# Temperature Measurement

Life gives us chances and choices, it will be up to us which to choose; whether the more efficient but harmful, or the less efficient but harmless.

Here’s to the first experiment for the second part of our Physics 103.1 course, Thermodynamics.

As cliché as it is, temperature is the measurement of the hotness and coldness of an object and thermometer is the device used to measure temperature. In a clinic or maybe in our house, whenever we get sick, we traditionally use an alcohol thermometer put in our tongue or armpit and then we will wait until  the temperature reading steadies. This “simple” process we always encounter in our daily lives is what we are gonna deal with this experiment scientifically, the principles behind how thermometers work.

Temperature measurement coheres with two principles; first, the Zeroth Law of Thermodynamics, and second, Newton’s Law of Cooling.

Zeroth Law of Thermodynamics

When two objects of different hotness or coldness, i.e. one is warmer and one is cooler, are put in contact, heat is exchanged until there are no further changes in the system. That is, the warmer object is cooled and the cooler object is warmed. This state is called thermal equilibrium. Figure 1. Zeroth Law of Thermodynamics

Consider systems A, B, and C. The Zeroth Law of Thermodynamics states that if B is initially in equilibrium with A and C then, a and C are also in thermal equilibrium with each other. 

This principle is used by thermometers wherein actual temperature of an object is measured.

Newton’s Law of Cooling

I first encountered this in Math 121 but yeah, I didn’t understand this that time ‘coz Sir Earlyseven just gave a differential equation about it given by (1) where the rate of change of temperature dT/dt is proportional to the difference of its temperature and the temperature of its surroundings To and the constant is the inverse of time 1/τ where the time constant is τ.

Thermometers don’t measure temperature at instant. That is, we still have to wait for a certain time until the temperature reading is steady.

Newton’s Law of Cooling describes the cooling of a warmer object to the cooler temperature of the environment. I won’t go with the detailed derivation anymore ‘coz it’s so hassle. This equation is a separable differential equation with the solution given by:

T(t) = T+ (T– Ti) (1 – e-t/τ)                                                (2)

When t = τ, the equation transform to

T(t) = T+ (T– Ti) (1 – e-1)                                                   (3)

T(t) = T+ 0.63(T– Ti)                                                          (4)

wherein the temperature T at a certain time t is given by the initial temperature Tplus the constant 0.63 multiplied to the change in temperature (T– Ti).

This principle works for first-order temperatures, that is, the time constants τ for both heating and cooling are the same.

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The materials for this experiment are pot, hot plate, ice, tap water, beaker, and different kinds of thermometers; alcohol thermometer and mercury thermometer. (Supposedly, there should be thermistor and thermocouple but then, the universe didn’t conspire and we, Physics majors, are not the top priority.)

The experiment was divided into two parts: heating and cooling. For both setups, the pot was filled with 3/4 pot water and then the pot was put to the hot plate to boil the water. Ice, on the other hand, was placed in the beaker. Temperature measurements are made after the temperature stops increasing for boiling water and decreasing for ice.

Heating

For heating, the temperature of the boiling water was measured and recorded as Tf while the temperature of the ice on beaker was measured and recorded as Ti. Then, we calculated T(t) using equation (4) and the values measured.  The thermometer was be put again to the boiling water, then the time τ it takes for the thermometer to reach the calculated value of T(t) was measured. For our case, we used stopwatch as timer.

This procedure was done for three trials for both alcohol and mercury thermometers. Cooling

For cooling, the temperature of the ice on beaker was measured and recorded as Tf then, the temperature of the boiling water was measured and recorded as Ti. Then, we calculated T(t) using equation (4) and the values measured.  The thermometer was put again to the ice water, then the time τ it takes for the thermometer to reach the calculated value of T(t) was measured. For our case, we used stopwatch as timer.

This procedure was also done for three trials for both alcohol and mercury thermometers.   Table 1 shows the values of T(t) and τ measured using an alcohol thermometer for both cooling and heating for three trials. On the other hand, Table 2 shows the values of T(t) and τ measured using an mercury thermometer for both cooling and heating fro three trials. It could be seen for both tables that the time constant τ for heating is lower than that of cooling. Table 3 shows the average time constants for heating and cooling of alcohol thermometer and mercury thermometer both for three trials. It could be seen from here that it takes more time to cool than to heat. Also, alcohol thermometers have higher time constants than mercury thermometer. This could be because alcohol expands when it absorbs temperature and contract in colder temperatures while mercury thermometers, with temperature sensitive fluid, are best at measuring higher temperatures than the lower temperatures. 

Moreover, this table could imply that the time constants for both thermometers are more or less accurate. Going back to Newton’s Law of Cooling, the thermometers are assumed to be first-order before we calculated T(t). Based on our results, the both the alcohol and mercury thermometers are first-order thermometers and linear even if the time constants for heating and cooling are not exactly the same. This is primarily due to errors that deviated; first, we used a stopwatch as timer so, the reaction time to start and stop the timer could have affected the measurements, second, is the uncertainty of the thermometers itself.

After all, this experiment is a success. We found out that mercury thermometers measure temperature faster, making it more linear than alcohol thermometers but, alcohol thermometers are safer to use than mercury thermometers because those are toxic, like love. Kidding.

References:

 Young, Hugh and Roger Freedman. University Physics with Modern Physics 13th edition. San Francisco: Pearson education Inc., 2012.

 Experiment 1: Temperature Measurement. Physics Laboratory Manual. National Institute of Physics, UP Diliman, 2013.