### Temperature is a measure of heat

Temperature is a measure of the heat content of a substance or system. It is an imperfect measure of the total heat content of a system because it doesn't directly measure all of the rotational or vibrational energy of molecules, but those can be calculated fairly accurately from the temperature.

We exploit the property of thermal expansion of materials to make devices to measure heat as temperature.

The figures below illustrate what happens to a gas of atoms as the heat is added to it. In the figure below, gas atoms are enclosed in a container with a movable piston. The piston is weighted with a mass, which causes a certain pressure inside. Pressure is the avarage force of the impacts of the atoms on the piston (a function of their speed and the frequency with which they strike the piston). This setup insures a constnat pressure because the mass always balances the internal pounding of of the piston by the atoms. The atoms are, relatively-speaking, closely packed at low temperature.

As more heat is added (next figure), the average speed of the atoms increases, which makes collisions more energetic, resulting in two phenomena.

First, the collisions with the movable piston are more forceful, so the piston moves up until the frequency of atom-piston collisions reduces enough to re-balance the pressure: fewer more forceful collisions = more less-forceful collisions. The second is that the average spacing between the atoms increases.

Most materials expand over most temperature ranges as their internal heat energy is increased. Some materials expand more rapidly than others, a property that is tabulated as the coefficient of thermal expansion.

Mercury (Hg) has a large thermal expansion coefficient and was once widely used in thermometers (see below) until it was understood that toxic mercury from broken thermometers could be a significant health hazard, especially to children. Now liquid-based thermometers use colored alcohols.

### Thermometers

Bulb thermometers (glass thermometers) exploit the property of thermal expansion of materials to put a number on the amount of internal heat of a substance.

The illustration shows a thermometer at a relatively low temperature (left). When the bulb is exposed to a material with more internal energy (heat), the atoms/molecules of the material collide with the glass of the bulb, which transfers the collision energy to the molecules of the thermometer liquid, which absorb that energy and expand.

All that is left is to calibrate the stem of the thermometer with degree markings. That is done thermometer-by-thermometer for higher precision instruments in order to compensate for differences in forming the narrow tube and bulb.

Another kind of thermometer, the bimetallic device, exploits the difference in thermal expansion properties of two fused metals.

There are other kinds of thermometers, as well. See for example, thermocouple, thermistor, liquid-crystal thermometer and others.

### Temperature scales

#### Kelvin

The Kelvin scale is the most modern of the temperature scales, and the one recommended for scientific reporting. Zero Kelvin (we don't say "degrees Kelvin" with this scale — a Kelvin is a unit) is the temperature at which all atomic and molecular motion stops. It is not possible to reach this temperature, so far as we know, as it would violate an important principle of quantum mechanics. It makes sense that zero should be absolute zero. There is no lower temperature and a substance at 0K contains no heat energy whatsoever. Temperatures of below 1 nanoKelvin (10-9 K) have been achieved in the lab, however.

#### Celsius

Also (but less frequently) called "centigrade", the Celsius (C) scale is a base-ten scale calibrated to the melting and boiling temperatures of pure water at sea level. Zero Celsius is the temperature of melting ice; a convenient calibration of a Celsius thermometer is to place it in water containing ice. By definition, that system is at 0˚C (ice alone can

be much colder than 0˚C). The boiling temperature of water is 100˚C.

The size of the Celsius degree is the same as the Kelvin degree, and 0˚C is 273K, so conversions between degrees-Celsius and Kelvin are very simple, just add 273 to the Celsius temperature or subtract 273 from the Kelvin temperature.

#### Fahrenheit

The Fahrenheit (F) temperature scale is a relic that many nations have abandoned, but there's a lot of inertia for keeping it if it's what you grew up with. The melting temperature of ice is 32˚F and the boiling point of water is 212˚F, both inconvenient when compared with the Celsius scale. Additionally, the size of the Fahrenheit degree is 5/9 the size of the Celsius degree or Kelvin. Because of this difference, the Celsius and Fahrenheit scales are the same at one temperature: -40˚F = -40˚C

#### Terminology

Fahrenheit and Celcius are names of degrees, therefore we refer to temperatures of 27 "degrees Fahrenheit" and 37 "degrees Celsius."

The Kelvin is a unit of temperature, so we refer to 77K as "77 Kelvin," and we don't say the word "degrees."

#### Celsius — Fahrenheit conversion

Conversions between Fahrenheit and Celcius temperatures are a little cumbersome because the size of the degree is different in the two systems. The Fahrenheit degree is about 5/9 the size of the Celcius degree. The conversions are:

#### Some benchmark temperatures on 3 scales

Here are some benchmark temperatures so you can compare the temperature scales.

Phenomenon ˚F ˚C Kelvin
All atomic motion stops -459 -273 0
Liquid helium evaporates -452 -269 4
Liquid nitrogen evaporates -321 -196 77
Sublimation of dry ice -109 -78 195
Water freezes 32 0 273
Water boils 212 100 373
Surface of the Sun 9940 5504 5778

#### Temperature differences, ΔT

Many functions in thermodynamics depend on differences between a final and an initial temperature: ΔT = Tf - Ti .

Because the size of the Celcius degree and the Kelvin is the same, when we're calculating ΔT, it doesn't matter which unit we use — just don't use Fahrenheit, and be aware of when you need to use the absolute temperature as opposed to a temperature difference.

### Interesting fact

The Arctic Fox (Vulpes lagopus) is at thermal equilibrium with its environmental temperatures as low as -30˚C.

That means the Arctic Fox doesn’t need to increase its metabolism to generate more internal heat until the temperature drops below -30˚C. And when the temperature is above -30˚C, the fox starts panting a bit to cool itself.

Humans are in thermal equilibrium at about 72˚F (22˚C)

### Practice problems

 1 Convert 37˚C to both Fahrenheit degrees and Kelvin. Solution To convert ˚C to K, add 273. Remember that the size of the Celsius degree and the Kelvin are the same. $$37 + 273 = \text{310 K}$$ To convert to ˚F, $$37 \left( \frac{9}{5} \right) + 32 = 98.6˚ \, F$$ 2 Convert 80 K to both Celsius and Fahrenheit degrees. Solution To convert K to ˚C, subtract 273. Remember that the size of the Celsius degree and the Kelvin are the same. $$80 \, K - 273 \, K = \text{-193˚C}$$ That's about the temperature of liquid nitrogen. To convert ˚C to ˚F, $$-193˚C \left(\frac{9}{5}\right) + 32 = -315˚F$$ 3 Calculate the difference between 97˚F and 40˚F in both Celcius degrees and Kelvin. Solution The difference is $$97˚F - 40˚F = \text{57 F degrees}$$ Now convert that to the size of Celsius degree: $$57 \left(\frac{5}{9} \right) = \text{31.7 C degrees}$$ There are fewer Celsius degrees between these two temperatures than Fahrenheit degrees. The F degree is smaller than the C degree. The Celsius degree and the Kelvin (it's not "Kelvin degrees") are the same size.
 4 Convert -40˚F to Celsius degrees. Solution $$\left( -48˚F - 32 \right) \frac{5}{9} = -44.4˚C$$ Note: -40˚F = -40˚C, the one place where the two scales are equal. 5 The boiling temperature of liquid oxygen (O2) is -297.3˚F. Convert this temperature to Celsius degrees and Kelvin. Solution $$\left( -297.3˚F - 32 \right) \frac{5}{9} = \text{-183˚C}$$ $$-183˚C + 273.15 = \text{90 K}$$

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