Temperature Measurement

  One of the interesting principal branches in physics is thermodynamics, which is study of energy transformations involving heat, mechanical work, and other aspects of energy and how these transformations relate to physical properties. Thermodynamics can study microscopically that talks about the behavior of individual atoms and molecules and the other one is macroscopically properties in bulk such as volume, pressure, and etc..

      Temperature is one of the central concepts of thermodynamics. It gauges the "hardness" and "coldness" of a system. Thermometer is the device use to measure the temperature. It works by the principle of thermal equilibrium, which means further interaction between the body & the thermometer causes no further changes. In layman's term when two objects, one warm and one cold, are placed in contact with each other, the warmer object cools while the cooler object warms up. Eventually, no more changes in the warmness or coldness would occur and they would have the same temperature.

Zeroth Law of Thermodynamics
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      The concepts of thermal equilibrium was formally explained by the zeroth law of thermodynamics. Consider three systems A, B, and C. Also, a conductor permits thermal interactions between systems and an insulator prevents the thermal interactions between the two systems (this is an ideal insulator, in real world, an insulator also permits thermal interactions slowly). In the first picture, A&B is separated by an insulator(ideal), A&C and B&C are separated by conductor. Since, conductor permits the thermal interactions, in a time t A&C and B&C will be at thermal equilibrium. Therefore, A and B are also in thermal equilibrium with each other. In the second picture says that A&B will be at thermal equilibrium with each other at time t, and both A and B will not be in thermal equilibrium with C because they don't have any thermal interactions.

         Heat from the hotter object will transfer to the colder one until they both reach a common temperature that is different from their original temperatures. If a thermometer is placed in thermal contact with a hot body, what we actually read is the temperature of the thermometer itself! The act of touching the body with a thermometer changes the temperature of the body. It is therefore necessary that the transfer of heat to or from the thermal sensor is minimal such that it does not change the temperature of the object significantly. We have to wait until thermal equilibrium is reached before we can reliably read the temperature and it takes time to achieve this. How fast a thermal sensor achieves thermal equilibrium depends on the thermal time constant τ of the sensor.
          Consider a temperature sensor that has a reading T(t) at any time t. Initially, the sensor has a temperature T(t = 0) = Ti and is placed in contact with an object that is maintained at a constant temperature. After a sufficient time, the temperature sensor would have a final reading Tf. The difference between Tf and T(t) is ∆T(t) = Tf −T(t). As time progresses, the difference between the sensor reading and its final reading vanishes. If the sensor is a first-order linear device, the rate of change of ∆T can be assumed to be proportional to the difference of Ti and Tf,

d∆T dt = −k∆T. (1)

where k is a positive constant. The negative sign here implies that the rate of change of ∆t decreases in time. The left side of Equation (1) has dimensions [Temperature]/[Time]. To keep the dimensions the same as on the right side, the constant k must have dimensions of 1/[Time]. If we let k = 1/τ, where τ has units of time. Equation (1) becomes

d∆T dt= −∆T τ. (2)

Solving for T(t) we obtain,

T(t) = Ti + (Tf −Ti)(1−e^(−t/τ)).

        Temperature can be felt but cannot be measured directly. Instead we measure the degree of change in the properties of a material to heat and cold. For example, metals expand when heated. We can then measure how much the expansion is and equate it to a certain temperature. This and other measurable responses to heat are called thermometric properties.

        In this experiment we will using different temperature sensors and we will be interested with its thermal time constant, and this is dependent to the thermometric properties of the material.

       The experiment is divided in two procedure, the heating procedure and the cooling procedure. Basically, the two procedure is the same but will have different initial and final temperature.  First, we need a beaker or a pot with water up to 3/4 full and let the water boil. Keep the water boiling throughout the activity.  Take the thermometer and dip it in hot water. When the temperature stops increasing, record the final temperature as Tf in the worksheet. Take the thermometer and dip it in ice water. Wait for the temperature reading to stop decreasing then record it as Ti in the worksheet.  Compute the values T(τ) and tabulate in the worksheet. With initial temperature Ti, dip the thermometer in the boiling water then measure the time it takes for the reading to reach each T(τ). Record these values in the worksheet. Repeat the process three times each of the temperature sensor.

       The table above shows the data recorded from the experiment with three trials. The table above tells that the data is precise, the heating time constant recorded is 5.8 for heating and 6.0 for cooling in alcohol thermometer. Using alcohol thermometer, it takes six seconds before it reach thermal equilibrium. In simple words, approximately six seconds contact time is needed before we read the alcohol thermometer reading to be sure that its already the right temperature of the object we are measuring.

       The second table is the recorded data with three trials. Compare to the previous thermometer which is alcohol thermometer, mercury thermometer has approximately two seconds before it will reach thermal equilibrium. Therefore, mercury thermometer will have faster results compare to the alcohol thermometer. If we want to have a faster reading of the temperature we are testing it is advisable to use mercury thermometer than alcohol thermometer.

        The third table shows the data with three trials for thermocouple. The data shows that thermocouple only required approximately .2 seconds before it reach thermal equilibrium. If we want to measure a temperature with a fluctuating temperature it is better to use thermocouple because as fast as .2 second it will give you the reading of the object your trying to measure. Thermocouple has the least time constant because its operate with electric current which is very fast.

         The last table shows the time constant of each temperature measurement devices. The table shows that the thermocouple has least time constant which is .2 seconds, followed by the mercury thermometer with approximate 2 seconds and the last is alcohol thermometer with time constant approximately 6 seconds.

P.S. Hindi ko magraph yung exponential function using its equation, hindi ako makahanap ng app
kaya walang graph huhuhu

Reference
1.       “Experiment 7 Temperature Measurement,” Laboratory Manual for Physics (Physics 73.1), (2013).
2.       Walker, J., Halliday, D., & Resnick, R. (2011). Fundamentals of physics. Hoboken, NJ: Wiley.