# 15.4 Carnot’s perfect heat engine: the second law of thermodynamics  (Page 4/7)

 Page 4 / 7

Can improved engineering and materials be employed in heat engines to reduce heat transfer into the environment? Can they eliminate heat transfer into the environment entirely?

Does the second law of thermodynamics alter the conservation of energy principle?

## Problem exercises

A certain gasoline engine has an efficiency of 30.0%. What would the hot reservoir temperature be for a Carnot engine having that efficiency, if it operates with a cold reservoir temperature of $2\text{00}\text{º}\text{C}$ ?

$\text{403}\text{º}\text{C}$

A gas-cooled nuclear reactor operates between hot and cold reservoir temperatures of $\text{700}\text{º}\text{C}$ and $\text{27}\text{.}0\text{º}\text{C}$ . (a) What is the maximum efficiency of a heat engine operating between these temperatures? (b) Find the ratio of this efficiency to the Carnot efficiency of a standard nuclear reactor (found in [link] ).

(a) What is the hot reservoir temperature of a Carnot engine that has an efficiency of 42.0% and a cold reservoir temperature of $\text{27}\text{.}0\text{º}\text{C}$ ? (b) What must the hot reservoir temperature be for a real heat engine that achieves 0.700 of the maximum efficiency, but still has an efficiency of 42.0% (and a cold reservoir at $\text{27}\text{.}0\text{º}\text{C}$ )? (c) Does your answer imply practical limits to the efficiency of car gasoline engines?

(a) $2\text{44}\text{º}\text{C}$

(b) $\text{477}\text{º}\text{C}$

(c)Yes, sinceautomobiles enginescannot gettoo hotwithout overheating,their efficiencyis limited.

Steam locomotives have an efficiency of 17.0% and operate with a hot steam temperature of $\text{425}\text{º}\text{C}$ . (a) What would the cold reservoir temperature be if this were a Carnot engine? (b) What would the maximum efficiency of this steam engine be if its cold reservoir temperature were $\text{150}\text{º}\text{C}$ ?

Practical steam engines utilize $\text{450}\text{º}\text{C}$ steam, which is later exhausted at $\text{270}\text{º}\text{C}$ . (a) What is the maximum efficiency that such a heat engine can have? (b) Since $\text{270}\text{º}\text{C}$ steam is still quite hot, a second steam engine is sometimes operated using the exhaust of the first. What is the maximum efficiency of the second engine if its exhaust has a temperature of $\text{150}\text{º}\text{C}$ ? (c) What is the overall efficiency of the two engines? (d) Show that this is the same efficiency as a single Carnot engine operating between $\text{450}\text{º}\text{C}$ and $\text{150}\text{º}\text{C}$ . Explicitly show how you follow the steps in the Problem-Solving Strategies for Thermodynamics .

(a) ${\mathit{\text{Eff}}}_{\text{1}}=1-\frac{{T}_{\text{c,1}}}{{T}_{\text{h,1}}}=1-\frac{\text{543 K}}{\text{723 K}}=0\text{.}\text{249}\phantom{\rule{0.25em}{0ex}}\text{or}\phantom{\rule{0.25em}{0ex}}\text{24}\text{.}9%\text{}$

(b) ${\mathit{\text{Eff}}}_{2}=1-\frac{\text{423 K}}{\text{543 K}}=0\text{.}\text{221}\phantom{\rule{0.25em}{0ex}}\text{or}\phantom{\rule{0.25em}{0ex}}\text{22}\text{.}1%\text{}$

(c) $\begin{array}{l}{\mathit{\text{Eff}}}_{1}=1-\frac{{T}_{\text{c,1}}}{{T}_{\text{h,1}}}⇒{T}_{\text{c,1}}={T}_{\text{h,1}}\left(1,-,{\mathit{\text{eff}}}_{1}\right)\end{array}$ $\begin{array}{l}\text{similarly,}\phantom{\rule{0.25em}{0ex}}{T}_{\text{c,2}}={T}_{\text{h,2}}\left(1-{\mathit{\text{Eff}}}_{2}\right)\end{array}$ $\begin{array}{l}\text{using}\phantom{\rule{0.25em}{0ex}}{T}_{\text{h,2}}={T}_{\text{c,1}}\phantom{\rule{0.25em}{0ex}}\text{in}\phantom{\rule{0.25em}{0ex}}\text{above}\phantom{\rule{0.25em}{0ex}}\text{equation}\phantom{\rule{0.25em}{0ex}}\text{gives}\end{array}$ $\begin{array}{l}{T}_{\text{c,2}}={T}_{\text{h,1}}\left(1-{\mathit{\text{Eff}}}_{1}\right)\left(1-{\mathit{\text{Eff}}}_{2}\right)\equiv {T}_{\text{h,1}}\left(1-{\mathrm{Eff}}_{\text{overall}}\right)\\ \therefore \left(1-{\mathrm{Eff}}_{\text{overall}}\right)=\left(1-{\mathit{\text{Eff}}}_{1}\right)\left(1-{\mathit{\text{Eff}}}_{2}\right)\\ {\mathrm{Eff}}_{\text{overall}}=1-\left(1-0\text{.}\text{249}\right)\left(1-0\text{.}\text{221}\right)=\text{41}\text{.}5%\text{}\end{array}$

(d) ${\text{Eff}}_{\text{overall}}=1-\frac{\text{423 K}}{\text{723 K}}=0\text{.}\text{415}\phantom{\rule{0.25em}{0ex}}\text{or}\phantom{\rule{0.25em}{0ex}}\text{41}\text{.}5%\text{}$

A coal-fired electrical power station has an efficiency of 38%. The temperature of the steam leaving the boiler is $\text{550}\text{º}\text{C}$ . What percentage of the maximum efficiency does this station obtain? (Assume the temperature of the environment is $\text{20}\text{º}\text{C}$ .)

Would you be willing to financially back an inventor who is marketing a device that she claims has 25 kJ of heat transfer at 600 K, has heat transfer to the environment at 300 K, and does 12 kJ of work? Explain your answer.

The heat transfer to the cold reservoir is ${Q}_{\text{c}}={Q}_{\text{h}}-W=\text{25}\phantom{\rule{0.25em}{0ex}}\text{kJ}-\text{12}\phantom{\rule{0.25em}{0ex}}\text{kJ}=\text{13}\phantom{\rule{0.25em}{0ex}}\text{kJ}$ , so the efficiency is $\mathit{Eff}=1-\frac{{Q}_{\text{c}}}{{Q}_{\text{h}}}=1-\frac{\text{13}\phantom{\rule{0.25em}{0ex}}\text{kJ}}{\text{25}\phantom{\rule{0.25em}{0ex}}\text{kJ}}=0\text{.}\text{48}$ . The Carnot efficiency is ${\mathit{\text{Eff}}}_{\text{C}}=1-\frac{{T}_{\text{c}}}{{T}_{\text{h}}}=1-\frac{\text{300}\phantom{\rule{0.25em}{0ex}}\text{K}}{\text{600}\phantom{\rule{0.25em}{0ex}}\text{K}}=0\text{.}\text{50}$ . The actual efficiency is 96% of the Carnot efficiency, which is much higher than the best-ever achieved of about 70%, so her scheme is likely to be fraudulent.

Unreasonable Results

(a) Suppose you want to design a steam engine that has heat transfer to the environment at $\text{270ºC}$ and has a Carnot efficiency of 0.800. What temperature of hot steam must you use? (b) What is unreasonable about the temperature? (c) Which premise is unreasonable?

Unreasonable Results

Calculate the cold reservoir temperature of a steam engine that uses hot steam at $\text{450}\text{º}\text{C}$ and has a Carnot efficiency of 0.700. (b) What is unreasonable about the temperature? (c) Which premise is unreasonable?

(a) $\text{–56.3ºC}$

(b) The temperature is too cold for the output of a steam engine (the local environment). It is below the freezing point of water.

(c) The assumed efficiency is too high.

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