Mass balance and equation of state.

You have a $1m^3$ container filled with $N_2$ that is at an initial pressure $P=5$ bars and temperature of $T=300K.$ You can consider the $N_2$ to be an ideal gas $)$ for the entirety of this problem. A leak develops in the container and you will learn in transport that the leak rate often depends linearly on the pressure difference between the inside and outside, or mathematically $Q=K(P-P_0),$

where $Q$ is the flow rate $($moles$/$sec$),P_0$ is the pressure outside the tank, and $K$ is the leak rate coefficient. In our case we will take a leak rate of $K=0\mathrm{.}01$mole$\frac{s}{sec}$ bar $.$ The outside pressure is always atmospheric, $P_0=1\mathrm{.}01$ bar $($you can approximate $P_0$ to be $1$ bar$).$

a$)$ Calculate the initial flow rate out of the tank $($moles $\frac{?}{sec}).$

b$)$ If the leak continues at this flow rate at all times $($i$.$e$.$ it does not slow down with the decreasing pressure$),$ calculate the time it takes for the pressure in the tank to reach $P=3$ bars. Hint: Do a molar balance, and calculate the number of moles you start with at $P=5$ bars, and how much is left in tank at $P=3$ bars.

c$)$ Part b is a decent estimate, but not the best estimate as the flow rate will slow down as the pressure is decreased. Using a molar balance approach with the instantaneous molar leak rate, calculate the amount of time it takes to reach $P=3$ bars. You will need to set up your equations correctly, and perform an integration.