Measuring Pressure: Barometers and Manometers – Atmospheric Pressure and Archimedes' Principle (Senior 3 Physics) | Ganzaa

Atmospheric Pressure and Archimedes' Principle

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Senior 3 · Physics

Atmospheric Pressure and Archimedes' Principle

5 lessons

Lesson 2 of 5

Measuring Pressure: Barometers and Manometers

25 min

How does a phone count stairs?

Many smartphones contain a pressure sensor smaller than a grain of rice. Climb three floors of a school building, about 10 m, and a fitness app records "3 floors climbed". It knows because the air pressure at the top of the stairs is lower, by roughly $10 \times 1.05 \times 9.8 \approx 100\ \text{Pa}$ , about a thousandth of the whole atmospheric pressure. Measuring pressure that precisely is a modern achievement. This lesson follows how it is done, from the mercury barometer to the metal box inside an aneroid barometer, and how a manometer compares a gas's pressure with the air's.

The mercury barometer revisited

In S2 you saw that the atmosphere holds up a column of mercury in an inverted tube, with a vacuum above it, so that

p_{atm} = h\,\rho\,g

where $h$ is the vertical height of the mercury column, $\rho = 13\,600\ \mathrm{kg\,m^{-3}}$ and $g = 9.8\ \mathrm{N\,kg^{-1}}$ . The same rule tells you how tall a barometer made with any other liquid would have to be.

Worked example 1: a barometer on Karisimbi

At the summit of Karisimbi, a mercury barometer stands at 432 mm. Find the atmospheric pressure.

p = h\,\rho\,g = 0.432 \times 13\,600 \times 9.8 \approx 57\,600\ \text{Pa} \approx 57.6\ \text{kPa}

This is only 57% of the sea-level value.

Worked example 2: a water barometer in Kigali

How tall a column of water ( $\rho = 1000\ \mathrm{kg\,m^{-3}}$ ) could the atmosphere hold up in Kigali, where $p = 84\ \text{kPa}$ ?

h = \frac{p}{\rho\,g} = \frac{84\,000}{1000 \times 9.8} \approx 8.6\ \text{m}

A water barometer would need a tube taller than a two-storey house. This number also sets the limit for the lift pump in lesson 3.

Warning

Mercury is poisonous. Its vapour damages the nervous system and kidneys. Pupils must never touch spilt mercury or a broken mercury barometer or thermometer: move away, keep others out, open the windows and tell a teacher, who must follow the school's spill procedure. Never sweep up or vacuum spilt mercury; that spreads its vapour through the room. This is why most schools and weather stations now use aneroid or electronic barometers.

The aneroid barometer

"Aneroid" means "without liquid". The heart of the instrument is a thin, flat, corrugated metal box from which most of the air has been pumped out. A strong spring stops the atmosphere from crushing it flat.

When the atmospheric pressure rises, the lid of the box is pushed in a little further against the spring.
When the pressure falls, the spring pulls the lid out a little.
The lid moves by only a fraction of a millimetre, so a lever and a chain magnify the movement and turn a spindle; the pointer on the spindle moves across a scale marked in hPa (or mmHg).

An aneroid barometer must be calibrated against a mercury or electronic standard, and a good one is adjusted for the height of the place where it hangs.

Definition

Barometer: an instrument that measures atmospheric pressure. A mercury barometer balances the atmosphere against a column of mercury; an aneroid barometer measures how much the atmosphere squeezes a partly evacuated metal box.

Altimeter: an aneroid barometer whose scale is marked in metres of height instead of units of pressure, used in aircraft and by climbers.

Mercury or aneroid?

	Mercury barometer	Aneroid barometer
Accuracy	Very accurate: a standard for checking others	Less accurate; must be calibrated
Size and transport	Tall (about 1 m), fragile, must stay upright	Small, robust, works in any position
Safety	Contains poisonous mercury	No liquid at all
Uses	Laboratories, calibration	Homes, ships, aircraft altimeters, hikers' watches

Worked example 3: reading an altimeter

During a climb, an aneroid barometer's reading falls from 840 hPa to 810 hPa. Taking the air's average density as $1.05\ \mathrm{kg\,m^{-3}}$ , estimate the height climbed.

\Delta p = 840 - 810 = 30\ \text{hPa} = 3000\ \text{Pa}

\Delta h = \frac{\Delta p}{\rho\,g} = \frac{3000}{1.05 \times 9.8} \approx 290\ \text{m}

An altimeter does this calculation for you, using a scale drawn from the standard atmosphere.

The manometer: pressures above and below atmospheric

A U-tube manometer compares the pressure $p$ of a gas with the atmospheric pressure $p_{0}$ acting on its open arm. In S2 you used one on a biogas digester, where the gas pressure was higher than atmospheric and pushed the liquid down on the gas side: $p = p_0 + h\rho g$ . If the gas pressure is lower than atmospheric, for example in a flask that is being pumped out, the atmosphere pushes the liquid up on the gas side instead:

Worked example 4: a flask being emptied

A mercury manometer is joined to a flask that is being pumped out. The mercury stands 250 mm higher in the arm joined to the flask. Atmospheric pressure is 84 kPa. Find the pressure of the gas in the flask.

h\rho g = 0.250 \times 13\,600 \times 9.8 = 33\,320\ \text{Pa}

p = p_0 - h\rho g = 84\,000 - 33\,320 \approx 50\,700\ \text{Pa} \approx 51\ \text{kPa}

Worked example 5: how hard can you blow?

A pupil blows into one arm of a water manometer and holds the water 45 cm lower on her side than on the open side. Find (a) the gauge pressure of her breath; (b) its absolute pressure in Kigali.

\text{(a) } h\rho g = 0.45 \times 1000 \times 9.8 \approx 4400\ \text{Pa} = 4.4\ \text{kPa}

\text{(b) } p = 84\,000 + 4410 \approx 88\ \text{kPa}

The gauge pressure is the pressure above atmospheric; the absolute pressure includes the atmosphere.

Gauges read the difference

A tyre-pressure gauge or the dial on a gas cylinder (a Bourdon gauge, in which a curved tube straightens as the pressure inside it rises) reads gauge pressure, the excess over atmospheric. A tyre gauge reading 2.2 bar in Kigali means the absolute pressure in the tyre is about $2.2 + 0.84 \approx 3.0\ \text{bar}$ .

Tip

Which side is the gas? In a manometer the liquid is always pushed away from the higher pressure. Liquid lower on the gas side: $p = p_0 + h\rho g$ . Liquid higher on the gas side: $p = p_0 - h\rho g$ .

Aneroid barometer (Shiksha House, Best for NEET). A 3D animation of the vacuum box, spring, levers and pointer; watch how a small movement of the box becomes a large swing of the pointer.

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Practice

Give two reasons why mercury, not water, is used in a liquid barometer.
A mercury barometer in Kigali stands at 629 mm. Find the atmospheric pressure ( $\rho = 13\,600\ \mathrm{kg\,m^{-3}}$ , $g = 9.8\ \mathrm{N\,kg^{-1}}$ ).
What height of water column could the atmosphere support at the summit of Karisimbi, where $p = 58\ \text{kPa}$ ?
Explain how an aneroid barometer works. Name the part that stops the box from being crushed.
A water manometer is joined to a gas tap. The water is 12 cm lower on the gas side. Find the gas pressure if atmospheric pressure is 84 kPa.
A mercury manometer on a flask shows the mercury 180 mm higher on the flask side. Find the gas pressure in the flask ( $p_0 = 84\ \text{kPa}$ ).
A climber's altimeter shows the pressure falling from 850 hPa to 800 hPa. Taking the average density of the air as $1.0\ \mathrm{kg\,m^{-3}}$ , estimate the height climbed. Why is this only an estimate?
Compare the mercury barometer and the aneroid barometer for use by a hiker on Karisimbi. Which would you choose, and why?

Show answersHide answers

Mercury is very dense, so the column is short (about 0.63 m in Kigali instead of 8.6 m of water); it does not wet the glass and evaporates very little, so the space above the column stays almost a vacuum and the reading is clear.
$p = 0.629 \times 13\,600 \times 9.8 \approx 83\,800\ \text{Pa} \approx 83.8\ \text{kPa}$ .
$h = \frac{58\,000}{1000 \times 9.8} \approx 5.9\ \text{m}$ .
A thin, corrugated metal box with most of the air removed is squeezed more when the atmospheric pressure rises and less when it falls; a strong spring stops it collapsing. Levers and a chain magnify the tiny movement of its lid and turn a pointer over a scale.
$p = 84\,000 + 0.12 \times 1000 \times 9.8 = 84\,000 + 1176 \approx 85.2\ \text{kPa}$ .
$p = 84\,000 - 0.180 \times 13\,600 \times 9.8 = 84\,000 - 23\,990 \approx 60\ \text{kPa}$ .
$\Delta h = \frac{5000}{1.0 \times 9.8} \approx 510\ \text{m}$ . The air's density changes with height and temperature, so a single "average" density gives only an estimate.
The aneroid barometer (or altimeter): it is small, light and robust, works in any position and contains no poisonous mercury. A mercury barometer is about a metre tall, fragile and dangerous if broken, even though it is more accurate.

Key points

Mercury barometer: $p_{atm} = h\rho g$ ; in Kigali $h \approx 630$ mm of mercury, or 8.6 m of water.
Aneroid barometer: a partly evacuated metal box, a spring, levers and a pointer; portable, but needs calibration. Scaled in metres, it is an altimeter.
Manometer: liquid lower on the gas side, $p = p_0 + h\rho g$ ; higher on the gas side, $p = p_0 - h\rho g$ .
Gauges read gauge pressure; absolute pressure = gauge pressure + atmospheric pressure.

Sources and credits

11.6: Gauge Pressure, Absolute Pressure, and Pressure Measurement, from College Physics (1st edition) by Paul Peter Urone and Roger Hinrichs, OpenStax, hosted by LibreTexts, licensed under CC BY 4.0. Consulted for the physics; no text was copied.
Photo: Barometer 20191005 210534.jpg by Ka23 13, Wikimedia Commons, licensed under CC BY 4.0.
Video: ICSE Class 9 Physics, Aneroid Barometer, Best for NEET (Shiksha House), embedded from YouTube.

Written by Ganzaa: the lesson text, the aneroid and manometer diagrams, the table, worked examples and practice exercises.

1How does a phone count stairs?

2The mercury barometer revisited

3The aneroid barometer

4Mercury or aneroid?

5The manometer: pressures above and below atmospheric

6Gauges read the difference