Chemical & Aerospace Engineering (Non-Linear Fluids)

An aerospace propulsion engineer is modeling the expansion kinetics of a dense supercritical hydrocarbon propellant in a high-pressure rocket combustion chamber. Because the fluid operates near its critical point, it deviates significantly from ideal gas laws and obeys a simplified Van der Waals equation of state:

where $P$ represents pressure (in atmospheres) and $V$ represents the molar volume (in liters per mole, $\text{L/mol}$).

1. Differentiate the implicit equation of state with respect to volume $V$ using the Product Rule, treating pressure $P$ as an implicit function $P(V)$.
2. Derive a symbolic expression for the rate of pressure change with respect to volume ($\frac{dP}{dV}$).
3. Calculate the instantaneous rate of pressure change ($\frac{dP}{dV}$) when the gas expands through a volume of $V = 3\text{ L/mol}$ at an internal pressure of $P = 5\text{ atm}$.