# Measuring Temperature

#### Prerequisites

The critical new concept in our study of heat and cold is temperature. We now know that this is a measure of the average motion (kinetic energy) of an object's molecules. There is a temperature, absolute 0 on the Kelvin scale, at which molecular motion stops. However, above this temperature, molecules are always moving. The hotter an object is, the greater the molecular motion.

Temperature is critical in our conception of hot and cold because it determines the way the internal random energy will naturally flow — from hot to cold, that is, from the object with a higher temperature to the object with the lower temperature. This is a density-like construct, not a quantity-like one. If you put a tablespoon of nearly boiling water into a cool lake, energy will flow from the tablespoon of water into the water of the lake, even though there is far more thermal energy in the lake than in the tablespoon of hot water.

Although we know what thermal energy means in terms of molecular motion, because the description of thermal energy and temperature was developed phenomenologically (from observation) before the molecular structure of matter was well understood, independent measures of temperature were developed.

## The dimensionality of temperature

Although we have an internal sense of how hot and cold things are (though, as we shall see, that's not really what our senses measure), if we want to describe the phenomenon of hot and cold quantitatively, we have to find a way to measure how hot or cold something is. As is often the case when we are starting to define a concept quantitatively, we begin by creating an operational definition -- a process by which we assign a number to the phenomenon. Examples of this are measuring a length or a distance by counting how many standard measuring sticks can be laid end-to-end to match the spatial extent we are trying to measure, using the stretch (a length measurement) of a standard spring to define how to measure force, and using the acceleration of an object feeling a standard force compared to that of a standard mass in order to infer the object's mass.

When we introduce an operational definition for a new quantity, we have a new measuring device and therefor a new dimension to include in our dimensional analysis of symbols and equations. We will use the symbol $\Theta$ (capital theta) to represent a symbol measured using this process. Thus, if $T$ is a temperature, we would write

$$[T] = \Theta$$

What we did to measure force was we built on our everyday qualitative understanding of what force does and we found an object that responded to a force with a change that we could measure in terms of things we already knew how to measure -- a spring changes its length when pulled by a force from opposite sides, stretching more if more force is applied. With a spring, we converted a length measurement into a force measurement by adopting a standard (spring).

For hot and cold, we can begin to generate a way to define how hot or cold something is by making two observations:

1. When two objects are placed in close contact, they "share" their hotness or coldness and come to the same degree of warmth.
2. Some objects expand when they get warmer.

Although the first assumption doesn't appear to always be true (try touching the metal and plastic parts of an object made of both) we'll see that it's not only a good starting point, it turns out to be truer than we expect from our sensory experience -- and that our sensory experience can be misleading!

## Devices to measure temperature

So to measure how hot or cold something is, we might put a liquid like colored alcohol or mercury in a thin transparent glass tube. We could then observe that as the measuring device was put in increasingly warm liquids, it expanded up the tube. We then calibrate our thermometer by defining two temperatures that anyone can create -- such as melting ice and boiling water -- and defining them as our standards: 0°C and 100°C. We can then divide the length difference up into equal parts and get a result for any temperature in between.

While this is a good starting point, it raises some questions. If we start with different liquids -- say colored alcohol and mercury (both commonly used) -- even if we define the same 0 and 100, does that mean that we will get the same result for temperatures in between? Mightn't the alcohol say, go up faster for the first numbers and slower for the later ones than mercury? This just has to be tested. And eventually, we will get a theoretical understanding of temperature as the average kinetic energy of the molecules of a substance and be able to refine our measuring devices.

Today, there are many more ways to measure temperature as we find more measurable phenomena that change a property as their temperature changes.  For example, the voltage across a thermocouple made by joining two dissimilar metals (in a digital thermometer) is one way.  Another is to observe that every object emits electromagnetic radiation (mostly infrared light) and the the amount of the different colors of light given off is a measure of their temperature. A conceptually simpler (though not easier to implement) measure is the  pressure of a dilute gas at a constant volume using the Ideal Gas Law.

## Temperature Scales

There are a variety of temperature scales in common and scientific use.

• The Fahrenheit Scale: This is mostly used in the USA. Originally, 0 oF was defined as the coldest temperature that could be created in the lab using ice and salt mixture and 100 oF was defined as the temperature of a human body (but he got it a bit wrong).
• The Celsius Scale: This is mostly used in all other countries in the world and in science. 0 oC is defined as the temperature at which ice melts and 100 oC is defined as the temperature at which water boils (with various pressure and purity conditions). It is related to the Fahrenheit scale through the equation $$F = {9 \over 5}C + 32$$
• The Kelvin Scale: This is a more scale based on the observation that if the pressure vs temperature graph of an dilute (nearly ideal) gas is extrapolated, the pressure goes to 0 at -273.15 oC. With this temperature scale, the ideal gas law simplifies -- as do many other scientific equations. It is related to the Celsius scale through the equation $$C = K + 273$$ The size of a Celsius and a Kelvin degree are the same.

Since we are interesting in biological organisms, an amusing meme that has been running round the internet reminds us that the Fahrenheit scale was constructed to match human experience while the others were not.

Joe Redish, Karen Carleton, and Julia Svoboda 11/20/11 and 3/6/19

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