Initially two gas samples 1 and 2 are at the same condition. The volume of the two are halved, one isothermally and the other adiabatically. What is the relation between the final pressure P1 and P2 ?

Question

Initially two gas samples 1 and 2 are at the same condition. The volume of the two are halved, one isothermally and the other adiabatically. What is the relation between the final pressure P1 and P2 ? (A) P1 = P2 (B) P1 > P2 (C) P2 > P1 (D) Cannot be determined

Tardigrade Admin · Accepted Answer

For isothermal process PV = K (constant) P =( K / V ) ldots(1) =( K / V / 2)=2 K For adiabatic process PV &gamma;= K (constant) ∴ P =( K / V &gamma;) ldots . .(2) =( K /( V / 2)&gamma;)- K (2&gamma;) From Eqs. (1) and (2), we have P 2 > P 1

Q. Initially two gas samples $1$ and $2$ are at the same condition. The volume of the two are halved, one isothermally and the other adiabatically. What is the relation between the final pressure $P_{1}$ and $P_{2}$ ?

$P_{1} = P_{2}$

$P_{1} > P_{2}$

$P_{2} > P_{1}$

Cannot be determined

For isothermal process
$P V = K$ (constant)
$P = \frac{K}{V} \dots (1)$
$= \frac{K}{V /2} = 2 K$
For adiabatic process
$P V^{γ} = K$ (constant)
$∴ P = \frac{K}{V ^{γ}} \dots .. (2)$
$= \frac{K}{( V /2 ) ^{γ}} - K (2^{γ})$
From Eqs. (1) and (2), we have
$P_{2} > P_{1}$

Q. Initially two gas samples 1 and 2 are at the same condition. The volume of the two are halved, one isothermally and the other adiabatically. What is the relation between the final pressure P1​ and P2​ ?

P1​=P2​

P1​>P2​

P2​>P1​