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The two most commonly used temperature scales are the Celsius scale, also known as the Centigrade scale , and the Fahrenheit scale . The first is used in most of the world, especially in those countries that have adopted the international system of units. It is also, along with the Kelvin scale or absolute scale , the temperature scale used in science everywhere. The second, on the other hand, is mainly used in those countries that still use the imperial system of units.
There are fundamental differences regarding the way in which these two scales are defined; however, the differences in the sizes of degrees Fahrenheit and degrees Celsius means that there is a temperature at which both scales mark the same number.
Let’s first see how both scales are defined to make it easier to understand.
Celsius scale
The Celsius scale is a relative temperature scale in which all temperatures are set relative to the freezing point of pure water at standard pressure . This means that positive temperatures on the Celsius scale correspond to temperatures higher than the melting (or freezing) point of water, while a negative temperature corresponds to a temperature lower than said reference point.
The size of the Celsius degree or centigrade degree (ºC) is determined by the temperature difference between the freezing point and the boiling point of water, to which the value of 100 ºC was assigned. This is the reason why it is called the centigrade scale.
Fahrenheit scale
On the other hand, the Fahrenheit scale, sometimes misspelled as Fahrenheit, uses different reference points, as well as a different way of dividing the temperature intervals. There are different versions, but they all seem to agree that Fahrenheit decided to set the lowest temperature he could record as the zero of his scale, in order to avoid negative temperatures. To do this, he used a mixture of water, ice, and ammonium chloride. This mixture has the particularity that it reaches a quite stable and quite low equilibrium temperature, for which reason it was assigned zero on the scale.
Some say that the next reference temperature was that of his own body when suffering from a mild fever. Others indicate that he used the freezing point of water as a second reference temperature, dividing the scale into four groups of eight units each to give a temperature of 32ºF for the freezing point of pure water. Using this scale, water would boil at 212ºF.
At what temperature do the Celsius and Fahrenheit scales coincide?
Determining when the Fahrenheit and Celsius temperature scales have the same value can be done in two different ways:
- Through the algebraic method, using the scale conversion equation.
- Through the graphic method.
Determine the intersection of the Celsius and Fahrenheit scales with the algebraic method
The conversion equation from the Celsius scale to the Fahrenheit scale is very simple, and is presented below:
In the above equation, ºF represents the temperature in degrees Fahrenheit and ºC represents the same temperature, but on the Celsius scale.
To find the temperature at which both scales agree, all that is needed is to establish the condition that ºF = ºC = X . In other words, we make both temperatures equal and define X as the value we’re looking for, to make the equation simpler.
Now, the previous equation remains:
Just solve this equation for X (in other words, solve for X):
The solution of the equation is X= – 40 , so the temperature of – 40 ºC will be equal to – 40 ºF .
This solution was found using the Celsius to Fahrenheit conversion equation. It should be noted that, since the conversion equation from Fahrenheit to Celsius is the same equation but solved to find ºC, then that equation could have been used instead of the first one to obtain exactly the same result (since, in reality , are the same equation).
Graphical determination of the point at which the Celsius and Fahrenheit scales have the same measure
An alternative way to find the point of agreement between the two scales or their intersection is to plot both scales on the same graph and find the point at which they agree.
This can be done in two different ways. You can plot the temperature in Celsius and the temperature in Fahrenheit as a function of the absolute temperature in Kelvin, in which case the intersection would be in Kelvin, or you can plot the temperature in Fahrenheit as a function of the temperature in degrees Celsius, which is much easier.
The procedure consists of drawing a coordinate system in which the abscissas correspond to the Celsius scale, and the ordinates can represent either the temperature in Celsius or the temperature in Fahrenheit:
We now represent the Celsius scale on this graph. Simply place just two points and draw a straight line between them. Each point is located by crossing the temperature on the X axis with its equivalent temperature on the Y axis. In this case, since the Celsius scale is being plotted, all the values in X will be equal to the values in Y. A point can be ( 0;0) and another can be (100;100)
Now we do the same thing to graph the Fahrenheit scale. Remembering the aforementioned, 0 ºC (the freezing point of water) is equivalent to 32 ºF and 100 ºC (the boiling point of water) is equivalent to 212 ºF. We graph those two points and draw a line.
Once this is done, the point at which both lines cross or intersect is the temperature at which both scales have the same value. As can be seen in the figure, both lines intersect at the point (-40;-40), that is, – 40 ºC = – 40 ºF .
This same process can be carried out for any pair of temperature scales, but there is a limitation. In the case of temperature in Celsius and in Kelvin, the unit size on both scales is the same, so both graphs will be parallel and will never intersect. The same is true of the Fahrenheit scale and the Rankine scale.
References
The mathematical treatment of measurement results . (n.d.). Retrieved from https://espanol.libretexts.org/@go/page/1798
measurements . (2020, October 30). Retrieved from https://espanol.libretexts.org/@go/page/1796
The Editors of Encyclopaedia Britannica. (2018, October 25). Fahrenheit temperature scale | Definition, Formula, & Facts . Retrieved from https://www.britannica.com/science/Fahrenheit-temperature-scale
