A sealed carbon-fiber tank contains compressed air that behaves as an ideal gas

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A sealed carbon-fiber tank contains compressed air that behaves as an ideal gas. At a particular instant the state of the gas is described by • Pressure

P=4.80\text{ MPa}

and decreasing at a rate of

\dfrac{dP}{dt}=-0.060\text{ MPa\,s}^{-1}

, • Temperature

T=300\text{ K}

and increasing at a rate of

\dfrac{dT}{dt}=1.8\text{ K\,s}^{-1}

, • Volume

V=13.2\text{ L}

(the tank walls are flexible enough to expand or contract). Assume the amount of gas (in moles) remains constant and the gas obeys

PV=nRT

. (a) Derive an expression for

\dfrac{dV}{dt}

in terms of

P,\,V,\,T,\,\dfrac{dP}{dt},\,

and

\dfrac{dT}{dt}

. Show all differentiation steps and clearly indicate how the ideal-gas relation is used. (b) Using the numerical data provided, calculate

\dfrac{dV}{dt}

\text{L\,s}^{-1}

. Express your answer to three significant figures. (c) State whether the volume of the tank is increasing or decreasing at this instant and justify your conclusion based on the sign of

\dfrac{dV}{dt}

. (d) The tank engineer wants the volume to remain momentarily constant despite the falling pressure. Determine the temperature rate of change

\dfrac{dT}{dt}

that would accomplish this, assuming the pressure rate remains

-0.060\text{ MPa\,s}^{-1}

. Express your result in

\text{K\,s}^{-1}

Related theory

Browse the problem library