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 = ρ_Hggh
  • ρ_Hg ≈ 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

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 = ρ_Hggh

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 + ρ_Hggh. Since the pressure pₜₒₚ in the vacuum is negligible, pₐₜₘ = ρ_Hggh.

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)

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
  • ρ_Hg ≈ 13.6 × 10³ kg m⁻³

6. Worked Examples

7. Mind Stretchers

8. Practice and next step

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