.Allow the apparatus to cool and clean the beaker. Repeat the experiment
at least two more times with the same settings to check reproducibility.
Optionally, repeat with different starting temperatures or with other
liquids (e.g., oil, glycerol) keeping volume and ambience constant.
E. Variations and Controlled Comparisons
13.To study the effect of insulation, repeat the experiment with the beaker
wrapped in insulating material (cotton wool or foam) and compare the
cooling constant k.
14.To study the effect of surface area, use beakers of different diameters but
the same volume and compare the cooling curves.
15.To study forced convection, place a small fan at a fixed distance and
direction and record how the cooling constant changes.
F. Data Handling and Good Practices
16.For manual data, immediately transfer the recorded values to a
spreadsheet (Excel/Google Sheets) and compute T − T_env and ln(T −
T_env) for each time point.
17.Plot temperature vs. time to obtain the cooling curve. Then plot ln(T −
Tenv) vs. time; fit a straight line and determine the slope (−k)
18.Calculate the time constant (τ = 1/k) and compare results between
different trials and conditions.
G. Safety and Cleanup
19.Use heat-resistant gloves and safety goggles while handling hot beakers.
Keep a heat-proof mat and tongs available.
 20.Ensure electrical cords and gas flames are handled safely. After the
experiment, switch off heaters, let glassware cool, and wash all
equipment. Dispose of liquids according to school safety rules.
H. Notes on Minimising Errors
 Use an electronic temperature probe and data logger where possible to
reduce human reaction-time errors.
 Ensure the thermometer bulb is immersed but not in contact with the
beaker walls.
 Maintain a steady environment — avoid drafts and keep ambient
temperature constant.
 Repeat trials and report mean values and standard deviation for k to
reflect experimental uncertainty.

Observation
-Data Table
 Time (s) Temperature T−T env Ln(T−T env )
(°C) (°C)
0
30
60
90
120
150
180
240
300
360
...
Note: intervals are adjusted based on how fast the liquid cools and timer
accuracy.

Initial Conditions: (Expected and taken for calculations)

 Liquid used: Water (150 mL)
 Initial temperature (T₀): 80.0 °C
 Average Ambient room temperature (T_env): 25.0 °C
 Time interval for readings: Every 30 seconds for first 5 minutes, then
every 60 seconds
 Thermometer least count:0.5 °C
Notes:
1. The cooling rate is faster in the first few minutes due to a higher
temperature difference between the liquid and the surroundings.
2. As temperature approaches the ambient value, the cooling slows down
significantly.
3. The graph of Temperature vs. Time is a smooth downward curve, while
the graph of ln(T − T_env) vs. Time is nearly a straight line, confirming
Newton’s Law of Cooling.
Analysis and Calculations
Plot temperature (°C) vs. time (s) — this is the cooling curve (exponential-like
decay).
To find the cooling constant kk:
Compute y=ln(T−Tenv) for each data point.
Plot y vs. t. According to Newton's law, this should be a straight line with slope
−k.
Fit a best-fit straight line to the ln plot and find its slope. Then k=−slope.
Using k, you can predict temperature at later times, or compute characteristic
time constant τ=1/k (time to reduce the temperature difference by a factor e).

Graphs
Graph 1: Temperature (°C) vs. Time (s).
Graph 2: ln(T−Tenv) vs. Time (s)

Results and Precautions

Result
Cooling constant k=
Time constant τ=1/k=
The graph Temperature (°C) vs. Time (s) is:
The graph ln(T−Tenv) vs. Time (s) is

Precautions
Make sure thermometer bulb is fully immersed but not touching container walls.
Remove heater quickly and place beaker on insulating support to avoid extra
heat pathways.
Keep ambient conditions (drafts, fans) constant.
Repeat measurements and average results.

Error Analysis and Sources of

Uncertainty
Thermometer least count / probe accuracy contributes to temperature
uncertainty (±0.1 °C typical for digital probes).
Heat loss to surroundings not perfectly uniform; drafts or airflow change results.
Contact of thermometer with beaker walls affects reading.
Non-uniform temperature (incomplete mixing) — stirring helps but may alter
cooling.
Human reaction time when recording values — use data logger for improved
accuracy.
Estimate percent error where possible. For instance, if temperature accuracy
±0.2 °C and typical measured temperature difference ~30 °C, fractional error
~0.7% in temperature difference.

Conclusion
From the experiment, it is observed that the temperature of a hot liquid
decreases over time in a manner consistent with Newton’s Law of Cooling.
Initially, when the temperature difference between the liquid and the
surroundings is large, the rate of cooling is significantly higher. As time
progresses and the liquid’s temperature approaches that of the surrounding
environment, the cooling rate decreases noticeably. This gradual reduction in
cooling speed confirms the exponential nature of heat loss predicted by the
theoretical model.
The plotted cooling curve clearly shows a steep slope at the beginning,
indicating rapid heat transfer, followed by a progressively flatter section as the
liquid nears room temperature. The plot of ln(T − T_env) against time produces a
nearly straight line, which validates the relationship described by the equation:

where k is the cooling constant. The experimental data’s close alignment with
this linear relationship reinforces the reliability of the method and the accuracy
of the readings taken.
This study highlights that the cooling process depends not only on the
temperature difference but also on environmental factors such as air movement,
humidity, and the thermal conductivity of the container. Slight deviations from
perfect linearity in the logarithmic plot can be attributed to these uncontrolled
variables, along with possible heat loss through conduction to the surface on
which the container is placed.
In practical terms, understanding cooling curves has applications in fields like
food preservation, industrial heat treatments, climate control systems, and even
forensic science (estimating time of death from body cooling). This experiment
bridges theoretical concepts and real-world applications, allowing students to
see the direct connection between physics laws and daily life phenomena.

Bibliography
Class 12 Physics practical manual (CBSE)
Textbook sections on heat transfer and Newton's law of cooling