THERMOMETRIC SCALES: FORMULAS, EXAMPLES, SOLVED EXERCISES - PHYSICAL - 2024

The thermometric scales are those used in the measurement of temperature, magnitude scale used to quantify the thermal energy of a system. The device used to measure temperature, that is, a thermometer, must incorporate a scale to be able to take the reading.

To construct an appropriate scale, you have to take two reference points and divide the interval between them. These divisions are called degrees. In this way, the temperature of the object to be measured, which can be the temperature of the coffee, the bath or the body temperature, is compared with the reference marked on the instrument.

Figure 1. Thermometer graduated in Celsius scale. Source: Pixabay.

The most commonly used temperature scales are the Celsius, Fahrenheit, Kelvin and Rankine scales. All are equally suitable for measuring temperature, since the points selected as reference points are arbitrary.

In both the Celsius scale and the Fahrenheit scale, the zero of the scale does not indicate the absence of temperature. For this reason they are relative scales. On the other hand, for the Kelvin scale and the Rankine scale, 0 represents the cessation of molecular activity, hence they are considered absolute scales.

Celsius scale

This scale was invented by the 18th century Swedish astronomer Anders C. Celsius (1701–1744), around 1735. Very intuitive, this scale uses the freezing point and the boiling point of water at normal atmospheric pressure (1 atm) as reference points.

Water is a universal substance very suitable for this, and its values ​​are easy to obtain in the laboratory.

On the Celsius scale, the freezing point of water is that which corresponds to 0 ° C and the boiling point to 100 ° C, although originally Celsius had proposed them the other way around and later the order was reversed. Between these two reference values ​​there are 100 identical divisions, which is why it is sometimes known as the centigrade scale.

Equivalences

To establish an equivalence between degrees Celsius and other temperature scales, two aspects must be taken into account:

-The relationship between the Celsius scale and the other scale is linear, therefore it is of the form:

y = mx + b

-You have to know the reference points of both scales.

Example: equivalence between the Celsius and Fahrenheit scales

Let T _{ºC be} the temperature on the Celsius scale and T _ºF the temperature on the Fahrenheit scale, therefore:

It is known that 0ºC = 32ºF and 100ºC = 212ºF. We substitute these values ​​in the previous equation and we obtain:

This is a system of two linear equations with two unknowns, which can be solved by any of the known methods. For example, by reduction:

________________

Knowing m, we obtain b by substitution:

Now we plug the values ​​of m and b into our equivalence equation to obtain:

T _{° C} = (5/9). T _ºF - (160/9) = (5T _ºF -160) / 9

Equivalently: T _{° C} = (5/9). (T _ºF - 32)

This equation allows passing degrees Fahrenheit to degrees Celsius directly, just by typing the value where T _ºF appears.

Example: equivalence between the Celsius and Kelvin scales

Many experiments have been carried out to try to measure the absolute zero of temperature, that is, the value for which all molecular activity disappears in a gas. This temperature is close to -273 ºC.

Let T _{K be} the temperature in kelvin –the word “degree” is not used for this scale-, the equivalence is:

That is, the scales differ in that the Kelvin scale does not have negative values. In the Celsius - Fahrenheit relationship, the slope of the line is equal to 5/9 and in this case it is equal to 1.

Kelvins and degrees Celsius are the same size, only that the Kelvin scale, as can be seen from the foregoing, does not include negative temperature values.

Fahrenheit scale

Daniel Fahrenheit (1686–1736) was a Polish-born physicist of German origin. Around 1715, Fahrenheit made a thermometer with a scale based on two arbitrarily chosen reference points. Since then it is widely used in English speaking countries.

Originally Fahrenheit chose the temperature of a mixture of ice and salt for the lower set point and set it as 0 °. For the other point, she selected the human body temperature and set it at 100 degrees.

Unsurprisingly, he had some trouble determining what the "normal" body temperature is, because it changes throughout the day, or from one day to the next, without the person necessarily being ill.

It turns out that there are totally healthy people with a body temperature of 99.1ºF, while for others it is normal to have 98.6ºF. The latter is the average value for the general population.

So the Fahrenheit scale benchmarks had to change for the freezing point of water, which was set at 32ºF and the boiling point at 212ºF. Finally, the scale was divided into 180 equal intervals.

Convert degrees Fahrenheit to degrees Celsius

From the equation shown above, it follows that:

In the same way, we can consider it like this: the Celsius scale has 100 degrees, while the Fahrenheit scale has 180 degrees. So, for each increase or decrease of 1 ºC, there is an increase or decrease of 1.8 ºF = (9/5) ºF

Example

Using the previous equations, find a formula that allows you to go from Fahrenheit to Kelvin scale:

Knowing that: T _ºC = T _K - 273 and substituting in the equation already deduced, we have:

T _ºC = T _K - 273

Therefore: T _ºF = (9/5) (T _K - 273) + 32 = (9/5) T _K - 459.4

Kelvin scale

William Thomson (1824–1907), Lord Kelvin, proposed a scale without arbitrary reference points. This is the absolute temperature scale that bears his name, proposed in 1892. It does not have negative temperature values, since absolute 0 is the lowest possible temperature.

At the temperature of 0 K any movement of the molecules has completely ceased. This is the International System (SI) scale, although the Celsius scale is also considered an accessory unit. Remember that the Kelvin scale does not use "degrees", so any temperature is expressed as the numerical value plus the unit, called "kelvin".

So far it has not been possible to reach absolute zero, but scientists have gotten pretty close.

Indeed, in laboratories specialized in low temperatures, they have managed to cool sodium samples to 700 nanokelvin or 700 x 1010 ^-9 kelvin. On the other hand, towards the other end of the scale, it is known that a nuclear explosion can generate temperatures of 100 or more million kelvin.

Each kelvin corresponds to 1 / 273.16 parts of the temperature of the triple point of water. At this temperature the three phases of water are in equilibrium.

Kelvin scale and Celsius and Fahrenheit scales

The relationship between the Kelvin and Celsius scales is -rounding 273.16 to 273-:

Similarly, by substitution, a relationship between the Kelvin and Fahrenheit scales is obtained:

Rankine scale

The Rankine scale was proposed by William Rankine, a Scottish-born engineer (1820-1872). A pioneer of the Industrial Revolution, he made great contributions to thermodynamics. In 1859 he proposed an absolute temperature scale, setting zero at −459.67 ° F.

On this scale the size of the degrees is the same as on the Fahrenheit scale. The Rankine scale is denoted as R and as with the Kelvin scale, its values ​​are not called degrees, but rather rankine.

Thus:

0 K = 0 R = −459.67 ° F = - 273.15 ºC

In summary, here are the conversions necessary to go to the Rankine scale from any of the ones already described:

Figure 2. Rankine temperature scale conversions. Source: F. Zapata.

Réaumur scale

Another previously used temperature scale is the Réaumur scale, which is denoted as degrees or ºR. It is currently in disuse, although it was widely used in Europe until it was displaced by the Celsius scale.

It was created by René-Antoine Ferchault de Réaumur (1683-1757) around 1731. Its references are: 0 ° R for the freezing point of water and 80 ° R for the boiling point.

As can be seen, it coincides with the Celsius scale at zero, but certainly not at the other values. It is related to the centigrade scale by:

And since the temperatures must coincide, then T _ºC = T _ºF = x, it follows that:

When T _ºC = -40 ºC, also T _ºF = -40 ºF

Exercise 2

The steam that comes out of a boiler is at a temperature of 610 ºR. Find the temperature in degrees Fahrenheit and in degrees Celsius.

Solution

The equivalences found in the section of the Réaumur scale are used, therefore: T _ºC = (5/4) T _ºR = (5/4). 610 ° C = 762.5 ° C.

You can then convert this found value to degrees Fahrenheit, or use another of the conversions mentioned:

Or this other one, which gives the same result: T _ºR = (4/9) (T _ºF - 32)

It is solved: T _ºF = (9/4) T _ºR + 32 = (9/4) 610 + 32 ºF = 1404.5 ºF.

Conversions Summary

In summary, the following table provides the conversions for all the scales described:

Figure 3. Table of conversions for the temperature scales. Source: F. Zapata.

References

1. Temperature Scales. Recovered from: thales.cica.es.
2. Knight, R. 2017. Physics for Scientists and Engineering: a Strategy Approach. Pearson.
3. Tillery, B. 2012. Physical Science. McGraw Hill.
4. Wikipedia. Degree Celsius. Recovered from: es.wikipedia.org
5. Wikipedia. Fahrenheit degree. Recovered from: es.wikipedia.org.
6. Wikipedia. Rankine. Recovered from: es.wikipedia.org.