When air expands adiabatically (without gaining or losing heat), its pressure and volume are related by the equation where is a constant. Suppose that at a certain instant the volume is cubic centimeters and the pressure is kPa and is decreasing at a rate of kPa/minute. At what rate in cubic centimeters per minute is the volume increasing at this instant?
To find the rate at which the volume is increasing, we can differentiate the given equation that relates pressure and volume adiabatically.
The equation is:
PV^γ = constant
where P is the pressure, V is the volume, and γ is a constant.
Differentiating both sides of the equation with respect to time (t), we get:
P(dV/dt) + V(dP/dt) = 0
Where (dV/dt) represents the rate at which the volume is changing and (dP/dt) represents the rate at which the pressure is changing.
Given information:
Volume (V) = cubic centimeters
Pressure (P) = kPa
(dP/dt) = kPa/minute
(dP/dt) = -kPa/minute (as the pressure is decreasing)
We need to find (dV/dt), the rate at which the volume is increasing.
Now, substitute the given values into the equation:
P(dV/dt) + V(dP/dt) = 0
(P)(dV/dt) + (V)(-kPa/minute) = 0
Plugging in the values:
(kPa)(dV/dt) + (cubic centimeters)(-kPa/minute) = 0
Now, isolate (dV/dt) to find the rate at which the volume is increasing:
(kPa)(dV/dt) = (cubic centimeters)(kPa/minute)
(dV/dt) = (cubic centimeters)(kPa/minute) / (kPa)
(dV/dt) = cubic centimeters/minute
Therefore, at this instant, the volume is increasing at a rate of cubic centimeters per minute.