Chapter 14: The Key of Torricelli, the Door of Boyle

Continuation of this

1. The Invisible Wall Named "Nature"
Under the scorching sun, the apprentice was gasping for air. On the high ground of the settlement, a long, long "suction pipe" extended from the bottom of a well that the two had spent months digging, intended to draw up water.
"Master... I can't do it anymore. No matter how much I pull the piston, the lever feels as heavy as a rock. The water should be right there!"
Where the apprentice pointed, inside the glass tube, the water column had stopped dead at a height of 10.3 meters from the ground. Just a few dozen centimeters more. With just a little more, the water would have overflowed onto the ground and moistened the parched village.
The primitive man stared intently at the water surface while flipping through the tattered pages of the "Principia".
"Nature abhors a vacuum... is that so? Old man Aristotle said as much. He claimed that if you empty the inside of a tube, nature would rush to push the water up. But it seems this 'nature' fellow's aversion has a 'bottom' at exactly 10 meters."
"A bottom, you say?"
"Yes. It abhors a vacuum and lifts water up to 10 meters, but beyond that, it doesn't care if it becomes a vacuum or not. ...Don't you find it strange? That nature's selfishness should have such a precise 'numerical limit'."

2. The Silver Magician
The primitive man stood up and brought over a jar containing the "silver liquid" he had been storing—mercury.
"Apprentice. Stop messing with the 10-meter water column. If there is a limit to nature's power to abhor a vacuum, we shall test it with something 'heavier'."
They filled a thin glass tube, closed at one end, to the brim with mercury. "Listen, precision is life from here on out. Do not forget the 'unit of length' we decided on."
The primitive man firmly blocked the mouth of the tube with his thumb and plunged the tube upside down into the vessel containing mercury.
"Here we go. I'm letting go."
His heart skipped a beat. The silver column inside the tube slid down under its own weight. At the top of the tube, a complete "nothingness"—a vacuum—where nothing in this world existed, was about to be born.
However, the silver column came to a stop at a certain point, as if supported by an invisible hand.
"Measure it. ...Right now!"
The apprentice applied the scale with trembling hands. "...76 centimeters. Master, it's exactly 76 centimeters! Even if I tilt the tube, the vertical height doesn't move by even a millimeter!"

3. We Are at the Bottom of an "Ocean"
The two stood speechless for a while before that 76-centimeter silver column. This was the "true form" they had reached after much trial and error.
"Apprentice. We had it backwards. The pump is not 'sucking up' the water. ...It is this transparent 'air' around us. It has weight."
The primitive man looked up at the sky.
"We are fish crawling at the bottom of a massive, heavy, invisible ocean called air. This ocean of air, with its weight, is pushing the mercury in the vessel and the water in the well from the outside with all its might. That 'pushing force' is exactly balanced by the force that lifts 10 meters of water or 76 centimeters of mercury."
Scales fell from the apprentice's eyes. It was not the vacuum pulling it up; the atmosphere was pushing it up.
"...Does that mean, Master, that this height of 76 centimeters becomes a scale to measure the 'strength of the atmosphere' directly?"
"That's right. That intangible, ghost-like word that Newton predicted in the Principia... 'Pressure'. It has now taken physical form before our eyes as the height of this silver column."
The primitive man's eyes shone sharper than ever before.
"Boyle must have known this too. Now that the weight of the air has become a numerical value, the seal on 'Book 2, Chapter 5' of that book can finally be broken. ...Hey, apprentice. Prepare a J-shaped tube immediately."
"Eh, right now!?"
"Of course! The truth is waiting there. Run! Apprentice!"

4. The Prison of Glass
As the sun began to sink below the western horizon and the world was dyed in a gradient of red and black, the two men were moving like madmen.
"Increase the fire! Bend it before the glass cools!"
The primitive man's roar echoed. Covered in sweat and soot, the apprentice carefully, yet boldly, bent the red-hot glass tube like candy work. Before long, a strange "J"-shaped glass apparatus was completed before their eyes.
It was a transparent prison designed to trap and interrogate the "invisible demon" called air.
"Listen, apprentice. We will leave a little air at the end of this bend and seal it with mercury. This is today's protagonist, the sacrificial lamb."
They carefully poured in the mercury. Air, cut off from the outside world, was trapped in the shorter end of the J-tube. The heights of the mercury on the left and right were level. In other words, the trapped air was currently pushing back from the inside with the exact same pressure as the "ocean of atmosphere" outside (the weight of 76 centimeters of mercury).
The primitive man measured the length of the trapped air with a wooden ruler.
"Initial volume, recording complete. ...Now, it is time for judgment."

5. The Scream of the Invisible "Spring"
"Pour it!"
At the master's signal, the apprentice began pouring the silver liquid into the longer opening of the J-tube.
A heavy weight was transmitted to the hand holding the tube. The trapped air was being squeezed tightly by the weight of the poured mercury.
"Master, look! The air... the air is in pain!"
The volume of air seen through the glass was visibly shrinking. It was as if an invisible giant was bearing down on it from above.
"Fool, it's not in pain. ...It's angry."
The primitive man's eyes were bloodshot.
"They are desperately 'pushing back' right now. In addition to the outside atmospheric pressure of 76 centimeters, they are supporting the weight of the mercury you just added. They are supporting both with that tiny body. What a stubborn lump of a 'spring' air is!"
They continued to add mercury. The longer mercury column rose higher and higher than the shorter one. The difference in height (h) was the true identity of the "pressure" they had added.

6. Calculation of the Starry Sky
Before long, the primitive man raised his hand. "Stop!"
The volume of the trapped air was exactly "half" of what it had been initially.
"Measurement. Don't let your hands shake, apprentice. We dug that damn deep well for this very moment."
The apprentice swallowed hard and applied the ruler. First, the volume of the trapped air (V). It was indeed half of the initial amount. Next, the difference in height between the left and right mercury surfaces (h).
"...Se-seventy... 76 centimeters! Master, the height of the added mercury is exactly the same as the atmospheric pressure we measured earlier, 76 centimeters!"
The primitive man snatched the "Principia" that had been thrown on the ground. With mud-stained fingers, he traced the page where complex geometric figures were drawn.
"...Heh, hehehe. So that's it, so that was it, Mr. Newton!"
He let out a beast-like cry toward the night sky.
"Look, apprentice! When the volume of air became 'half', the pressure crushing it became exactly 'double', adding the 'original atmospheric pressure (76)' to the 'added mercury (76)'!"
If the volume is halved, the pressure doubles. If the volume becomes one-third, the pressure will become three times as much.
"It's inverse proportion...! The magical formula written in that book. 'Density (pressure) is inversely proportional to volume.' It has now been perfectly proven right before our eyes, inside this glass tube covered in mud and mercury!"

Epilogue: Dialogue with the Universe
The two were slumped in front of a bonfire under the starry sky. At hand lay the great fruits of today's battle—a wooden stick carved with "76cm" and the J-tube filled with mercury.
The apprentice muttered while staring at the blazing flames.
"Master. We did something amazing today. We translated the words of that difficult book into our own language."
The primitive man, still unable to cool his excitement, beat his own hairy chest.
"Yes. Until today, we were just hole-digging savages. But tonight, we obtained a common language called 'pressure', and we conversed as equals with the god named Newton, and with the very laws of this universe."
He grinned and added firewood to the bonfire. The flames danced vigorously.

To be continued

【Exercises: The Ocean of Atmosphere and the Spring of Air】
Part 1: The True Identity of Atmospheric Pressure — Visualizing "Weight"

【Question 1】 Calculation of Mercury Column and Atmospheric Pressure
In the Torricelli experiment conducted by the primitive man, the mercury column came to a rest at a height of 760 mm (0.76 m).
(1) When the density of mercury is $${13.6 \times 10^3 \, \text{kg/m}^3}$$ and the gravitational acceleration is $${g = 9.8 \, \text{m/s}^2}$$, find the atmospheric pressure $${P_0}$$ (answer with two significant figures).
(2) If this experiment were conducted using "water" instead of mercury, at what height in meters would the water column come to a rest? Calculate using a water density of $${1.0 \times 10^3 \, \text{kg/m}^3}$$.

【Question 2】 Thought Problem: Invitation to a Vacuum
What would happen if a small hole were opened in the "vacuum" space created at the top of the Torricelli tube? Explain so that the difference between the Aristotelian interpretation that "nature abhors a vacuum" and the primitive man's fluid-mechanical interpretation that "the atmosphere is pushing it up" is clear.

Part 2: Boyle's Law — The Trapped Demon

【Question 3】 "Interrogation" of the J-tube
The primitive man and the apprentice confirmed Boyle's Law using a J-tube.
(1) Initially, the mercury levels on the left and right were equal, and the length of the trapped air (spring) was 20 cm. At this time, the pressure of the air is exactly equal to atmospheric pressure (76 cmHg).
(2) Next, when mercury was added from the longer opening, the difference in height (h) between the left and right mercury surfaces became 76 cm. At this time, what is the volume (length) of the trapped air in cm? Calculate using Boyle's Law $${P_1 V_1 = P_2 V_2}$$.

【Question 4】 Reading Graphs
When pressure P is on the horizontal axis and volume V is on the vertical axis, draw the "inverse proportion curve (isothermal curve)" described by Boyle's Law on a clay tablet. Also, if the same experiment were conducted at a higher temperature, answer how that curve would change relative to the original curve (whether it shifts outward or inward).

【Answers and Explanations】

【Question 1】
(1) $${1.0 \times 10^5 \, \text{Pa}}$$
Explanation: From $${P = \rho g h}$$, $${13.6 \times 10^3 \times 9.8 \times 0.76 \approx 1.01 \times 10^5 \, \text{Pa}}$$.
(2) Approximately 10.3 m
Explanation: Since mercury is 13.6 times heavier than water, the height it can support is 1/13.6. Conversely, if it were water, $${0.76 \, \text{m} \times 13.6 \approx 10.3 \, \text{m}}$$ would be required. This is the true identity of the "limit of the well".

【Question 2】
When a hole is opened, the "weight of the ocean of atmosphere" enters directly from there. The mercury column loses its support and flows down into the vessel all at once. There is no such thing as a "force of a vacuum sucking it up"; it is merely that the balance of "pushing forces" between the outside and inside has been broken.

【Question 3】
10 cm
Explanation: Initial pressure $${P_1 = 76}$$. Pressure after addition $${P_2 = 76 (atmosphere) + 76 (mercury difference) = 152}$$. Since the pressure doubled, according to Boyle's Law, the volume becomes 1/2, which is 10 cm.

【Question 4】
The graph draws a "hyperbola". When the temperature is raised, the volume becomes larger even at the same pressure (Charles's Law), so the curve shifts to the **upper right (outward)**.