A rigid tank (volume V) containing an ideal gas is initially at T₁ and P₁. At time zero, an exit pipe (area A) is opened and gas flows out of the tank at velocity V = K(P – Patm)1/2, where P is the absolute pressure in the tank, Patm is the pressure of the atmosphere outside the tank, and K is a constant. The temperature of the gas in the tank is maintained at T₁ during the process. The pressure and the temperature of the gas exiting the tank are Patm and T₁, respectively.

Assume that, inside the tank, the pressure and specific volume do not vary with position.

A. Derive a differential equation for the tank pressure P as a function of time t.

B. Determine the time required for the tank pressure to reach P_atm.