1 mm^3 of a gas is compressed at 1 atmospheric pressure and temperature 27°C to 627°C. What is the final pressure under adiabatic condition

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

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