Since we were children, we’ve known what thermometers did. These simple devices are used in a multitude of fields — from medicine, to chemistry, to the culinary arts. Many often overlook the science behind the measurement of temperature, because it is such a basic concept.
When an instrument, such as a thermometer, comes in contact with an object, heat from the object transfers to the thermometer, due to thermal equilibrium. This change in temperature is not instantaneous. A graphical representation of this relationship may be seen below.
This is described by the following function:
Where τ is the thermal time constant, t is the time elapsed, and T is the temperature. Substituting t for τ, this function takes the following form:
We may also substitute multiples of τ into the equation. The following functions result:
Because of the exponential nature of the function, as the time value gets higher, the change of the function with respect to time decreases. As time approaches infinity, T(t) = Tf. However, if too much time elapses, too much heat may escape, and error becomes more likely. Since the change in temperature is grows insignificant as time elapses, it is good practice to read the measurement when the time elapsed is between 3τ and 5τ. The goal of the experiment is to find the value of τ for an alcohol thermometer.
For this experiment, the temperature function of an alcohol thermometer was found by testing its behavior for both heating and cooling. For heating, a cup filled with tap water was placed on top of a small stove, then thermometer was held in place in the cup until the temperature ceased to rise. This value was recorded as Tf. The thermometer was then submerged in a cup of ice, and its temperature was recorded as Ti. Using these values, τ and its multiples were calculated using the above table. The thermometer was then submerged back into the hot cup, and when the thermometer read each temperature value in the table, the time was recorded.
These results were then linearized, where the negative of the inverse of the slope is the value for τ. Representative graphs for both heating and cooling may be seen below, as well a table showing the τ of each trial.
At first glance, both seem to be quite accurate, with high R2 values. This indicated a high degree of precision. However, the resulting values for τ indicate otherwise.
In this table, all values for τ are indicated in seconds. This result is quite alarming, considering that the inherently linear nature of measurement of the alcohol thermometer should indicate a uniform τ value across all trials. The values for heating are quite consistent, with an average deviation of only 0.071. For cooling, on other other hand, the average deviation was 3.739. This is quite a large variation in values, and is indicative of inaccurate results with low precision. From this data, the value of τ cannot be conclusively determined.
Several sources of error may have caused this. The set-up itself was quite crude, and the thermometer was held in place by a person holding one end. Since the set-up was an open system, the temperature may have fluctuated due to the loss of heat to the surroundings. The source of heat, the stove, for the heating trials was consistently on. Since a constant source of heat was applied, the results were more precise. Compare this to the cooling set-up, which consisted of an ice cube that was melting, and it is easy to tell that the source of heat for the cooling set-up was not at all consistent. The thermometer may have also been placed incorrectly, since the ice cubes created many spaces that may have resulted in fluctuating temperatures within the cup itself.
In conclusion, the experiment was not successful because the τ values for heating and cooling were inconsistent. For future experiments, it is highly recommended that a cooling set-up with a more consistent temperature is used. For example. a cup of liquid water surrounded by ice may suffice.