Q: 1 mm3 of a gas is compressed at 1 atmospheric pressure and temperature 27°C to 627°C. What is the final pressure under adiabatic condition (γ for the gas = 1.5)
(a) 27 × 105 N⁄m2
(b) 80 × 105 N⁄m2
(c) 36 × 105 N⁄m2
(d) 56 × 105 N⁄m2
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Ans: (a)
Sol: P1 = 1 atm , T1 = 27°C = 27 + 273 = 300 K
T2 = 627°C = 627 + 273 = 900 K, P2 = ?
$\large \frac{T_1^{\gamma}}{P_1^{\gamma -1}} = \frac{T_2^{\gamma}}{P_2^{\gamma -1}} $
$\large (\frac{P_2}{P_1})^{\gamma -1} = (\frac{T_2}{T_1})^{\gamma} $
$\large \frac{P_2}{P_1} = (\frac{T_2}{T_1})^\frac{\gamma}{\gamma -1} $
$\large \frac{P_2}{P_1} = (\frac{900}{300})^\frac{3/2}{\frac{3}{2} -1} $
$\large \frac{P_2}{P_1} = 3^3 = 27 $
P2 = 27 P1
= 27 × 1 = 27 atm. = 27 × 105 N⁄m2