Question

The pressure and volume of an expanding gas are related by the formula PV^b=C, where b and C are constants (this holds in adiabatic expansion, with or without loss).
Find dP/dt if b=1.5, P=7 kPa, V=110 cm^3, and dV/dt=40 cm^3/min.

(Use symbolic notation and fractions where needed.)

Answer 1

The dP/dt of the adiabatic expansion is -42/11 kPa/min

How to calculate dP/dt in an adiabatic expansion?

An adiabatic process is a process in which there is no exchange of heat from the system to its surrounding neither during expansion nor during compression

Given b=1.5, P=7 kPa, V=110 cm³, and dV/dt=40 cm³/min

PVᵇ = C

Taking logs of both sides gives:

ln P + b ln V = ln C

Taking partial derivatives gives:

$\frac{1}{P}\frac{∂P}{∂t} + \frac{b}{V}\frac{∂V}{∂t} = 0$

Substitutituting the values b, P, V and dV/dt into the derivative above:

1/7 x dP/dt +  1.5/110 x 40 = 0

1/7 x dP/dt +  6/11 = 0

1/7 x dP/dt = - 6/11

dP/dt = - 6/11 x 7

dP/dt = -42/11 kPa/min

Therefore, the value of  dP/dt is -42/11 kPa/min

Learn more about adiabatic expansion on:

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