A monatomic ideal gas initially at $27^{\circ}\mathrm{C}$ is compressed adiabatically to $1/27$ of i

Question

A monatomic ideal gas initially at $27^{\circ}\mathrm{C}$ is compressed adiabatically to $1/27$ of its original volume. The final temperature after compression will be

Professor PC Thomas Classes · Answer

(d) : Initial temperature $T_1 = 27^{\circ}\mathrm{C} = 300 \mathrm{~K}$. For a monatomic gas, $\gamma = 5/3$. The adiabatic relation is $T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1}$. Thus, $T_2 = T_1 (V_1 / V_2)^{\gamma-1} = 300 	imes (27)^{5/3 - 1} = 300 	imes (27)^{2/3}$. Since $27^{2/3} = (3^3)^{2/3} = 3^2 = 9$, the final temperature $T_2 = 300 	imes 9 = 2700 \mathrm{~K}$. In Celsius, this is $2700 - 273 = 2427^{\circ}\mathrm{C}$.