How does changing the method of cooling hot water affect the rate at which the water cools?

Factors other than the cooling method that could have potentially altered experiment results were kept constant. These factors are –

1. Amount of water in the beaker The volume of water was kept at a constant 450 cm3 or 450 g (for density 1 g cm-≈3)
2.  Material of beakerMaterial of the beaker used was Borosilicate for all experiments.
3.  Dimensions of beakerConsistently used cylindrical beaker of radius and height.
4.  Background TemperatureBackground temperature was maintained at 20oC.
5. The liquid used tap water from the same source was consistently used.

When experimenting, I observed a few safety and environmental risks and ensured these risks were avoided for optimal, safe outcomes.

The variable being altered is the method used behind cooling the beaker of hot water. These methods have been defined in Figure 3.

A few ecological issues were noticed with this experiment, and I economized these resources for optimal outcomes.

1. The flame source required thermal energy from gas. The low flame was used to prevent overheating, which would result in the wastage of gas.
2. Water wastage was minimized by using 450 mL of water per trial.

Only the averaged values of the observed temperatures have been displayed below for ease of readability. All values and calculations have been done considering 3 significant figures for accuracy.

The table below defines the methods used, and the symbols used to denote these methods throughout the experiment.

The variable being measured is the temperature of the water in the beaker at the twenty-second interval for ten minutes.

Newton conjectured that the rate of change of temperature of an object is directly proportional to the temperature gradient, or temperature difference, between the object and its surroundings. This statement can be transposed into a differential equation as follows.

\(\frac{dT}{dt}\propto(T-T_{background})\ (Eqn\ 1)\ [\therefore,\ \frac{dT}{dt}=temperature\ decay\ rate]\)

The proportional sign can be removed by using a constant of proportionality k, where k is a positive constant. However, a negative sign is also introduced as the (T - T _background) value is constantly decreasing with time, therefore decreasing the rate of cooling. Hence, we derive.

\(\frac{dT}{dt}=-k(T-T_{background})\ \ (Eqn\ 2)\)

After arranging the like terms temperature and time on LHS and RHS, respectively, the differential can be solved by integrating concerning time. Once solved, we derive a function that follows a form similar to the exponential decay form.

\(T(t)=e^{-kt}(T-T_{background})+T_{bakground}\ (Eqn\ 3)\)

The table below summarizes the used variables in this exploration.

I predict that as the methods are varied, the method that produces the highest value of cooling coefficient would be the most efficient for cooling future samples. From the above differential in Background Information,

\(\frac{dT}{dt}=-k(T-T_{background})\)

The cooling rate is  \(\frac{dT}{dt}\)​​​​​​​ directly proportional to the constant of proportionality k (also known as the coefficient of cooling). Hence, the higher the value of k, the more influential the method.

• A probe was connected to the laptop via Bluetooth, and data analysis software was activated.
• Background temperature was recorded, and the thermostat was adjusted to maintain a constant temperature.
• Water was poured into the Borosilicate glass beaker. Beaker was then placed on a clamp stand, supported by metal gauze.
• The Bunsen burner was placed below the stand and ignited. A probe was placed in a beaker.
• A flame source was switched off when the probe reading was 90oC.
• Different methods to cool the hot water were used, which have been defined later in the investigation.
• A probe was configured to measure the temperature every thirty seconds for ten minutes. Readings were consequently noted.
• This procedure was repeated for five trials in each method.

Note that each iteration of the procedure took approximately 16 minutes to finish. Hence, two trials were conducted simultaneously to avoid the monotony of work and complete it under time constraints. Given the shortage of temperature probes to carry out two trials simultaneously, a stopwatch and digital thermometer had to be used for the second simultaneous trial.

Hot beverages are consumed by many on a regular basis. Worldwide, 25,000 cups of tea are consumed per second, while 2.16 billion cups of coffee are consumed every year. However, there are problems regarding the consumption of these liquids. For example, the coffee served in cafeterias is approximately 85 – 90oC, almost 25o C higher than the optimal drinking temperature of 60o C. Hence, consuming this coffee at high temperatures could potentially cause damage to a person’s throat and food pipe. Additionally, it often takes a long time for the cup to cool down to the optimal temperature, posing a heavy inconvenience in terms of time, especially for commuters and travellers. I have also had such unfortunate experiences with coffee and tea. Hence, I was determined to explore the process of cooling and understand how thermal transfer and cooling could determine optimal methods for cooling. Additionally, preliminary research showed me that the cooling process of any beverage is exponential in nature, and I was eager to use my math knowledge regarding exponential processes to confirm this natural cooling process.

Exponential growth and decay are concepts that I have learned in my Mathematics class but come across even in Physics (Radioactive decay) and Computer Science (Efficiency of certain searching algorithms). I am very fascinated with how this concept is a fundamental property in many aspects of nature and was additionally motivated to pursue this specific investigation when I discovered that cooling has an exponential nature. Exponential growth can be defined as a phenomenon where the growth rate of a mathematical function increases proportionally to the function’s current value. Similarly, exponential decay is where the decay rate of a mathematical function increases directly proportional to the function’s current value. The shapes of these curves are depicted below.

There are some forms of minor hazards, which include –

1. After heating, care was taken to handle the hot beakers with tongs or tissues to avoid physical contact and hence skin burns.
2. Beaker was transferred from the stand to the table carefully and slowly to minimize the risk of spillage.
3. The magnetic stirrer was set at a low speed (speed = “2”) on the machine to avoid the splashing of hot water.
4. The filled beaker and apparatus were heavy, and there was a risk of falling from either one. To prevent this, I was aware to place these objects away from the edge of the surface.
5. Care was taken to provide a low amount of gas so the flames wouldn’t cause burn damage.

• 450 mL of tap water in a 500 mL Borosilicate glass beaker
• Bunsen Burner and gas source to provide thermal energy
• PASCO Temperature probe to record temperatures
• SPARKvue software on a laptop to record readings and analyze data
• Laboratory tripod to hold beaker above the Bunsen burner
• Wire gauze to diffuse heat to the cup and hence protect glassware from thermal shock
• Magnetic stirrer for slow, uniform stirring
• Tiny electric table fan at speed “1” (low speed), where 1 is the lowest speed, and 3 is the highest speed.
• Metal spoon
• Stopwatch and digital thermometer for the second consecutive trial.