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

Question

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.

Accepted Answer

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:

[latex]P_i V_i^\gamma=P_f V_f^\gamma[/latex] (20.39)

[latex]\begin{aligned}P_f & =P_i\left(\frac{V_i}{V_f} ight)^\gamma=(1.00 ext{ atm})\left(\frac{800.0 cm^3}{60.0 cm^3} ight)^{1.40} \& =37.6 ext{ atm}\end{aligned}[/latex]

Use the ideal gas law to find the final temperature:

[latex]\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 ext{ atm})\left(60.0 cm^3 ight)}{(1.00 ext{ atm})\left(800.0 cm^3 ight)}(293 K) \& =826 K=553^{\circ} C\end{aligned}[/latex]

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.

Question 20.3: A Diesel Engine Cylinder Air at 20.0°C in the cylinder of a ...

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