Atmospheric and gauge pressure
Torricelli’s barometer
We live at the bottom of an ocean of air, many kilometres deep. The weight of the air above each square metre of ground pushes down on it. That is atmospheric pressure.
In 1643 Evangelista Torricelli found a way to measure it. He filled a long glass tube, closed at one end, completely with mercury.
Then he turned it upside down, with the open end under the mercury in a dish. Watch: the mercury column drops, then stops about 76 centimetres above the level in the dish.
The space left at the top of the tube has almost nothing in it, only a little mercury vapour. It is called the Torricelli vacuum, and its pressure is nearly zero.
Why does the column stop there? Take point A on the open surface in the dish, and point B inside the tube at the same level. Points at the same level in a still liquid have the same pressure.
At A, the air pushes down with P a. At B, the pressure is just the weight of the mercury column: ρ g h, with nothing on top. So P a equals ρ g h.
Put in the numbers: 13 600 times 9.8 times 0.76 gives about 1.013 times 10⁵ pascals. This is one atmosphere, the average air pressure at sea level.
So pressure is often quoted as a height of mercury: 1 atm is 760 millimetres of mercury, a unit also called the torr. Weather reports use the bar, which is 10⁵ pascals, and the millibar.
Why mercury? It is 13.6 times denser than water. A water barometer would need a column about 10.3 metres tall, taller than a three-storey building.
Carry the barometer up a mountain, to Shimla or Leh. There is less air above you, so the column gets shorter. A falling barometer at one place can also warn of a coming storm.
Gauge pressure and the manometer
At a petrol pump, the air gauge on a car tyre might read 30 psi, about 2 atmospheres. That is not the full pressure in the tyre: it is how much the pressure exceeds the atmosphere.
The pressure above atmospheric, P minus P a, is called the gauge pressure. The full pressure, P, is called the absolute pressure.
A simple instrument to measure gauge pressure is the open-tube manometer: a U-shaped tube with some liquid in it. One arm is joined to a container of gas, and the other is open to the air.
If the gas is at atmospheric pressure, the liquid stands at the same level in both arms. Now raise the gas pressure. It pushes its side down, and the liquid rises in the open arm.
Take the lower liquid level on the gas side. The point at the same level in the open arm has a column of height h above it, with the atmosphere on top: P a plus ρ g h. So P equals P a plus ρ g h.
The level difference h measures the gauge pressure directly: P minus P a equals ρ g h. If the gas is below atmospheric pressure, the liquid rises on the gas side instead.
Worked example: in a mercury manometer the open arm stands 15 centimetres higher. The gauge pressure is 13 600 times 9.8 times 0.15, about 20 000 pascals, or 20 kilopascals. The absolute pressure is about 121 kilopascals.
Mercury suits large pressures. For small pressure differences, a light liquid like water or oil is used, because the same pressure then gives a much bigger, easier-to-read h.
A doctor’s blood pressure reading, like 120 over 80, is also a gauge pressure in millimetres of mercury. 120 millimetres of mercury is about 16 kilopascals above the atmosphere.
Try it: a mercury barometer
- At sea level the mercury column is 76 cm tall. Lower Pₐ to see what happens up a mountain.
- Switch to water. How tall does the column need to be now?
- A and B are at the same level, so they have the same pressure: Pₐ = ρgh.
Try it: an open-tube manometer
- Raise the gauge pressure of the gas: the liquid rises in the open arm.
- Make the gauge pressure negative (gas below Pₐ): now the gas side rises.
- Switch to water: the same pressure gives a 13.6 times bigger h.
Why mercury?
Why did Torricelli fill his barometer with mercury instead of water?
- Water would not rise in a glass tube at all
- Mercury is much denser, so the column is only about 76 cm instead of about 10 m
- Mercury feels a larger atmospheric pressure than water
- Water evaporates, so the column would slowly disappear
Answer
B. The air balances the column: Pₐ = ρgh, so h = Pₐ/ρg. Mercury is 13.6 times denser than water, so its column is 13.6 times shorter: 0.76 m instead of about 10.3 m. The atmosphere pushes on both liquids with the same pressure.
Reading a manometer
An open-tube mercury manometer is connected to a gas cylinder. The mercury in the open arm stands 20 cm higher than in the arm joined to the gas. The atmospheric pressure is 76 cm of Hg. What is the absolute pressure of the gas?
- 20 cm of Hg
- 56 cm of Hg
- 76 cm of Hg
- 96 cm of Hg
Answer
D. The open arm is higher, so the gas is above atmospheric pressure. Its gauge pressure is h = 20 cm of Hg, and the absolute pressure is 76 + 20 = 96 cm of Hg. In pascals: gauge = 13 600 × 9.8 × 0.20 ≈ 26.7 kPa, absolute ≈ 128 kPa. (56 cm would be right if the gas side were higher.)
Summary
- Barometer:
Pₐ = ρgh - 1 atmosphere:
1 atm = 1.013 × 10⁵ Pa = 76 cm of Hg = 760 torr - Other units:
1 bar = 10⁵ Pa - Manometer (gauge pressure):
P − Pₐ = ρgh
- Atmospheric pressure is caused by the weight of the air above us. It is lower at higher altitudes.
- In Torricelli’s barometer the air holds up a 76 cm column of mercury, with a near vacuum above it.
- Gauge pressure is the pressure above atmospheric (P − Pₐ). Tyre gauges and blood pressure readings are gauge pressures.
- In an open-tube manometer the level difference h gives the gauge pressure. A denser liquid gives a smaller h.