K323 / 2027
Pressure overview

Lesson 6 of 6

Manometers and pressure differences

  • I can apply liquid-column pressure = height x density x gravitational field strength to new situations.Syllabus

    Syllabus K323, 5(e). Apply liquid-column pressure = height x density x gravitational field strength to new situations.

  • I can explain how a manometer measures pressure difference.Syllabus

    Syllabus K323, 5(g). Explain how a manometer measures pressure difference.

A manometer uses the level difference in a connected liquid to compare two pressures. First identify which side's liquid surface is lower.

At the same horizontal level in a connected stationary liquid, pressure is equal. A vertical difference h gives a pressure difference of ρgh.

An open-end manometer has one arm connected to a gas and the other open to the atmosphere. The greater pressure pushes its side's liquid surface lower. The two surface pressures are therefore the gas pressure and the atmospheric pressure.

Magnitude of pressure difference = ρghh is the vertical separation of the two liquid surfaces. Use liquid density in kg/m3, g in N/kg and h in m for a difference in Pa.

Compare the two liquid surfaces first

The same connected water has density 1000 kg/m3; g = 10 N/kg. The right-hand end is open to air at 101000 Pa.

Gas pressure is higher than atmospheric

Gas pressure is higher than atmosphericThe left limb of a U-tube is connected to the gas; the right limb is open to atmospheric pressure, 101000 Pa. The water is continuous through the base of the tube. The gas-side water surface is lower. At horizontal reference level A and B, A is at the gas-side surface and B is 0.120 metres below the open-side surface. Thus gas pressure equals atmospheric pressure plus 1200 Pa, or 102200 Pa. The bracket h measures the vertical difference of the two water levels, not either whole liquid column. Pressures at A and B are equal in the stationary connected liquid.Gas connectionOpen to airp gas101000Pah =0.120 mABA and B have equal pressure

Pressure difference = 1000 x 10 x 0.120 = 1200 Pa.

At A: gas pressure. At B: atmospheric pressure + 1200 Pa.

Gas pressure = 101000 + 1200 = 102200 Pa.

Gas pressure is lower than atmospheric

Gas pressure is lower than atmosphericThe left limb of a U-tube is connected to the gas; the right limb is open to atmospheric pressure, 101000 Pa. The water is continuous through the base of the tube. The gas-side water surface is higher. At horizontal reference level A and B, A is 0.120 metres below the gas-side surface and B is at the open-side surface. Thus gas pressure plus 1200 Pa equals atmospheric pressure, giving gas pressure 99800 Pa. The bracket h measures the vertical difference of the two water levels, not either whole liquid column. Pressures at A and B are equal in the stationary connected liquid.Gas connectionOpen to airp gas101000Pah =0.120 mABA and B have equal pressure

Pressure difference = 1000 x 10 x 0.120 = 1200 Pa.

At A: gas pressure + 1200 Pa. At B: atmospheric pressure.

Gas pressure = 101000 - 1200 = 99800 Pa.

The blue dashed line marks the equal-pressure reference level. Use the vertical level difference to compare pressures, then use the known atmospheric value.

Both examples have water levels separated vertically by 0.120 m. A lower gas-side surface means gas pressure is above atmospheric pressure; a higher gas-side surface means it is below atmospheric pressure. The horizontal reference lines compare equal-level points in each connected liquid.

Gas-side surface lower: add the difference

Choose the horizontal reference at the lower, gas-side surface. Its pressure is the gas pressure. At that same level in the open arm, the point is h below the atmosphere-exposed surface, so its pressure is atmospheric pressure plus ρgh.

Worked example

Gas pressure above atmospheric pressure

Use water density 1000 kg/m3, g = 10 N/kg, h = 0.120 m and atmospheric pressure 101000 Pa.

  1. Find the difference: ρgh = 1000 x 10 x 0.120 = 1200 Pa.
  2. Use the surface positions: the gas-side surface is lower, so gas pressure is greater.
  3. Add to the reference: pgas = 101000 + 1200 = 102200 Pa.

Try it step by step

Work it out in steps: Pressure difference rho g h in Pa, then Gas pressure higher or lower than atmospheric?, then Gas pressure in Pa.

  1. Problem 1. An open water manometer is connected to a gas supply. The water surface on the gas side is 0.150 m lower than the surface on the open side. Water has density 1000 kg/m3, g = 10 N/kg and atmospheric pressure is 101000 Pa. Find the gas pressure.
  2. Problem 2. The water surface on the gas side is now 0.080 m higher than the surface on the open side. Water has density 1000 kg/m3, g = 10 N/kg and atmospheric pressure is 100000 Pa. Find the gas pressure.
  3. Problem 3. A mercury manometer is used instead. The mercury surface on the gas side is 0.050 m lower than the surface on the open side. Mercury has density 13600 kg/m3, g = 10 N/kg and atmospheric pressure is 100000 Pa. Find the gas pressure.
Show the answer

Problem 1.

Pressure difference rho g h in Pa: change in p = rho g h = 1000 x 10 x 0.150 = 1500 Pa

Gas pressure higher or lower than atmospheric?: The gas pushes its side down, so the gas pressure is higher than atmospheric pressure.

Gas pressure in Pa: p = 101000 + 1500 = 102500 Pa, which is 103000 Pa to 3 sf

Problem 2.

Pressure difference rho g h in Pa: change in p = 1000 x 10 x 0.080 = 800 Pa

Gas pressure higher or lower than atmospheric?: The gas side is higher, so the gas pushes less hard than the air: gas pressure is lower than atmospheric pressure.

Gas pressure in Pa: p = 100000 - 800 = 99200 Pa (3 sf)

Problem 3.

Pressure difference rho g h in Pa: change in p = 13600 x 10 x 0.050 = 6800 Pa

Gas pressure higher or lower than atmospheric?: The gas side is lower, so the gas pressure is higher than atmospheric pressure.

Gas pressure in Pa: p = 100000 + 6800 = 106800 Pa, which is 107000 Pa to 3 sf

Gas-side surface higher: subtract the difference

Now the open-side surface is lower. At that horizontal level, atmospheric pressure equals the gas pressure plus the pressure of the liquid column below the higher gas-side surface:

patmosphere = pgas + ρgh
pgas = patmosphere - ρghThe open end supplies the pressure reference. The sign follows from which surface is higher.

With the same supplied water density, g and 0.120 m separation, the difference is still 1200 Pa. Gas pressure is now 101000 - 1200 = 99800 Pa. The gas pressure is positive but lower than atmospheric pressure.

If both levels are equal, the pressure difference is zero. That means the gas and atmosphere have equal pressures, not that their pressures are both zero.

Measure the full vertical separation

Read both water menisci from the same vertical scale and use the bottom of each meniscus at eye level. Wait for a steady liquid. These example readings are consistent with the higher-gas-pressure case:

Example levels measured from one common vertical reference
Water surfaceHeight reading / mm
Gas side134
Open side254

The separation is 254 - 134 = 120 mm = 0.120 m. Use the difference between the two surfaces, not the length around the U-bend or the movement of just one surface from an earlier position.

If h is overestimated, the magnitude of the pressure difference is overestimated. In the higher-gas-pressure case this makes the calculated gas pressure too high; in the lower-gas-pressure case it makes it too low because too much is subtracted. A vertical scale and eye-level readings address a specific measurement cause.

A difference needs a reference before it gives a gas pressure. The height and liquid density give ρgh. To find gas pressure relative to a vacuum, also use the atmospheric pressure or another supplied reference. Do not assume that ρgh alone is the absolute gas pressure.

Check your understanding

An open-end water manometer has its gas-side surface 0.10 m higher than the open-side surface. Use water density 1000 kg/m3, g = 10 N/kg and atmospheric pressure 100000 Pa. What is the gas pressure?

  1. 1000 Pa
  2. 99000 Pa
  3. 101000 Pa
Show the answer

99000 Pa

The higher gas-side surface means gas pressure is lower than atmospheric pressure. The difference is 1000 x 10 x 0.10 = 1000 Pa, so gas pressure = 100000 - 1000 = 99000 Pa.