Question 20.3: A Diesel Engine Cylinder Air at 20.0°C in the cylinder of a ...
A Diesel Engine Cylinder
Air at 20.0°C in the cylinder of a diesel engine is compressed from an initial pressure of 1.00 atm and volume of 800.0 cm³ to a volume of 60.0 cm³. Assume air behaves as an ideal gas with γ = 1.40 and the compression is adiabatic. Find the final pressure and temperature of the air.
Conceptualize Imagine what happens if a gas is compressed into a smaller volume. Our discussion above and Figure 20.8 tell us that the pressure and temperature both increase.
Categorize We categorize this example as a problem involving an adiabatic process.
Analyze Use Equation 20.39 to find the final pressure:
P_i V_i^\gamma=P_f V_f^\gamma (20.39)
\begin{aligned}P_f & =P_i\left(\frac{V_i}{V_f}\right)^\gamma=(1.00 \text{ atm})\left(\frac{800.0 cm^3}{60.0 cm^3}\right)^{1.40} \\& =37.6 \text{ atm}\end{aligned}Use the ideal gas law to find the final temperature:
\begin{aligned}\frac{P_i V_i}{T_i} & =\frac{P_f V_f}{T_f} \\T_f & =\frac{P_f V_f}{P_i V_i} T_i=\frac{(37.6 \text{ atm})\left(60.0 cm^3\right)}{(1.00 \text{ atm})\left(800.0 cm^3\right)}(293 K) \\& =826 K=553^{\circ} C\end{aligned}Finalize The temperature of the gas increases by a factor of 826 K/293 K = 2.82. The high compression in a diesel engine raises the temperature of the gas enough to cause the combustion of fuel without the use of spark plugs.
