A 100 m rigid tank at 25 C initially contains 1 atm of argon (MW= 40 g/mol). We want to know how long it will take to fill the tank with an additional 1,000 kg of argon. Assume for this problem isothermal operation (25 C). The inlet valve to the tank operates in a way such that when it is opened, the flow rate into the tank is inversely proportional to the pressure of the tank, i.e., n = k/P, where n is in mol/s, P is the pressure in the tank (in atm), and k is given by the valve manufacturer as 50 (mol atm)/s. Argon can be treated as an ideal gas under these conditions. All pressures are absolute. a) (2 pt) What is the final pressure, in atm, of the tank assuming no leaks occur and the 1,000 kg of Ar is added isothermally? b) (3 pt) Set up, but do not solve, a differential equation that describes the accumulation of either mols or kg of Ar in the tank as a function of time. Your answer should only contain constants and the total moles of gas in the tank at any given time, N(). State the initial condition. c) (2 pt) Integrating your answer from part b, how long, in seconds, does it take to fill the tank with 1,000 kg Ar? Immediately after transferring the 1,000 kg of Ar to the tank, we notice that the pressure gauge reads 6.8 atm (absolute), which is below the value I expected (as calculated in part b). I suspect that the tank is leaking. It is known that any leak rate will be proportional to the pressure in the tank, so nek = P, where & is proportionality constant related to the size of the leak (for no leak, a would be 0 mol s atm). d) (3 pt) Set up a new mass or mol balance for Ar in the tank during the filling process, including the leak. What are the initial and final conditions for this balance? e) (7 pt) Integrate this balance and solve the resulting nonlinear equation to find the value for x (in mol s atm'). f) (3 pt) Using the value of a calculated in part e., from the end of filling (i.e., when the inlet valve is shut off), how long would it take to lose an additional 500 kg of Ar from the leak? If you can't find a value for x in part f., just use 1 mol/(s atm).