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Figure 1: The length of the metal in this rail way expanded due
to the heat, causing the supposed to be straight path to bend and
curve. (Source: http://www.mylargescale.com)
|
However, there are materials out there which can be used to measure temperature more accurately, and further, be calibrated so as for us to have some basis. Some metals expand and contract in different temperatures and we can relate those to the change of its length, and volume. Some liquids' volume changes depending on its temperature, and we can measure the change of volume. Even gas can be used - when they are in a container, the pressure changes when the gas particles are exposed to different temperatures. There are a lot of materials that have the potential to be used to measure temperature.
The classic apparatus used to measure temperature is known as the thermometer. Thermometers come in different types. There are glass thermometers, infrared thermometers, digital thermometers, probe thermometers, etc. However we will only focus on glass thermometers for now.
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| Figure 2: An illustration of a glass thermometer. (Source: http://labsuppliesusa.com) |
Glass thermometers are such which consists of a bulb containing some liquid, which are usually mercury or liquid containing some organic compound attached to a narrow glass tube. Doctors and nurses use it to measure the temperature of their patient's body. Others use it to measure the temperatures of liquids and certain gases. Glass thermometers are commonly used as it requires little to no maintenance and because it is easy to read the temperature off of it.
When we were sick and the doctor takes our temperature, we notice that we usually have to wait for some time before the doctor takes out the thermometer, and reads our actual temperature. Other times as well, when we are measuring the temperature of water, we see the column of liquid rise and fall before it stops and only then will we be able to get the temperature reading.
How do we know that at the temperature it stops expanding or contracting, is the actual temperature we are seeking? Why do we have to wait before actually reading the measured temperature?
To measure the temperature of some object, one must place the thermometer in contact with the object. When this happens the thermometer and that object interact. If the thermometer is initially hotter than the object, it cools and the object in return becomes a bit hotter, and vice versa. The moment the reading in the thermometer does not change, the thermometer itself and the object are said to be in thermal equilibrium. That is, their interactions cause no further change in the state of the system.
Now, the time it takes for thermal equilibrium to occur depends on the properties of the corresponding materials. Thermal conductors permit thermal interaction between object hence thermal equilibrium occurs faster between conductors. Insulators, on the other hand, inhibit thermal interaction. Metals are common every day thermal conductors while glass, plastic, rubber, wood are examples of insulators.
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| Figure 3: Set-Up Discussed (Source: University Physics by Young and Freedman) |
Suppose we have three systems, System A, B, and C, which are initially not in thermal equilibrium. We surround the whole thing with a thick insulating material so as these systems will interact with themselves alone. Now, we separate System A and B using an insulator however we let both interact with System C through a conductor (as shown in Figure 3). After waiting for some time so that the whole system attains thermal equilibrium, we let System A and B interact with each other by replacing the insulator in Figure 3 with a conductor. Furthermore, we do not let any of these systems interact with System C by replacing the conductors with insulators.
When we now try to measure the interaction between System A and B by measuring their temperatures, we find that the temperature of System A is the same as that of System B. This result is more commonly known as the Zeroth Law of Thermodynamics.
Explicitly, the Zeroth Law of Thermodynamics states that:
"If two bodies are in thermal equilibrium with a third body, then the two bodies are in thermal equilibrium with each other."
Further, since we said that at thermal equilibrium, the temperature of System A is the same as that of System B, therefore, we can say that two objects are be in thermal equilibrium if and only if they have the same temperature.
This means that the thermometer is actually measuring its own temperature. So when the thermometer is in thermal contact with another object, then the temperature of the object must be the same as the temperature of the thermometer!
The time it takes before we can accurately determine the temperature of an object is dependent on the time constant $\tau$ of that thermometer. Usually, it is preferred to wait around 3 to 5 time constants before reading the temperature on the thermometer. However, how does one get this time constant?
Consider a thermometer about to measure the temperature of an object. Initially it reads a constant temperature $T_i$. After placing the thermometer in thermal contact with that object, the temperature $T(t)$ changes after some time $t_0$, and after a long time achieves the final temperature $T_f$. It can be said that the difference between the final and current temperature reading on the thermometer $\Delta T(t_0)=T_f-T(t_0)$ vanishes as time increases. Now, assuming that the thermometer is a first-order linear device, then the rate of change of $\Delta T(t)$ is proportional to $\Delta T=T_f-T_i$; explicitly:
\begin{equation*}
\dfrac{\mathrm{d}\Delta T}{\mathrm{d} t}=-k\Delta T
\end{equation*}
\begin{equation*}
\dfrac{\mathrm{d}\Delta T}{\mathrm{d} t}=-k\Delta T
\end{equation*}
where $k$ is the proportionality constant. Now, we let $k=\dfrac{1}{\tau}$ where $\tau$ has units of time (to be consistent with units on the left hand side of the equation). Solving this differential equation, one gets that the temperature reading on the thermometer as a function of time can be expressed as
\begin{equation}
\label{mainequation} T(t)=T_i+(T_f-T_i)(1-\mathrm{e}^{-t/\tau})
\end{equation}
Equation \eqref{mainequation} agrees with our initial assumption. When $t=0$, $T(t)$ evaluates to $T_i$, and when $t\to\infty$, then $\mathrm{e}^{-t/\tau}\to0$ and $T(t)$ evaluates to $T_f$. Hence this equation relates the temperature reading on the thermometer as it approaches the temperature of the object of interest.
In this experiment on Temperature Measurement, we obtained the time constants for heating and cooling for a linear thermometer, the alcohol thermometer.
Initially, we filled a beaker with water and bring the water to a boil. Also, we put ice in a Styrofoam cup, and added a small amount of water.
To get the time constant for heating, we got the temperature reading on the thermometer while it was immersed in the boiling water $T_f$. After which we immersed the same thermometer in the cup filled with ice and got the stable temperature reading $T_i$. Then, using Equation \eqref{mainequation} we calculated for the corresponding temperatures for one through five time constants. We did this calculation while the thermometer was immersed in the ice so as to keep the initial temperature constant. Afterwards, we submerged the thermometer in the boiling water and recorded the times at which the temperatures for the $n$-th time constant ($n$ ranged from one to five) occurred. This was done three times.
To get the time constant for cooling, we got the temperature reading on the thermometer while it was immersed in the ice $T_f$. Then we got its temperature while it was submerged in the boiling water $T_i$. Again, the corresponding temperatures for one through five time constants was calculated while the thermometer was immersed in the boiling water. Then the thermometer was dipped in ice and the time at which the temperature reading was that corresponding to the $n$-th time constant ($n$ ranged from one to five) occurred. Like in the first part, this was done three times.
Then for all the data gathered, the temperature was plotted as a function of time, and a fit of the form
\begin{equation}
\label{fit} T(t)=A(1-\mathrm{e}^{-t/B})+C
\end{equation}
was taken. This was done in correspondence to Equation \eqref{mainequation}. As in Equation \eqref{mainequation}, the time constant is taken to be the value of the constant $B$ in Equation \eqref{fit}. Then the average time constant was taken each for the heating and cooling.
After analysis, after plotting the temperature reading on the thermometer as a function of time for heating, one gets:
\begin{equation}
\label{mainequation} T(t)=T_i+(T_f-T_i)(1-\mathrm{e}^{-t/\tau})
\end{equation}
Equation \eqref{mainequation} agrees with our initial assumption. When $t=0$, $T(t)$ evaluates to $T_i$, and when $t\to\infty$, then $\mathrm{e}^{-t/\tau}\to0$ and $T(t)$ evaluates to $T_f$. Hence this equation relates the temperature reading on the thermometer as it approaches the temperature of the object of interest.
In this experiment on Temperature Measurement, we obtained the time constants for heating and cooling for a linear thermometer, the alcohol thermometer.
Initially, we filled a beaker with water and bring the water to a boil. Also, we put ice in a Styrofoam cup, and added a small amount of water.
 |
| Figure 4: Thermometers immersed in ice water. This was taken while the time constant for cooling was being obtained. |
To get the time constant for cooling, we got the temperature reading on the thermometer while it was immersed in the ice $T_f$. Then we got its temperature while it was submerged in the boiling water $T_i$. Again, the corresponding temperatures for one through five time constants was calculated while the thermometer was immersed in the boiling water. Then the thermometer was dipped in ice and the time at which the temperature reading was that corresponding to the $n$-th time constant ($n$ ranged from one to five) occurred. Like in the first part, this was done three times.
Then for all the data gathered, the temperature was plotted as a function of time, and a fit of the form
\begin{equation}
\label{fit} T(t)=A(1-\mathrm{e}^{-t/B})+C
\end{equation}
was taken. This was done in correspondence to Equation \eqref{mainequation}. As in Equation \eqref{mainequation}, the time constant is taken to be the value of the constant $B$ in Equation \eqref{fit}. Then the average time constant was taken each for the heating and cooling.
After analysis, after plotting the temperature reading on the thermometer as a function of time for heating, one gets:
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| Figure 5: The corresponding plot to get the time constants for heating In Trial 1, the fit has the equation $T_1(t)=105.946(1-\exp(-t/4.71683))-11.338$. In Trial 2, $T_2(t)=107.703(1-\exp(-t/5.00108))-12.7498$. Lastly, in Trial 3, $T_3(t)=100.956(1-\exp(-t/4.90984))-6.81594$ |
Further, after plotting the temperature reading on the thermometer as a function of time for cooling, one gets:
 |
| Figure 6: The corresponding plot to get the time constants for cooling In Trial 1, the fit has the equation $T_1(t)=-79.1353(1-\exp(-t/10.2651))+70.6856$. In Trial 2, $T_2(t)=-82.4988(1-\exp(-t/8.26271))+83.1465$. Lastly, in Trial 3, $T_3(t)=-78.4758(1-\exp(-t/9.47937))+80.2633$. From the values of $B$ for all trials, it is observed that they are not equal to those for heating. |
Tabulating the time constants for heating and cooling with the corresponding averages, one gets:
| Heating | Cooling | |
|---|---|---|
| Trial 1 | $4.71683$ s | $10.26510$ s |
| Trial 2 | $5.00108$ s | $8.26271$ s |
| Trial 3 | $4.90984$ s | $9.47937$ s |
| Average | $4.87592$ s | $9.33573$ s |
From Table (1) it is observed that the time constant for heating is $4.87592$ s, and that for cooling is $9.33573$ s. It was observed before that Equation \eqref{mainequation} works if the rate of change of $\Delta T(t)$ is proportional to $\Delta T$. This implies that the time constant $\tau$ for heating and cooling must be the same. Earlier, it was established that alcohol thermometers were linear thermometers, hence the time constant for heating should be the same as that for cooling. However, from Table (2) the time constant for cooling is $9.33573$ s, approximately twice that for heating.
There can be many reasons as to the discrepancy. One of these is due to the state of the boiling water and the ice in the Styrofoam cup. During the experiment, many thermometers were immersed in the boiling water and in the cup, which means that there are many objects absorbing the heat from the boiling water, and releasing heat into the ice. With this, the actual temperature of the boiling water and ice is constantly changing. With that, the final and initial temperatures read by the thermometer immediately changes. This increases the room for error in terms of temperature measured.
Another reason to the discrepancy is the reaction time for the person looking at the thermometer as the temperature reading changes and the person timing. Since it takes time for the eye to see the reading on the thermometer and also to speak, time measured by the person timing immediately changes. Add the reaction time of the person timing and this discrepancy increases further. Hence another human error can be part of the reason why there is such a big gap between the time constants for heating and cooling.
Overall, this experiment was still fun. It was just annoying that the whole class had to share the same beaker as thermometers do absorb heat from the water hence changing its temperature. If there were more materials, I guess we would have better results, and further, more precise data.
There can be many reasons as to the discrepancy. One of these is due to the state of the boiling water and the ice in the Styrofoam cup. During the experiment, many thermometers were immersed in the boiling water and in the cup, which means that there are many objects absorbing the heat from the boiling water, and releasing heat into the ice. With this, the actual temperature of the boiling water and ice is constantly changing. With that, the final and initial temperatures read by the thermometer immediately changes. This increases the room for error in terms of temperature measured.
Another reason to the discrepancy is the reaction time for the person looking at the thermometer as the temperature reading changes and the person timing. Since it takes time for the eye to see the reading on the thermometer and also to speak, time measured by the person timing immediately changes. Add the reaction time of the person timing and this discrepancy increases further. Hence another human error can be part of the reason why there is such a big gap between the time constants for heating and cooling.
Overall, this experiment was still fun. It was just annoying that the whole class had to share the same beaker as thermometers do absorb heat from the water hence changing its temperature. If there were more materials, I guess we would have better results, and further, more precise data.
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