14-3 Measuring Pressure#
Prompts
How does a mercury barometer measure atmospheric pressure? What physical principle relates the column height to pressure?
Why does the barometer reading (height \(h\)) not depend on the cross-sectional area of the tube?
How does an open-tube manometer measure the gauge pressure of a gas? What does the height difference \(h\) tell you?
When is gauge pressure positive? When is it negative? Give examples (tire, straw, blood pressure).
The torr unit equals 1 mm Hg. Why is mercury commonly used in barometers rather than water?
Lecture Notes#
Overview#
Barometer: measures atmospheric pressure \(p_0\) via the height of a liquid column.
Open-tube manometer: measures gauge pressure \(p_g = p - p_0\) of a confined gas.
Both rely on hydrostatic equilibrium: \(p = p_0 + \rho g h\) (see 14-2).
Energy view: Both devices convert pressure (energy density) into a measurable height \(h\).
Device |
Measures |
Principle |
|---|---|---|
Barometer |
Atmospheric pressure \(p_0\) |
Vacuum above column; \(p_0 = \rho g h\) |
Manometer |
Gauge pressure \(p_g\) of a gas |
Compare gas pressure to atmospheric; \(p_g = \rho g h\) |
Mercury barometer#
A barometer measures atmospheric pressure. A long glass tube is filled with mercury and inverted with its open end in a dish of mercury. The space above the mercury column contains only mercury vapor; its pressure is negligible at ordinary temperatures.
Level 1: air–mercury interface (pressure \(p_0\)). Level 2: top of mercury column (pressure \(\approx 0\)). From hydrostatic equilibrium:
\(\rho\): density of mercury
\(h\): height of the mercury column above the dish
Energy view: Atmospheric pressure \(p_0\) is the energy density that supports the column; \(\rho g h\) is the gravitational PE density of that column.
Important
The height \(h\) does not depend on the cross-sectional area of the tube. Only the vertical distance between the two mercury levels matters. A wide tube and a narrow tube give the same reading.
Units: 1 torr = 1 mm Hg. At sea level, \(p_0 \approx 760\;\text{mm Hg} \approx 1\;\text{atm}\). The height depends on \(g\) and on mercury density (which varies with temperature); corrections are needed for precise work.
Why mercury?
Mercury is dense (\(\rho \approx 13.6 \times 10^3\;\text{kg/m}^3\)), so a barometer is compact—\(h \approx 760\;\text{mm}\) for 1 atm. A water barometer would need \(h \approx 10\;\text{m}\).
Open-tube manometer#
An open-tube manometer measures the gauge pressure of a confined gas. It is a U-tube: one arm connects to the gas vessel, the other is open to the atmosphere.
Level 1: open arm, at atmospheric pressure \(p_0\). Level 2: same horizontal level in the other arm, at gas pressure \(p\). The liquid column has height difference \(h\) (measured from level 2 up to level 1).
\(p_g\): gauge pressure
\(\rho\): density of the liquid in the U-tube
Energy view: Gauge pressure \(p_g\) is the excess energy density over atmospheric; the height difference \(h\) measures this excess.
Sign of gauge pressure:
\(p > p_0\) (e.g., tire, blood pressure): \(p_g > 0\) (overpressure)
\(p < p_0\) (e.g., suction in a straw): \(p_g < 0\)
Summary#
Barometer: \(p_0 = \rho g h\); measures atmospheric pressure; height independent of tube area
Manometer: \(p_g = \rho g h\); measures gauge pressure of a gas; \(p_g\) positive or negative