$\frac{d{}_d}{dt}=\frac{(m^{})}{V}$

$\frac{d{P}_d}{dt}=\frac{(m^{})}{V}{a}_a^2$

where $m^{\text{'}}$ is the mass flow rate and the speed of sound in the atmospheric air is $a_a$

A small hole is perforated on the wall of a chamber of volume V that is initially void

of gas. The hole resembles a convergent nozzle of minimum cross$-$sectional area. Air from the

atmosphere flows through the hole filling up the chamber, as shown in the figure below. The

atmosphere is at pressure Pa and temperature Ta $.$

Show that mass and total internal$-$energy conservation constraints require that the time evolution

of the density d and pressure Pd of the gas in the chamber are given by