Challenge: mercury barometer

Problem

A Torricelli tube has mercury above which there is a vacuum chamber. Write the pressure balance that supports the mercury column and calculate its height for atmospheric pressure $p_\text{atm}=1{,}013\cdot 10^5\ \text{Pa}$ ($\rho_\text{Hg}=13600\ \text{kg/m}^3$).

Solution

Pressure balance. At the base of the column, the external atmospheric pressure must balance the weight of the mercury column. Above the mercury is a vacuum (zero pressure), so:

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

Height of the column. Isolating $h$:

$h=\frac{p_\text{atm}}{\rho_\text{Hg}\,g}=\frac{1{,}013\cdot 10^5}{13600\cdot 9{,}81}=\frac{1{,}013\cdot 10^5}{1{,}334\cdot 10^5}=0{,}759\ \text{m}$

This is the famous value of about 760 mm of mercury that defines standard atmospheric pressure.

$\ev{h\approx 0{,}759\ \text{m}=759\ \text{mm}}$

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Topics: Thermodynamics Concepts: Pressure Skills: Symbolic problem setup