Question

As per the International Standard Atmosphere (ISA), which statement about density variation with altitude in an isothermal layer is correct?

• A. remains constant
• B. increases linearly
• C. decreases linearly
• D. decreases exponentially

Hint:

In ISA: isothermal layer \(\Rightarrow\) \(T=\) const \(\Rightarrow\) \(p,\rho \propto e^{-kh}\). Linear variation occurs only in gradient (non-isothermal) layers for temperature, not for density.

Answer 1

The Correct Option is D

Step 1: Hydrostatic balance in an isothermal layer.
For a static atmosphere, \( dp/dh = -\rho g \). With temperature \(T\) constant (isothermal) and ideal gas \( p=\rho R T \Rightarrow \rho = p/(RT)\). Substitute into the hydrostatic equation: \[ \frac{dp}{dh} = -\frac{p}{RT}g \quad \Rightarrow \quad \frac{1}{p}\,dp = -\frac{g}{RT}\,dh. \] Step 2: Integrate and infer density law.
Integrating from a reference \(h_0\) to \(h\): \[ p(h) = p(h_0)\,\exp\!\left[-\frac{g}{RT}\,(h-h_0)\right]. \] Since \( \rho(h)=p(h)/(RT) \) with \(T\) constant, \[ \rho(h) = \rho(h_0)\,\exp\!\left[-\frac{g}{RT}\,(h-h_0)\right]. \] Thus density decays \emph{exponentially} with altitude in an isothermal ISA layer. \[ \boxed{\text{Density decreases exponentially with altitude.}} \]