Thermometry

Mercury thermometer (mercury-in-glass thermometer) for measurement of room temperature. Daniel Fahrenheit's application of mercury and a standardized temperature scale for liquid-in-glass thermometers ushered in a new era of accuracy and precision in thermometry. From the early 1710s until the introduction of electronic devices in the 1960s, mercury-in-glass thermometers were the world's most reliable and accurate thermometers, accounting for their widespread use.$^{[1]}$

The word "Thermometry" literately means measurement of temperature and various calculations based on conversion of temperatures from one scale to another. The knowledge of thermometry is also vital for daily life.

$\large \underline{\text{Measurement Of Temperature}}$

Just as we measure length of a body by a length scale we have to define temperature scale so that we can give numerical value to the temperature of a body. so a device used to measure temperature is called Thermometer. Property of material used to measure temperature is called thermometric property. To assign numerical value to temperature measured with any thermometer we have to initially calibrate the instrument. The SI unit of temperature is Kelvin $(K)$.

Calibration method :

We define two fixed points in thermometer. One is the freezing point $(t_0)$ and boiling point $(t_{100})$. The length of temperature column at freezing point is $l_0$ and at boiling point is $l_{100}$. Temperature corresponding to any length, $$\begin{aligned} t & = al + b \\ 0 = al_0 + b \quad & \quad 100 = al_{100} + b \\ t & = al_t + b \\ \dfrac{t - 0}{100 - 0} & = \dfrac{l_t - l_0}{l_100 - l_0} \\ t & = \dfrac{l_t - l_0}{l_100 - l_0} \times 100 \\ \end{aligned}$$ this means that if there is a change of one degree in temperature it will mean a change of $\dfrac{l_2 - l_1}{t_2 - t_1}$ in the length of the mercury column in mercury.

Now lets look at the freezing point of water, boiling point of water and Absolute zero temperatures of some standard temperature scales. The standard scales of temperature are Celsius, Fahrenheit, Kelvin, Rankine and Reaumer

• In both the scales Celsius and Fahrenheit we denote the value with degree $(^\circ)$ like $70^\circ C, 53 ^\circ F$ but for Kelvin scale we do not express the temperature in terms of degree and just write $K$ like $200~K$, etc.

$\large \underline{\text{Conversion Of Temperature From One Scale To Another}}$

Temperature can be converted from one scale to another scale. But the reading will be different. Suppose, if a Celsius thermometer reads $0^\circ C$ at the same temperature a Fahrenheit thermometer will read it as $32^\circ F$. In this example although the value of temperature will not change different scales will show different reading at this same temperature. At a certain temperature $T$ to convert reading one scale to another at that temperature there is only one standard formula,

$$\color{#3D99F6} \dfrac{\text{T - Lower Fixed Point on scale 1}}{\text{Upper Fixed Point on scale 1 - Lower fixed point on scale 1}} = \dfrac{\text{T - Lower Fixed Point on scale 2}}{\text{Upper Fixed Point on scale 2 - Lower Fixed Point on scale 2}}$$

Note that the lower fixed point and upper fixed points are nothing but the freezing point and boiling point of water in that scale.

Relation between Celsius Scale and Fahrenheit Scale :

For Celsius Scale : Lower Fixed Point = $0^\circ C$ and Upper Fixed Point = $100^\circ C$

For Fahrenheit Scale : Lower Fixed Point = $32^\circ F$ and Upper Fixed Point = $212^\circ F$

$$\begin{aligned} \left(\dfrac{T - 0}{100 - 0} \right)_{\text{in Celsius}} & = \left(\dfrac{T - 32}{212 - 32} \right)_{\text{in Fahrenheit}} \\ \boxed{T_C = \dfrac59 (T_F - 32)} & \\ \end{aligned}$$

Relation between Celsius Scale and Kelvin Scale :

For Celsius Scale : Lower Fixed Point = $0^\circ C$ and Upper Fixed Point = $100^\circ C$

For Kelvin Scale : Lower Fixed Point = $273.15~K$ and Upper Fixed Point = $373.15~K$

$$\begin{aligned} \left(\dfrac{T - 0}{100 - 0} \right)_{\text{in Celsius}} & = \left(\dfrac{T - 273.15}{373.15 - 273.15} \right)_{\text{in Kelvin}} \\ T_C = T_K - 273.&15 \\ \boxed{T_K = T_C + 273} & \\ \end{aligned}$$

Convert the temperature of a normal human being $(98.6^\circ F)$ into Celsius scale.

ANSWER

By using formula, $$\begin{aligned} T_C & = \dfrac59(T_F - 32) \\ & = \dfrac59(98.6 - 32) \\ & = 37^\circ C \\ \end{aligned}$$

At what temperature do the Celsius and Fahrenheit readings have same numerical value ?

ANSWER

Let $T_F = T_C = x$. By applying formula, $$\begin{aligned} T_C & = \dfrac59(T_F - 32) \\ 9x & = 5x - 160 \\ 4x & = - 160 \\ x & = -40 \\ \end{aligned}$$ So, $T_F = -40^\circ F$ and $T_C = -40^\circ C$ are the same temperatures.

A faulty thermometer reads freezing point of water at $20^\circ C$ and boiling point at $150^\circ$. What will be the thermometer reading when the actual temperature is $60^\circ C$ ?

ANSWER

Using formula, $$\begin{aligned} \left(\dfrac{60 - 0}{100 - 0} \right)_{\text{in Celsius}} & = \left(\dfrac{T - 20}{150 - 20} \right)_{\text{given scale}} \\ T & = \dfrac{130 \times 60}{100} + 20 \\ & = 78 + 20 = 98^\circ C \\ \end{aligned}$$ Thus, the faulty thermometer will show $98^\circ C$.

The lower point of a Fahrenheit scale thermometer is correct an its cross section is uniform. It reads $76.5^\circ F$ when a standard centigrade thermometer reads $25^\circ C$. What is its upper fixed point ?

ANSWER

As usual by applying formula, $$\begin{aligned} \dfrac{76.5 - 32}{T_{\text{upper}} - 32} & = \dfrac{25 - 0}{100 - 0} \\ \dfrac{76.5 - 32}{T - 32} & = \dfrac14 \\ T - 32 & = 44.5 \times 4 \\ T & = 210^\circ F \\ \end{aligned}$$

[1] : Image source - Wikipedia, Thermometer