Thermometers and the Celsius scale

Question

Why is the Celsius scale not useful for making extremely accurate temperature measurements?

We know that $\delta L/L=\alpha \delta T$. By setting $\delta T$ to $1°$C we can find how much mercury or alcohol increase in height when the temperature increases by $1°$C, and the thermometer can be calibrated accordingly.

But Halliday and Resnick say we find a slightly different temperature value. The discrepancies are especially large when the temperatures measured are far from the calibration points.

Is the above error due to uncertainty in the alpha value?

Please explain to me the meaning of the above sentence regarding the Celsius scale, and what he means by calibration points.

Answer 1

There is nothing wrong with the Celsius scale. It's equivalent to the Kelvin scale, but with a different zero point. That is, a change in temperature of 1 Celsius degree is exactly equal to a change of 1 Kelvin. The Kelvin scale is absolute, so its zero point is absolute zero. The Celsius scale uses a point very close to the freezing point of pure water for its zero point, 0°C = 273.15 Kelvin. (The exact definition of Celsius involves the triple point of water, which occurs at 273.16 Kelvin).

The problem discussed in Halliday and Resnick is that any thermometer based on thermal expansion is not perfect. The equation of linear expansion,

$$\delta L/L=\alpha \delta T$$

is only approximate. A good thermometer uses a substance like mercury or alcohol with an $\alpha$ that is fairly constant over a particular temperature range. If you make a graph of $\alpha$ for mercury and alcohol over a given range of temperatures, the graphs will be almost straight lines, but they aren't exactly straight, and they deviate from one another.

So the problem isn't that it's hard to measure $\alpha$ precisely, it's that $\alpha$ isn't actually perfectly constant.

To make a simple thermometer, you need to choose a value of $\alpha$ by measuring the slope of its graph over some temperature region. The points in that graph where the slope of the graph equals our chosen $\alpha$ are the calibration points mentioned by Halliday and Resnick.

You also have to take into account the expansion of the glass that contains the thermometer's working fluid, but that only introduces a small error, because the expansion coefficient of a solid is typically much smaller than that of a liquid.

Answer 2

The answer by @PM2Ring is good, but I want to add a historical note which might shed light on what could possibly be behind that quote.

While the Celsius scale today is tied to the Kelvin scale (which is based on thermodynamics) just as @PM2Ring explains, historically that was different and defined by expansion of mercury.

So possibly, what your quote means is not the present-day definition but the original definition by Anders Celsius (modified by Linnaeus).

Celsius defined the scale in 1742, when the thermodynamic theory of temperature was not yet developed. People at the time were trying to define temperature in some meaningful way and the best they could come up with is to define it by the expansion of materials.

Celsius originally only defined two fixed points (freezing and boiling of water) and the scale in between by dividing the arm of the mercury thermometer into 100 steps of equal length (he used the word "grader" as the Swedish plural form of Latin gradus=step).

In his definition, the freezing point of water would be 100 grader and the boiling point 0 grader. Note that this is the other way round from today's definition; this looks unintuitive to us today, but, again, at that time, without thermodynamics there was no reason to believe that "cold" means "less" of something (energy), so this made as much sense as any other scale.

Jean-Pierre Christin in 1743 (and, possibly independently, Carolus Linnaeus in 1744) then inverted the scale to our familiar 0° = freezing point and 100° = boiling point, and for a long time this scale was called "centigrade" ("hundred steps"); the name Celsius scale came much later.

At that time there was a lot of debate if mercury or alcohol is better; both have practical advantages. People realised their expansion is different so mercury and alcohol thermometers with linear scales don't match up, but there was no way of knowing which was "correct". So there were a lot of different scales based on different thermometers with different gradations, like Fahrenheit, Delisle, Reaumur and others. Today theses scales are no longer used or only in some marginal situations (Reaumur is still used in cheese production and Fahrenheit only in a few countries like the Cayman Islands, the Bahamas and some land mass to the Northwest of these islands). Today these scales have also, like the Celsius scale, been tied to the thermodynamic Kelvin scale, but originally they were all defined by different thermometers.