Gas at constant T and P is contained in a supply line connected through a valve to a closed tank containing the same gas at a lower pressure. The valve is opened to allow flow of gas into the tank, and then is shut again.

(a) Develop a general equation relating n₁ and n₂, the moles (or mass) of gas in the tank at the beginning and end of the process, to the properties U₁ and U₂, the internal energy of the gas in the tank at the beginning and end of the process, and H′, the enthalpy of the gas in the supply line, and to Q, the heat transferred to the material in the tank during the process.

(b) Reduce the general equation to its simplest form for the special case of an ideal gas with constant heat capacities.

(c) Further reduce the equation of (b) for the case of n₁ = 0.

(d) Further reduce the equation of (c) for the case in which, in addition, Q = 0.

(e) Treating nitrogen as an ideal gas for which C_P = (7/2)R, apply the appropriate equation to the case in which a steady supply of nitrogen at 25°C and 3 bar flows into an evacuated tank of 4m³ volume, and calculate the moles of nitrogen that flow into the tank to equalize the pressures for two cases:

1. Assume that no heat flows from the gas to the tank or through the tank walls.

2. Assume that the tank weighs 400 kg, is perfectly insulated, has an initial temperature of 25°C, has a specific heat of 0.46 kJ · kg⁻¹·K⁻¹, and is heated by the gas so as always to be at the temperature of the gas in the tank.