When air expands adiabatically (without gaining or losing heat), its pressure (P

Question

When air expands adiabatically (without gaining or losing heat), its pressure (P) and volume (V) are related by the equation PV^(1.4)=C where (C) is a constant. Suppose that at a certain instant the volume is 370 cubic centimeters and the pressure is 97 kPa and is decreasing at a rate of 9 kPa/minute. At what rate in cubic centimeters per minute is the volume increasing at this instant? (Pa stands for Pascal -- it is equivalent to one Newton/(meter squared); kPa is a kiloPascal or 1000 Pascals. )

Explanation / Answer

PV^1.4 = C, V = 370 cm^3, P = 91 KPa and dP/dt = KPa/min Since, P and V are implicitly related to time t, we have to differentiate using IMPLICIT DIFFERENTIATION. PV^1.4=C V^1.4 dP/dt + (1.4 V^0.4) P dV/dt = 0 .., V = 370 cm^3, P = 97 KPa and dP/dt = - 9 KPa/min 370^1.4 (- 9) + (1.4)(370^0.4) (97) dV/dt = 0 - 35458.87= - 1446.04dV/dt dV/dt = 24.52 cm^3/min

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