Question

# An ideal gas (1.0 mol) is the working substance in an engine that operates on the cycle shown in Figure. Processes BC and DA are reversible and adiabatic. (a) Is the gas monatomic, diatomic, or polyatomic?(b) What is the engine efficiency?

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An ideal gas (1.0 mol) is the working substance in an engine that operates on the cycle shown in Figure. Processes BC and DA are reversible and adiabatic. (a) Is the gas monatomic, diatomic, or polyatomic?(b) What is the engine efficiency?

2021-03-25
a) for process $$\displaystyle{D}\to{A}$$ gives
$$\displaystyle{p}_{{D}}{{V}_{{D}}^{\gamma}}={p}_{{A}}{{V}_{{A}}^{\gamma}}\rightarrow{\frac{{{p}_{{0}}}}{{{32}}}}{\left({8}{V}_{{0}}\right)}^{\gamma}={p}_{{0}}{{V}_{{0}}^{\gamma}}$$
which leads to $$\displaystyle{8}^{\gamma}={32}\Rightarrow\gamma=\frac{{5}}{{3}}$$. The result implies the gas is monatomic. b) The input heat is that absorbed during process $$\displaystyle{A}\to{B}$$
$$\displaystyle{Q}_{{H}}={n}{C}_{{p}}\triangle{T}={n}{\left({\frac{{{5}}}{{{2}}}}{R}\right)}{T}_{{A}}{\left({\frac{{{T}_{{B}}}}{{{T}_{{A}}}}}-{1}\right)}={n}{R}{T}_{{A}}{\left({\frac{{{5}}}{{{2}}}}\right)}{\left({2}-{1}\right)}={p}_{{0}}{V}_{{0}}{\left({\frac{{{5}}}{{{2}}}}\right)}$$
and the exhaust heat is that liberated during process $$\displaystyle{C}\to{D}$$
$$\displaystyle{Q}_{{L}}={n}{C}_{{p}}\triangle{T}={n}{\left({\frac{{{5}}}{{{2}}}}{R}\right)}{T}_{{D}}{\left({1}-{\frac{{{T}_{{L}}}}{{{T}_{{D}}}}}\right)}={n}{R}{T}_{{D}}{\left({\frac{{{5}}}{{{2}}}}\right)}{\left({1}-{2}\right)}=-{\frac{{{1}}}{{{4}}}}{p}_{{0}}{V}_{{0}}{\left({\frac{{{5}}}{{{2}}}}\right)}$$
where in the last step we have used the fact that $$\displaystyle{T}_{{D}}={\frac{{{1}}}{{{4}}}}{T}_{{A}}$$ (from the gas law in ratio form). Therefore, equation 20-12 leads to
$$\displaystyle\epsilon={1}-{\left|{\frac{{{Q}_{{L}}}}{{{Q}_{{H}}}}}\right|}={1}-{\frac{{{1}}}{{{4}}}}={0.75}={75}\%$$