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Chapter 20: Problem 37

An Otto engine has a maximum efficiency of \(20.0 \%\) find the compression ratio. Assume that the gas is diatomic.

Short Answer

Expert verified
Answer: The compression ratio of the Otto engine is approximately 2.378.

Step by step solution

01

Write down the efficiency formula for an ideal Otto cycle

The efficiency of an ideal Otto cycle is given by: \(\eta = 1 - \frac{1}{r^{(\gamma - 1)}}\) where \(\eta\) represents the efficiency of the engine, \(r\) is the compression ratio, and \(\gamma\) is the specific heat ratio of the gas.
02

Determine the specific heat ratio for diatomic gas

For a diatomic gas, the specific heat ratio \(\gamma\) is given by: \(\gamma = \frac{C_p}{C_v} = \frac{7}{5}\)
03

Rewrite the efficiency formula for an Otto engine using the given efficiency and specific heat ratio

Substitute the given efficiency and the specific heat ratio for a diatomic gas into the formula for the efficiency of an ideal Otto cycle: \(0.20 = 1 - \frac{1}{r^{(\frac{7}{5}- 1)}}\)
04

Simplify the formula and solve for the compression ratio

Simplify the formula as follows: \(0.20 = 1 - \frac{1}{r^{(\frac{2}{5})}}\) \(0.80 = \frac{1}{r^{(\frac{2}{5})}}\) Now, we need to solve for the compression ratio \(r\): \(r^{(\frac{2}{5})} = \frac{1}{0.80}\) \(r^{(\frac{2}{5})} = 1.25\) To get rid of the exponent, raise both sides of the equation to the power of \(\frac{5}{2}\): \((r^{(\frac{2}{5})})^{\frac{5}{2}} = (1.25)^{\frac{5}{2}}\) \(r = 1.25^{\frac{5}{2}}\)
05

Calculate the compression ratio

Perform the calculation to find the value of \(r\): \(r = 1.25^{\frac{5}{2}} \approx 2.378\) Thus, the compression ratio of the Otto engine is approximately \(2.378\).

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Most popular questions from this chapter

Which of the following processes (all constanttemperature expansions) produces the most work? a) An ideal gas consisting of 1 mole of argon at \(20^{\circ} \mathrm{C}\) expands from \(1 \mathrm{~L}\) to \(2 \mathrm{~L}\). b) An ideal gas consisting of 1 mole of argon at \(20^{\circ} \mathrm{C}\) expands from \(2 \mathrm{~L}\) to \(4 \mathrm{~L}\). c) An ideal gas consisting of 2 moles of argon at \(10^{\circ} \mathrm{C}\) expands from \(2 \mathrm{~L}\) to \(4 \mathrm{~L}\). d) An ideal gas consisting of 1 mole of argon at \(40^{\circ} \mathrm{C}\) expands from \(1 \mathrm{~L}\) to \(2 \mathrm{~L}\) e) An ideal gas consisting of 1 mole of argon at \(40^{\circ} \mathrm{C}\) expands from \(2 \mathrm{~L}\) to \(4 \mathrm{~L}\).

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