Barometer and atmospheric pressure

1. Definition

A. Atmospheric pressure

Atmospheric pressure is the pressure exerted by the weight of the air on a surface.

At sea level, pₐₜₘ ≈ 1.0 × 10⁵ Pa.

B. Barometer

A barometer measures atmospheric pressure. In a mercury barometer, the reading is the vertical height difference h between the mercury surfaces in the tube and reservoir.

For a mercury barometer with a vacuum above the mercury:

pₐₜₘ = ρ gh

2. Key Ideas

• Atmospheric pressure:
  • acts in all directions
  • is lower at higher altitudes (less air above you).
• A barometer works because atmospheric pressure pushes on the liquid in the reservoir and supports the liquid column in the tube.
• For a mercury barometer (vacuum at the top):
  • pₐₜₘ = ρHggh
  • ρHg ≈ 13.6 × 10³ kg m⁻³
  • h ≈ 0.76 m (76 cm) at 1 atm.
• Only the vertical height difference h matters. The reading does not depend on tube diameter or container shape.
• The length of mercury along a tilted tube is not the barometer reading.

3. Detailed Explanations

A. Atmospheric pressure and altitude

You can think of the atmosphere as a “sea of air”.

• lower altitude → more air above → larger weight per unit area → higher pressure
• higher altitude → less air above → smaller weight per unit area → lower pressure

B. Mercury barometer: how it works

Scroll diagram horizontally to read all labels.

To make a mercury barometer:

1. Fill a long glass tube completely with mercury.
2. Invert it into a mercury bath.
3. A vacuum forms at the top of the tube, so the pressure there is approximately zero.

Atmospheric pressure acts on the exposed mercury in the reservoir and supports the mercury column in the tube.

Using hydrostatic pressure (p = ρ gh):

pₐₜₘ = ρHggh

where h is the vertical height difference between:

• the mercury level in the tube, and
• the mercury level in the bath.

At the same horizontal level in the connected mercury, the pressures are equal. On the reservoir side the pressure is pₐₜₘ. Inside the tube it is pₜₒₚ + ρHggh. Since the pressure pₜₒₚ in the vacuum is negligible, pₐₜₘ = ρHggh.

C. Why mercury is used (instead of water)

At sea level, take pₐₜₘ ≈ 1.0 × 10⁵ Pa and g ≈ 10 N kg⁻¹.

• For water (ρ ≈ 1000 kg m⁻³): h = p/ρ g ≈ 1.0 × 10⁵/(1000)(10) = 10 m
• For mercury (ρ ≈ 13.6 × 10³ kg m⁻³): h ≈ 1.0 × 10⁵/(13.6 × 10³)(10) ≈ 0.74 m

So mercury produces a much shorter, more practical column.

D. What changes the reading (and what does not)

Scroll diagram horizontally to read all labels.

As long as the vacuum is maintained and the tube mouth stays under the mercury surface:

• changing the tube diameter does not change h
• tilting the tube does not change the vertical height h (but the mercury length along the tube changes)
• raising or lowering the tube slightly in the bath does not change h

E. Links (next steps)

4. Common Mistakes

• Using the length of mercury along a tilted tube instead of the vertical height difference.
• Using the wrong density (for mercury use ρ ≈ 13.6 × 10³ kg m⁻³).
• Forgetting unit conversions (e.g. 76 cm = 0.76 m).
• Writing p = ρ gh but using values in mixed units (keep to SI).

5. Exam Tips

• For a mercury barometer, assume the top space is a vacuum, so it exerts negligible pressure.
• Identify h carefully: it is the vertical height between the two mercury levels.
• If the question asks for “atmospheric pressure”, do not add atmospheric pressure again (the barometer is measuring it).
• Typical values:
  • pₐₜₘ ≈ 1.0 × 10⁵ Pa
  • ρHg ≈ 13.6 × 10³ kg m⁻³

6. Worked Examples

Example 1: Identify the correct height hCore

In a simple mercury barometer, describe precisely which height is used in pₐₜₘ = ρHggh.

Show Answer

h is the vertical height difference between the mercury surface in the tube and the mercury surface in the reservoir. It is not measured from the bottom of the reservoir or along a tilted tube.

Example 2: Water barometer heightCore

Instead of mercury, water is used in a barometer. Find the height h of the water column.

Assume:

• ρwₐₜₑᵣ = 1000 kg m⁻³
• g = 10 N kg⁻¹
• pₐₜₘ = 100 kPa

Show Answer

Use p = ρ gh: 100 kPa = 100 × 10³ Pa h = p/ρ g = 100 × 10³/(1000)(10) = 10 m

This is why water is impractical for barometers (the tube would need to be more than 10 m tall).

Example 3: Atmospheric pressure from mercury heightCore

A mercury barometer shows a height difference of 72 cm. Find the atmospheric pressure.

Take ρHg = 13.6 × 10³ kg m⁻³ and g = 10 N kg⁻¹.

Show Answer

Convert h: 72 cm = 0.72 m

pₐₜₘ = ρ gh = (13.6 × 10³)(10)(0.72) = 9.8 × 10⁴ Pa

Example 4: Change in atmospheric pressure from a change in heightCore

The height in a mercury barometer drops from 76 cm to 74 cm. Find the change in atmospheric pressure.

Take ρHg = 13.6 × 10³ kg m⁻³ and g = 10 N kg⁻¹.

Show Answer

Use Δ p = ρ g Δ h.

Δ h = 76 cm - 74 cm = 2.0 cm = 0.020 m

Δ p = (13.6 × 10³)(10)(0.020) = 2.72 × 10³ Pa

The atmospheric pressure decreased by 2.7 × 10³ Pa.

Example 5: Mercury height from atmospheric pressureCore

A mercury barometer is used on a day when atmospheric pressure is 98 kPa. Find the barometer height h.

Take ρHg = 13.6 × 10³ kg m⁻³ and g = 10 N kg⁻¹.

Show Answer

Convert pressure: 98 kPa = 98 × 10³ Pa

Use p = ρ gh: h = p/ρ g = 98 × 10³/(13.6 × 10³)(10) = 0.72 m

7. Mind Stretchers

Mind stretcher 1: Tilted barometer tubeExtension

A mercury barometer tube is tilted, but the top remains a vacuum and the tube mouth stays under the mercury surface. Does the barometer reading change? Explain.

Show Answer

The reading does not change.

Atmospheric pressure is given by: pₐₜₘ = ρ gh

h is the vertical height difference between the mercury levels. Tilting changes the length of mercury along the tube but does not change the vertical height difference.

Mind stretcher 2: Trapped air at the top (extension)Extension

Suppose some air is trapped at the top of the barometer tube (so it is not a vacuum). How will the height h compare with the vacuum case? Explain.

Show Answer

The height h will be smaller.

If the trapped air has pressure pₜₒₚ, then: pₐₜₘ = pₜₒₚ + ρ gh

Since pₜₒₚ is not zero, ρ gh is smaller for the same atmospheric pressure, so the observed height is smaller.

8. Practice and next step

Complete the Pressure structured questions, then compare atmospheric-pressure measurement with pressure-difference measurement in the Manometer lesson.