The air in an automobile engine at 20°C is compressed from an initial pressure of 1.0 atm and a volume of 200 cm³ to a volume of 20 cm³. Find the temperature if the air behaves like an ideal gas (γ = 1.4) and the compression is adiabatic. A) 730°C B) 460°C C) 25°C D) 50°C E) 20°C

Question

Quizplus · Accepted Answer

Since the compression is adiabatic, we can use the equation:

$PV^{\gamma} = $constant

where $\gamma = 1.4$ for air.

At the initial state, we have:

$P_1V_1^{\gamma} = P_2V_2^{\gamma}$

Substituting the given values, we have:

$1.0	imes10^5 	imes (200	imes10^{-6})^{1.4} = P_2 	imes (20 	imes 10^{-6})^{1.4}$

Solving for $P_2$ gives:

$P_2 \approx 3.33 	imes 10^{5}$ Pa

Next, we can use the equation:

$T_2 = T_1 	imes \left(\frac{P_2}{P_1}ight)^{\frac{\gamma - 1}{\gamma}}$

where $T_1 = 20$°C $= 293$ K.

Substituting the values, we have:

$T_2 = 293 	imes \left(\frac{3.33	imes10^5}{1.0	imes10^5}ight)^{\frac{1.4-1}{1.4}}$

Simplifying gives:

$T_2 \approx 732$ K $= 459$°C

Therefore, the temperature of the air at the final state is approximately 459°C, which is closest to choice B.

The Air in an Automobile Engine at 20°C Is Compressed