A gas is compressed adiabatically,it satisfies the equation PV^(1.4 )= k, where k is a constant,At what given instant the pressure is 40atms,volume is 28litres, and volume is decreasing by 2litre/min.

Question

Hi there, I am trying to solve this problem but I'm confuse, can anyone show me how to solve it?

I want to know at what rate is the pressure changing

Answer 1

PV^1.4 = k

Differentiate with respect to time. P and V are both functions of time, and so you need to use the product rule from Differential Calculus.

(dP/dt)V^1.4 + 1.4PV^0.4(dV/dt) = 0 The right-hand side is 0 since k is a constant.

At some instant, we are told that P = 40 atm, V = 28 L, dV/dt = -2.0 L/min where the negative sign because volume is decreasing in time. We do not know dP/dt, and we are being asked to find this.

Rearrange the last differential equation.

(dP/dt)V^1.4 = -1.4PV^0.4(dV/dt)

Multiply through by V^-1.4.

dP/dt = -1.4PV^-1(dV/dt)

dP/dt = -1.4(P/V)(dV/dt)

dP/dt = -1.4(40 atm / 28 L)(-2.0 L/min)

dP/dt = 4.0 atm / min