Question

When air expands adiabatically (without gaining or losing heat), its pressure P and volume V are related by the equation PV1.4 = C, where C is a constant. Suppose that at a certain instant the volume is 580 cm3 and the pressure is 83 kPa and is decreasing at a rate of 3 kPa/min. At what rate is the volume increasing at this instant? (Round your answer to the nearest whole number.)

Answer 1

To calculate the rate at which volume increases we use the given relation PV^1.4 = C, differentiate with respect to time, and use the known values of volume, pressure, and the rate of pressure change to solve for dV/dt.

Step-by-step explanation:

When air expands adiabatically, the pressure P and volume V are related by the equation PV1.4 = C, with C being a constant. Given that the volume is 580 cm3 and the pressure is 83 kPa and decreasing at a rate of 3 kPa/min, we can find the rate at which the volume is increasing by differentiating both sides of the equation with respect to time.

Let's differentiate the equation PV1.4 = C implicitly with respect to time t. This yields 1.4PV0.4dV/dt + V1.4dP/dt = 0. We can substitute the known values and solve for dV/dt, the rate of change of volume per minute. By doing this, we can conclude if the calculated volume change makes sense based on Boyle's Law, which states that for a fixed amount of an ideal gas kept at a fixed temperature, pressure and volume are inversely proportionate.