Filling gas tank at constant $T$ and constant $V.$ You have a tank that has a diameter of $4ft$ and a length of $6ft$ and contains the following gases: $N_2,H_2,N{H}_3$ and argon at mol fractions of $0\mathrm{.}2,0\mathrm{.}4,0\mathrm{.}36$ and $0\mathrm{.}04,$ respectively. At time equals zero, the pressure in the tank is $1$atm and the temperature is $298K.$ You start adding gas at $0\mathrm{.}5\frac{m^3}{m}in$ and remove heat, so the temperature stays the same at $298K.$ You can also assume that the volume is constant and that the gas continues to act as an ideal gas. You can assume that the pressure entering the tank is the same as the pressure inside the tank and both are changing with time.

a$.$ Write out an equation for the pressure with time. The maximum pressure allowed is $1000$ psig $($this is gauge pressure$).$ How long will it take to reach this pressure?

b$.$ You noticed that the gas is leaking out and the rate of gas leaving is equal to kP $(n_{\text{o}\text{u}\text{t}}^{}=kP)$ where $k$ is a constant and equals $2\mathrm{.}0{10}^{-7}mo\frac{l}{P}a-s.$ Write out a new material balance and solve for pressure vs$.$ time.

c$.$ How long will it take now to reach $1000$ psig?