Cooling, undercooling and Latent Heat

In this experiment we will analyze the temperature variation of a sample of water through melting and solidification. For this I used a resistive thermometer into a test tube and connected to an analog pin on the Arduino board. To cool the sample I prepared a freezing mixture (ice, water and salt) in a thermos steel, which was maintained at about -7 ° C for several hours. A good explanation of why these temperatures are achieved with this mixture, we can find in this  link .

 

This thermometer gives us the absolute values ​​of temperature, which varies its resistance compared to the temperature for it that we are going to optionally have a value of tension in the pin of the analog board. However, for the scope of the experience, we don’t need absolute values ​​of temperature, instead a reference of the behavior of this one, so we are going to handle with these voltage values.

It is expected that variations in temperature during cooling and heating follow the equation of Newton’s law of cooling, except during phase changes, during which the temperature does not change, since the energy as heat is delivered or received at this stage it is used exclusively for the solidification or melting, the amount varies according to what is known as latent heat of fusion.

An equation derived from Newton’s law is:

 \frac{\partial T}{\partial t}=-k (T - T_{a})

 

 


where to is a constant containing the heat exchange coefficient, to the amount of water, the contact surface and the specific heat of water. Ta is the ambient temperature, T is the body’s one, and t is the time. This equation indicates the rate of change in body temperature over time. By the method of separation of variables and integrating is:

\int {\frac{\partial T}{T - T_{a}}} = \int -k \partial t

solving the integral and applying the base and

e^{Ln(T-T{a})} = e^{-k t}

thus:

 T = T_{a} + e^{-k t}

In times of cooling is expected to have a negative exponential heating in a similar function with a negative sign.

The connection to the Arduino thermometer is between analog and 5v pin with a resistor  3k9\Omega between the pin and ground as a voltage divider:

The procedure was to insert the test tube with liquid water at room temperature and the thermometer inside the freezing mixture. Water began to cool, solidifying and cooling continued until the temperature of the mixture. Then he was taken out of this and left at room temperature. Thus began to heat up, again become liquid water and continued heating up to this temperature. Data were taken every two seconds and the result is this:

 

 

Point A indicates the moment when the tube was removed from the freezing mixture.

To make further analysis, we separate the process in the cooling part (up to point A) and heating (from point A).

In this graph three significant intervals are distinguished. The first, from the beginning to point A, shows the cooling of liquid water. At that point it can be seen that the temperature rises sharply to a value which remains constant to the point B. This value is the solidification temperature for the AB interval and water is passing from liquid to solid. The sudden jump in point A is due to the phenomenon of undercooling , which can cool a substance beyond its melting temperature until crystals begin to form in the solid state. At that time the temperature rises to the melting point of the substance and solidification continues at this temperature.

Point B is the approximate time when almost all the water turned to ice and begins to cool down to the temperature equilibrium with the freezing mixture, slightly lower than that of undercooling. Both cooling curves are shaped in a negative exponential function.

The following graph shows the heating process begins by removing the tube with ice freezing mixture:

Here are four interesting intervals. The first, to the point A, is the heating of ice up to the melting temperature. Note that it is the same as that of solidification. In the interval AB the phase shift occurs, and the temperature does not change. At point B, most of the ice has melted and liquid water begins to warm at the same time that is completed to melt the ice. The point C coincides with the time when all the ice has just melted and the water used throughout the absorbed heat to raise its temperature to reach equilibration at room temperature. You can see that in the two moments of warming before point A and after point C, the curve is a negative exponential function with a negative sign, the limit is the melting temperature in a case and the environment another.

The Arduino sketch used is simple:

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