Question

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

Short answer

Answer: The compression ratio for the Otto engine with a maximum efficiency of 20.0% is approximately 2.639.

Step-by-step solution

Set the values

Given efficiency: \(\eta_{otto} = 0.2\) Heat capacity ratio for a diatomic gas: \(\gamma = 1.4\) Now, we will use the efficiency formula and solve for the compression ratio:

Solve for compression ratio

We have the formula: $$\eta_{otto} = 1 - \frac{1}{(r_{compression})^{\gamma-1}}$$ Plug in the given values: $$0.2 = 1 - \frac{1}{(r_{compression})^{1.4 - 1}}$$ Now, let's solve for \(r_{compression}\): First, subtract \(1-0.2\) to get \(0.8\): $$0.8 = \frac{1}{(r_{compression})^{0.4}}$$ Next, take the reciprocal of both sides of the equation to isolate the term \((r_{compression})^{0.4}\): $$\frac{1}{0.8} = (r_{compression})^{0.4}$$ Now, take the power of both sides with the reciprocal of \(0.4\) which is \(2.5\): $$\left(\frac{1}{0.8}\right)^{2.5} = (r_{compression})^{0.4 \times 2.5}$$ This results in: $$r_{compression} = \left(\frac{1}{0.8}\right)^{2.5}$$ Now, calculate the numerical value of the compression ratio: $$r_{compression} \approx \left(\frac{1}{0.8}\right)^{2.5} \approx 2.639 $$

Interpret the result

The compression ratio for the Otto engine with a maximum efficiency of 20.0% is approximately 2.639.