Q.What is atmospheric pressure? How is it determined using a barometer?

What is Atmospheric Scale Height? The Intuition

Imagine you're standing at sea level. The air above you is a column of gas, and every square centimetre of your body is being pushed by that entire column. That's atmospheric pressure — roughly 1 kg per square centimetre at the ground.

Now climb a mountain. As you go up, there's less air above you, so pressure drops. But here's the key question: how fast does pressure fall with height?

If the atmosphere were a liquid (incompressible), pressure would drop linearly — every metre up, the same amount of pressure lost. But air is a gas: it compresses under its own weight. The lower layers are squished denser by the weight of the layers above. So near the ground, pressure falls quickly with height; higher up, it falls more slowly.

The scale height is the single number that captures this behaviour. It tells you: "If you go up by this height, the pressure drops by a factor of $e$ (about 2.718)."

The Precise Definition

For an isothermal atmosphere (constant temperature), the pressure $P$ at height $z$ follows an exponential decay:

where $P_0$ is the pressure at $z=0$ (usually sea level), and $H$ is the scale height.

Formally: The scale height $H$ is the vertical distance over which the atmospheric pressure decreases by a factor of $e$.

For Earth's lower atmosphere at 15°C (288 K), with $m \approx 4.8 \times 10^{-26}$ kg and $g = 9.8$ m/s²:

So every 8.5 km you go up, pressure drops to 37% of its value at that starting point.

Why Does This Formula Make Sense?

Think about the forces on a thin horizontal slice of air of thickness $dz$ at height $z$:

• Weight of the slice = $\rho g \, dz$ (density × gravity × thickness)
• Pressure difference across the slice = $P(z) - P(z+dz) = -dP$

For equilibrium: $-dP = \rho g \, dz$

Now use the ideal gas law: $P = \rho \frac{k_B T}{m}$ (since $\rho = \frac{mN}{V}$ and $P = \frac{N}{V}k_B T$)

Substitute $\rho = \frac{P m}{k_B T}$ into the equilibrium equation:

Rearrange:

Integrate from $z=0$ to $z$:

Exponentiate:

What Scale Height Tells You

Key insight: A larger $H$ means the atmosphere "stretches" higher — pressure falls more gradually. A smaller $H$ means the atmosphere is more compact, hugging the surface.

Common Mistake to Avoid …