The microchip diffusion furnace shown below contains air, which may be considered an ideal gas. The vacuum pump is pumping air out while the heating coils in the tank hold the temperature in the tank constant at $20C.$ The volumetric flow rate at the inlet of the pump, independent of pressure, is $1f\frac{t^3}{m}in.$ The tank has a leak of $0\mathrm{.}0001lb\frac{m}{m}in$ of air in$.$

Calculate the final density of the system after $50$min if the initial pressure is $1$atm. Given: $R=0\mathrm{.}287\frac{J}{g}.K$