First slide

Thermodynamic processes

Question

For an adiabatic change, the value of $\frac{dP}{P}$ is equal to ….(dv=change in volume, $δ = \frac{C_{p_{}}}{C_{v_{}}}$ )

Moderate

A

$δ^{1/2} . \frac{dV}{V}$
B

$\frac{-dV}{V}$
C

$-δ \frac{dV}{V}$
D

$- δ^{2} . \frac{dV}{V}$

Solution

${in an adiabatic process relation between pressure P and volume V is PV}^{δ} =Constant$
$d i f f e r e n t i a t e a b o v e e q u a t i o n w i t h r e s p e c t t o P \frac{dP}{dP} {.V}^{δ} +P. {\frac{dV}{dP}}^{δ} =0$
${1.V}^{δ} {+P.δ.V}^{δ-1} . \frac{dV}{dP} = 0 \Rightarrow V^{δ} {=-P.δ.V}^{δ-1} . \frac{dV}{dP}$
$\Rightarrow \frac{V^{δ}}{{δ.V}^{δ-1} .dV} = \frac{-P}{dP} \Rightarrow \frac{V^{δ-δ+1}}{δ.dV} = \frac{-P}{dP}$
$i n v e r t t h e r a t i o \Rightarrow -δ. \frac{dV}{V} = \frac{dP}{P}$
$⇒ \frac{dP}{P} = -δ. \frac{dV}{V}$

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