Manometer and Barometer

Stevin’s law is not only useful for predicting the pressure at a given depth: turning the reasoning around, it allows us to measure an unknown pressure by reading a difference in liquid level. Two classic instruments are based on this idea, the manometer and the barometer.

Open-tube manometer

A U-tube half filled with a liquid (water, oil or mercury) is connected on one side to a reservoir whose pressure $P$ we want to measure, and left open to the atmosphere on the other. By Stevin’s law, the reservoir pressure is related to the level difference $\Delta h$ between the two arms:

Law — Open-tube manometer

$\ev{P = P_\text{atm} + \rho\,g\,\Delta h}$

where $\Delta h$ is the height difference between the two arms. If the liquid in the arm connected to the reservoir is lower than the other, it means the reservoir pressure is above atmospheric; if it is higher, it is below.

Torricelli’s barometer

To measure atmospheric pressure directly, Torricelli’s barometer is used. A tube about $1$ m long is filled with mercury, turned upside down into a basin of the same liquid, and released. The mercury falls, leaving a near-perfect vacuum at the top, and stops when the column of height $h$ has, at its base, a pressure equal to the atmospheric one:

$P_\text{atm} = \rho_\text{Hg}\,g\,h$

At standard atmospheric pressure ($1{,}013\cdot 10^5\;\text{Pa}$) we get $h = P_\text{atm}/(\rho_\text{Hg}\,g) = 1{,}013\cdot 10^5/(13600\cdot 9{,}81) \approx 0{,}760\;\text{m}$, that is $760$ mm. This is where the unit mmHg, also called Torr, comes from — still used today for blood pressure and in vacuum technology.

Torricelli’s barometer: the mercury column stops at the height $h$ for which $\rho_\text{Hg}\,g\,h$ equals the atmospheric pressure; at sea level $h \approx 760$ mm.

Pourquoi pas d'eau?

Why did Torricelli choose mercury and not water? The formula is the same, but the density $\rho$ changes: with water we would get $h = 1{,}013\cdot 10^5/(1000\cdot 9{,}81) \approx 10{,}33$ m. A tube over $10$ metres tall full of water would be impractical, as well as fragile. Mercury, about $13{,}6$ times denser, brings the height down to a manageable value on a laboratory bench.

Links

Topics: Fluidostatica e fluidodinamica Concepts: Pressione · Legge di Stevino

Related exercises: Esercizio svolto — La colonna d’aria che schiaccia i nostri piedi · Problema — Derivazione di Stevino · Esercizio svolto — Pressa per la frutta