Question

Consider the following statements about vertical profile of atmosphere. Choose the correct statements.
A. Mean free path increases with increase in the height above sea.
B. Mean free path is independent of the height above sea.
C. Pressure decreases with increase in the height above sea.
D. Density decreases with increase in the height above sea.
E. Density increases with increase in the height above sea.
 

• A. A, C only
• B. B, C, E only
• C. B, C, D only
• D. A, C, D only

Hint:

Remember: Both pressure and density decrease with altitude (governed by the hydrostatic equation). Mean free path increases with altitude because fewer molecules per unit volume means each molecule travels farther before colliding.

Answer 1

The Correct Option is D

Concept: The vertical profile of the atmosphere describes how physical quantities vary with altitude. Key formulae used:
• Hydrostatic equation: \(\dfrac{dp}{dz} = -\rho g\), so pressure decreases with height.
• Ideal gas law: \(p = \rho R T\), so density \(\rho\) also decreases with height.
• Mean free path: \(\lambda = \dfrac{1}{\sqrt{2}\, n \pi d^2}\), where \(n\) is number density. As height increases, \(n\) decreases, so \(\lambda\) increases. These relationships allow us to evaluate each statement about the atmospheric vertical profile.

Step 1: Evaluate Statements A and B -- Mean Free Path.
Mean free path is given by: \[ \lambda = \frac{1}{\sqrt{2}\, n \pi d^2} \] As height increases, air becomes thinner, so number density \(n \downarrow\), which means \(\lambda \uparrow\).
\(\therefore\) Statement A is TRUE. Statement B is FALSE.

Step 2: Evaluate Statement C -- Pressure.
From the hydrostatic equation: \[ \frac{dp}{dz} = -\rho g < 0 \] Pressure decreases with increasing height.
\(\therefore\) Statement C is TRUE.

Step 3: Evaluate Statements D and E -- Density.
From the ideal gas law \(p = \rho R T\), since both \(p\) and \(T\) decrease with height in the troposphere, density \(\rho\) also decreases.
\(\therefore\) Statement D is TRUE. Statement E is FALSE.

Step 4: Select the correct combination.
Correct statements: A, C, D \(\Rightarrow\) Option (4).