Question

A parcel of dry air having an initial temperature of 30 °C at 1000 hPa level is lifted adiabatically. At what pressure (in hPa) its density reduces by half? (Take $C_p/C_v = 0.71$, rounded off to two decimal places).

Hint:

Use $\rho \propto p^{1/\gamma}$ for adiabatic processes. Density halves when pressure drops by about 62.5%.

Answer 1

Correct Answer: 611.3

Step 1: Recall Poisson’s relation for adiabatic process.
\[ p V^\gamma = \text{constant}, \gamma = \frac{C_p}{C_v} = 1.41 \]
Step 2: Relating density and pressure.
Density $\rho = \frac{p}{RT}$. For adiabatic process: \[ T \propto p^{(\gamma-1)/\gamma} \] So, \[ \rho \propto \frac{p}{T} \propto p^{1/\gamma} \]
Step 3: Density condition.
If $\rho_2 = \tfrac{1}{2}\rho_1$: \[ \frac{\rho_2}{\rho_1} = \left(\frac{p_2}{p_1}\right)^{1/\gamma} = \frac{1}{2} \] \[ \left(\frac{p_2}{1000}\right)^{1/1.41} = 0.5 \] \[ \frac{p_2}{1000} = 0.5^{1.41} = 0.375 \] \[ p_2 = 375 \, hPa \] Final Answer: \[ \boxed{375.00 \, hPa} \]