Question

In an atmosphere, temperature (T) decreases linearly with height above the ground (z), i.e., T(z) = T\(_0\) - \(\gamma\)z, where \(\gamma\) is a constant. Surface pressure is 900 hPa. If the atmosphere is at rest, then the value of z at which the pressure decreases to half of that at the surface is \(\underline{\hspace{2cm}}\) m (rounded off to the nearest integer).

Hint:

The pressure height relationship is governed by the barometric equation, which can be simplified for the case of an isothermal atmosphere.

Answer 1

Correct Answer: 4450

The pressure at a given height in the atmosphere is related to temperature and height by the equation: \[ P = P_0 \exp \left( \frac{-g \cdot M \cdot z}{R \cdot T_0} \right), \] where:
- \( P \) is the pressure at height \( z \),
- \( P_0 \) is the surface pressure,
- \( g \) is the acceleration due to gravity,
- \( M \) is the molar mass of air,
- \( R \) is the gas constant,
- \( T_0 \) is the temperature at the surface.
We are given that \( P = \frac{P_0}{2} \), and by solving for \( z \), we find: \[ z \approx 4450 \, \text{m}. \] Thus, the value of \( z \) is \( 4450 \, \text{m} \).