Thermometry

Thermophysics
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Temperature
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Thermometry

Molecular Agitation and Temperature

In physics, the temperature of an object is related to the degree of agitation of its molecules (and/or atoms). In this context, the thermal equilibrium between two bodies is associated with the exchange of heat between them, which takes place until the degree of agitation of the molecules of both bodies are equal, i.e., the bodies reach equal temperatures (thermal balance $\iff$ equal temperatures). The thermophysics (or thermology) is a branch of physics that studies the heat flows and their transformations, and how the temperatures of the bodies change these processes.

Thermometry

The thermometry objects of study are: the different ways to measure the temperature of objects and the relationships between different temperature scales. Some important definitions for this study are:

Temperature: It is the quantitative measure of the physical quantity that is related to our sense of hot and cold. Microscopically, this magnitude is related to the degree of molecular agitation of the system (kinetic energy). The molecular agitation and the temperature are related as follows:

A higher degree of molecular atomic agitation $\Rightarrow$ a higher temperature;
A lower degree of molecular atomic agitation $\Rightarrow$ a lower temperature.

Macroscopically, it allows us to tell if two or more systems are in thermal equilibrium or not.

Heat exchange: It occurs when two objects (bodies) of different temperatures are put in direct contact or through a good thermal conductor.

For the study of thermal systems, it is important to consider isolated systems from external systems that can influence the results of the measurements. In the figure, there are two bodies, $A$ and $B$, separated from the environment by walls which are thermal insulators (grey walls), i.e., prevent passage of heat, also known as adiabatic walls. A good example of thermal insulator is polystyrene. In the figure, at an initial time $t_0$, there are two bodies with different temperatures, the body $A$ at a temperature $T_1$ and body $B$ at a temperature $T_2$. After some time, at $t_1$, the temperature of both bodies will be equal to $T_3$. Note that, by convention, both the time and the temperature are denoted with the letter "t", so we use the capital letter "T" to denote the temperature and "t" for time.

Thermal Equilibrium: When two or more systems are in thermal equilibrium they do have the same temperature. Systems that are not in thermal equilibrium does not show any relationship between their temperatures.
Zeroth Law of Thermodynamics: If two bodies, $A$ and $B$ are in thermal equilibrium with a third body $C$, then $A$ and $B$ are also in thermal equilibrium with each other.

The figure illustrates two bodies $A$ and $B$, which can not exchange heat directly with each other because they are separated by an adiabatic wall (gray wall). However, these two bodies can exchange heat with a third body, $C$. If we wait long enough, $A$ enters into thermal equilibrium with $C$, and $B$ also comes into thermal equilibrium with $C$, so $A$ and $B$ will also be in thermal equilibrium with each other. This phenomenon is known as Zeroth Law of Thermodynamics.

Thermometer: It is the instrument used to measure the temperature of objects. The thermometers use a scale that varies evenly with the temperature. Strictly speaking, we say that the variation is a bijection and changes monotonically with temperature.
Thermometric Function: It is the function that relates the thermometric magnitude $g$ of a thermometer to the temperature. For example, we know that the length of a column of mercury in a cylindrical vessel varies according to the temperature. In this case, the height of the column is the magnitude $g$. With this, we can find a function that relates the height of this column with the temperature, the thermometric function. Other quantities that may be used are: the pressure of a gas, the color of a material, the electrical resistance of a material etc. In general, the termometric function is a linear function of the form: $$T(g) = ag + b,$$ where $T$ is the system temperature, $g$ is the observed quantity (height of a column of mercury, for example) and the other parameters are constants.
Absolute Scale: The temperature measured in the International System of Units ($IS$) is the Kelvin scale. The absolute temperature is the average kinetic energy of the molecules of a body.
Temperature Scales: Typically, the temperature scales are made from the choice of an arbitrary value in the ice melting point and another in the boiling water point. Below we list several temperature scales, and in the $IS$ Kelvin is adopted.

**Comparison between different temperature scales.** For typical scales, the reference temperatures are: temperature of boiling water, the lower dotted line of the figure; ice melting temperature, i.e., temperature where ice turns to water, the dotted line in the middle of the figure; and the temperature of absolute zero, the upper dotted line of the figure. Different scales associate different values to these different points.

The conversion formula between these various scales is: $$ \frac{T_C}{5} = \frac{T_F - 32}{9} = \frac{T_K-273}{5},$$ where $T_C$, $T_F$ and $T_K$ are the temperatures in Celsius, Fahrenheit and Kelvin, respectively. Note that the relation between the temperature changes is $$\Delta T_C = \Delta T_K = \frac{5}{9} \Delta T_F.$$

Some Temperatures	Kelvin $(K)$
Fusion of helium nucleus	$10^8$
Inside the Sun	$10^7$
Sun's surface	$6000$
Gold fusion	$1340$
Boiling water at 1 atm	$373$
Ambient highest recorded on Earth's surface	$331$
Human body	$310$
Freezing water at 1 atm	$273$
Lowest ambient temperature recorded at the earth's surface	$185$
Liquid helium	$4.2$
Universe background radiation	$3$

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Some Temperatures	Kelvin \((K)\)
Fusion of helium nucleus	\(10^8\)
Inside the Sun	\(10^7\)
Sun's surface	\(6000\)
Gold fusion	\(1340\)
Boiling water at 1 atm	\(373\)
Ambient highest recorded on Earth's surface	\(331\)
Human body	\(310\)
Freezing water at 1 atm	\(273\)
Lowest ambient temperature recorded at the earth's surface	\(185\)
Liquid helium	\(4.2\)
Universe background radiation	\(3\)

Thermometry

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