Plotting a cooling curve (relationship between the temperature of a hot body and time)

Introduction:

The cooling curve of a substance is obtained when a graph is drawn against the change of temperature and the time when the substance is allowed to cool.

To understand the cooling curve, we need to understand important terms:

Heat: It is a form of energy that is transferred between objects or systems, that varies the temperature of the system or the object.

The flow of heat energy is always from higher temperature to lower temperature.

Heat capacity: It is the amount of heat required to change the temperature of 1 mole of a substance by 1℃.

Specific heat capacity: it is the amount of heat required to change the temperature of 1 gram of a substance by 1℃.

The formula of specific heat capacity:

Q = mC ∆t

Here, Q = quantity of heat absorbed by a body

m = mass of the body

∆t = Rise in temperature

C = Specific heat capacity of a substance, that depends on the nature of the substance.

It’s SI unit is J kg^-1 K^-1.

The slope formed in the cooling curve depends on heat capacity, change in temperature, and the thermal conductivity of the substance. When more heat is required to change the temperature of the substance,it cools slower, and the gradient of the curve decreases. When the substance has a higher thermal conductivity, the faster the heat is transferred, so the substance cools faster.

Liquids have the highest heat capacity among all the phases.

Newton’s Law of Cooling

To plot the cooling curve, we need to understand Newton’s law of cooling, which states that the rate of heat loss is proportional to the temperature difference between the surroundings and the body when the temperature difference is minimal.

dQ/dt  ∝ (T – T₀)

Considering the body mass “m”, specific heat “c” and temperature “T” and kept in the surrounding temperature of “T₀”:

Q = mcT

Rate of heat loss or cooling, dQ/dt = mc dT/dt

mc dT/dt ∝  (T-T₀)

As ms = constant,

dT/dt ∝ (T-T₀)

Therefore, we can conclude that when T decreases, (T – T₀) decreases, and ultimately the rate of fall in temperature also decreases.

Experimental set-up

Newton’s law of cooling apparatus is used to analyze the relationship between the temperature of a hot body and time by drawing a cooling curve in an apparatus.

It is a slim-walled copper calorimeter that is enclosed in a double wall. There are two thermometers, a clamp, a stand, and a stopwatch.

Procedure

1. Fill the space between the double-walled enclosure with water.
2. Fill the calorimeter about two-thirds with 80℃ water.
3. Suspend the calorimeter within the enclosure along with the stirrer in it. Cover the setup with a wooden lid having a hole in the center.
4.  Take a thermometer and suspend it in the double-walled enclosure and the other one in the calorimeter setup using a clamp stand.
5.  Keep a note on the least count of the thermometers.
6. Set up a stopwatch and note its least count.
7. Note the temperature of water in the enclosure, this is To.
8. Begin stirring the water in the calorimeter to cool it evenly.
9. When the temperature of the water in the calorimeter reaches a certain temperature, say 70℃, note it and start the stopwatch.
10. Continue stirring and note down the fall in temperature every 1 minute. The temperature decreases fast.
11. Simultaneously, note down the temperature of water in the enclosure every 5 minutes.
12. When the fall in temperature slows down, that is, in a time interval of 5 or 10 minutes, stop the experiment.

Observation and calculation:

The least count of the water enclosed by double-wall =

The Least count of the thermometer in the calorimeter water =

The least count of the stopwatch =

Table:

Sl no.

Time allowed for cooling t ( min)

Temperature of water in calorimeter T (℃)

Temperature of water in enclosure T0 (℃)

Difference in temperature T – T0 (℃)

Interpretation of the cooling curve

The temperature difference between the surroundings and the object decreases as the object cools, therefore, the lines are curved. Because the temperature differential between the surroundings and the object decreases as the object cools, the lines are curved. This slows down the pace at which heat is transmitted out of the substance, slowing down the cooling process.

For most pure substances, phase transitions occur at particular temperatures.

The temperature remains constant during the phase transition because the energy produced by the formation of new bonds in the substance neutralizes the cooling.

Phase transitions appear as horizontal lines as a result of this.

Phase transitions of mixtures occur over a wide temperature range, therefore, instead of a flat segment, the cooling curves of mixtures undergo a gradient change during a phase transition.

Limitations of Newton’s law of cooling:

1. It is applicable only at lower temperatures.
2. It can only give the difference in temperature of the initial and final within the range of 50-80 ℃ not more than that.
3. The temperature of the surrounding area should be kept constant.

Cooling curves:

The cooling curve of the mixture is used in the preparation of alloys in exact proportions. As cooling can be constructed for pure metals. The temperature at which they form a molten state and then solidify can be clearly noted by these curves obtained by the change in phase.

As we know, all pure metals have a definite melting point and freezing point. Therefore, the graph obtained for the pure metals is a horizontal straight line.

When the mixture of metals is added, there are no sharp points. There is always a range of temperature that is noted, as in when they are cooled or heated.

The cooling curve for solidification of eutectic alloy (mixed molten state) ,shows two stages of solidification. As  this alloy exhibits a definite melting and freezing point, these temperatures are referred to as eutectic temperatures.

Conclusion:

To plot the cooling curve, first we had to understand the concept of heat energy, specific heat capacity, and how the thermal conductivity takes place between objects and the surrounding.

The next step was to understand Newton’s law of cooling which gave a specific equation to understand how the increase or decrease in temperature modifies the graph.

Later in this article, we understood how to set up Newton’s cooling apparatus and how to carry out the experiment. The result obtained from the experiment is then plotted as a graph.

Interpretation of the graph is important to understand the whole concept of the cooling curve.

The cooling curve of pure substances and mixture play an important role in metallurgy.

As in for separating the metals from ores and minerals and also for making alloys.