Atmospheric pressure — how does it support mercury column in barometer

Question

Explain how atmospheric pressure supports the mercury column in a barometer. Derive the expression for atmospheric pressure in terms of the height of the mercury column.

Solution — Step by Step

Evangelista Torricelli invented the barometer in 1643. Take a glass tube (about 1 metre long), closed at one end. Fill it completely with mercury (no air). Invert it and place the open end in a trough of mercury.

What happens: mercury falls slightly — but stops at about 76 cm above the mercury surface in the trough. The space above the mercury column in the sealed end becomes a vacuum (called Torricelli’s vacuum — no air, negligible mercury vapour pressure).

Consider two points at the same horizontal level — one inside the tube at the mercury surface in the trough (Point A) and one outside in the trough at the same level (Point B).

At Point B (outside): pressure = atmospheric pressure $P_0$ acting down on the trough surface, transmitted through the mercury to point B.

At Point A (inside): pressure = weight of mercury column above it per unit area + pressure of vacuum above the column.

Vacuum pressure ≈ 0 (ideal barometer). So pressure at A = pressure due to mercury column of height $h$ .

Since both points are at the same level in the connected mercury, their pressures must be equal (Pascal’s law).

P_0 = \rho g h

where:

$\rho$ = density of mercury = $13,600$ kg/m³
$g$ = $9.8$ m/s²
$h$ = height of mercury column = 0.76 m (at standard atmospheric pressure)

P_0 = 13600 \times 9.8 \times 0.76 = 101{,}292.8 \text{ Pa} \approx 1.01 \times 10^5 \text{ Pa}

This is standard atmospheric pressure = 1 atm = 101.325 kPa = 760 mmHg (76 cmHg).

The atmosphere presses down on the mercury surface in the trough with pressure $P_0$ . This pressure is transmitted through the mercury and pushes mercury up the tube. The mercury rises until the pressure it exerts (due to its own weight per unit area) exactly balances the atmospheric pressure pushing from below.

If atmospheric pressure increases (e.g., in good weather), the mercury column rises. If it decreases (e.g., at high altitude or stormy weather), the column falls.

Why This Works

The fundamental principle is pressure at the same horizontal level in a connected fluid is the same. The mercury in the tube and the trough is connected, so at the level of the trough surface, pressures must balance.

This is why the diameter of the tube doesn’t matter — a fat tube and a thin tube will both show the same height of 76 cm. The pressure balance depends only on height, not area.

Alternative Method — Energy/Work Perspective

Think of it as the atmosphere doing work to push mercury up the tube. The atmosphere will push mercury up until the work done lifting more mercury costs more energy than the atmosphere can supply. At equilibrium, $P_0 \times A = \rho g h \times A$ (force balance on the mercury column’s cross-section $A$ ), giving the same result.

Common Mistake

A very common error is saying “mercury is heavy, so atmospheric pressure is high enough to support it.” This is conceptually backwards. Atmospheric pressure supports the mercury BECAUSE pressure acts upward from below. Pressure at any point in a fluid acts in all directions — the upward component at the bottom of the mercury column supports it against gravity. Mercury is used precisely because it is heavy: a column of water would need to be 10.3 m tall to balance 1 atm — impractically large for a barometer.

If the question asks “why is mercury used in barometers and not water?”: (1) mercury is 13.6 times denser, so the column is 13.6 times shorter (76 cm vs 10.3 m), (2) mercury has very low vapour pressure so the “vacuum” is nearly perfect, (3) mercury does not wet glass, so it moves freely and gives accurate readings.

Question

Solution — Step by Step

Why This Works

Alternative Method — Energy/Work Perspective

Common Mistake

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