When an ideal gas with pressure P and volume V is compressed isothermally to one fourth of its volume, the pressure is P1. When the same gas is compressed polytropically according to the equation PV1.5 = constant to one fourth of its initial volume, the pressure is P2. The ratio (P1/P2) is

Question

When an ideal gas with pressure P and volume V is compressed isothermally to one fourth of its volume, the pressure is P1. When the same gas is compressed polytropically according to the equation PV1.5 = constant to one fourth of its initial volume, the pressure is P2. The ratio (P1/P2) is (A) (1/2) (B) (1/21.5) (C) 2 (D) 21.5

Tardigrade Admin · Accepted Answer

For isothermal process: PV = constant or PV=P1 (V/4) ⇒ P1=4P For polytropic process: PV1.5 = constant PV1.5=P2((V/4))1.5 ⇒ P2=(22)3/2 P=8 P ∴ (P1/P2)=(1/2)

$\frac{1}{2}$

$\frac{1}{2 ^{1.5}}$

$2$

$2^{1.5}$

For isothermal process : $P V =$ constant
or $P V = P_{1} \frac{V}{4} \Rightarrow P_{1} = 4 P$
For polytropic process: $P V^{1.5} =$ constant
$P V^{1.5} = P_{2} (\frac{V}{4})^{1.5}$
$\Rightarrow P_{2} = (2^{2})^{3/2} P = 8 P$
$∴ \frac{P _{1}}{P _{2}} = \frac{1}{2}$

21​

21.51​

2

21.5

For isothermal process : PV= constant or PV=P1​4V​⇒P1​=4P For polytropic process: PV1.5= constant PV1.5=P2​(4V​)1.5 ⇒P2​=(22)3/2P=8P ∴P2​P1​​=21​

$\frac{1}{2}$

$\frac{1}{2 ^{1.5}}$

$2$

$2^{1.5}$

For isothermal process : $P V =$ constant
or $P V = P_{1} \frac{V}{4} \Rightarrow P_{1} = 4 P$
For polytropic process: $P V^{1.5} =$ constant
$P V^{1.5} = P_{2} (\frac{V}{4})^{1.5}$
$\Rightarrow P_{2} = (2^{2})^{3/2} P = 8 P$
$∴ \frac{P _{1}}{P _{2}} = \frac{1}{2}$