How does this author conclude that the volume of the gas is zero at -273.15 degree Celsius?

Question

While reading Gaskell and Laughlin's Intro to ThermoD, I was stuck on something. The relevant passage is:

"In 1802, Joseph-Luis Gay-Lussac (1778–1850) observed that the thermal coefficient of what were called permanent gases was a constant. Previously, we noted that the coefficient of thermal expansion, α, is defined as the fractional increase of the volume of the gas, with the change in temperature at constant pressure; that is,

$$α = 1/v * (∂V/∂T)$$

where $V$ is the volume of 1 mole of the gas at 0°C. Gay-Lussac obtained a value of 1/267 for α, but more refined experimentation by Henri Victor Regnault (1810–1878) in 1847 showed α to have the value 1/273. Later, it was found that the accuracy with which Boyle’s and Charles’ laws describe the behavior of different gases varies from one gas to another. Generally, gases with lower boiling points obey the laws more closely than do gases with higher boiling points. It was also found that the laws are more closely obeyed by all gases as the pressure of the gas is decreased. It was thus found convenient to invent a hypothetical gas which obeys Boyle’s and Charles’ laws exactly at all temperatures and pressures. This hypothetical gas is called the perfect or ideal gas, and it has a value of α =1/ . 273 15.

The existence of a finite coefficient of thermal expansion therefore sets a limit on the thermal contraction of the ideal gas; that is, since α = 1 / . 273 15, then the fractional decrease in the volume of the gas, per degree decrease in temperature, is 1/ . 273 15 of the volume at 0°C. Thus, at –273.15°C, the volume of the gas would be zero, and hence the limit of temperature decrease, –273.15°C, is the absolute zero of temperature. This defines an absolute scale of temperature called the ideal gas temperature scale, which is related to the arbitrary Celsius scale by the equation:

T (degrees absolute) = T (degrees Celsius) + 273.15 "

Here how does he conclude that the volume of the gas is zero at -273.15 degree Celsius? Here we are defining the absolute scale so we shouldn't use the concept of absolute 0 to arrive at that conclusion. It can't be experimental because the word "thus" is used indicating a reasoning.

In short I was confused as the very last line of the extract is supposed to be a revelation, but I don't get what we concluded and how.

Kindly excuse the amateurish question and correct me wherever needed.

Answer 1

I used to buy milk, from a local dairy, that was packaged in glass bottles with slip-on plastic caps. When we finished a bottle, we'd rinse it out and return it to the dairy, who would sterilize it and re-use it. (Before cheap plastic, this is how everybody bought milk.)

Sometimes I would pull a nearly-empty bottle from the fridge, pour the last of the liquid into my glass, and then set the empty on the kitchen counter to rinse out after breakfast. If I set it on the counter with its plastic lid on, it would do a remarkable thing: after about ten minutes, I'd hear a musical "thoom," and look to see the plastic cap flying in the air above the jug. The height of the launch was surprisingly consistent: never less than a foot, never more than two feet.

The reason, of course, was that the refrigerated air in the bottle was warming up to room temperature. If its volume were held constant by the inflexible glass, then its pressure had to increase with its temperature, as misattributed to Gay-Lussac, eventually providing enough force to slip-off the lid. Plastic jugs generally don't do this, because they are formed with a flexible dimple on the side that moves before the lid does.

If I had replaced the rigid plastic lid with a balloon or a piston, I might have actually measured the constant-pressure volume change as the air in the jug heated up. I would have found that going from fridge temperature (3ºC) to an outdoor temperature of 30ºC, at constant pressure, would increase the volume of the gas by almost exactly 10%.

If I were to make a plot of the volume of the gas at constant pressure, I would find that its variation with temperature is a straight line, which gets straighter at high temperatures and low pressures. The place where this straight line crosses the zero-volume axis is the temperature $-$273 K, regardless of what species the gas is. Expressing this relationship purely in terms of the slope of the line, as your text does, is kind of a tedious way to do it.

Answer 2

Keep in mind that your discussion concerns the development of the concept of an “ideal” gas. For a gas at high density, calculations must correct for the actual volume of the molecules (which is never actually zero) and the force of attraction between them. Also, in an ideal gas, the kinetic energy of each molecule is zero at absolute zero, but quantum mechanics tells us that can not happen.