# 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.

how to prove that Newton's law of universal gravitation F = GmM ______ R²
sir dose it apply to the human system
prove that the centrimental force Fc= M1V² _________ r
prove that centripetal force Fc = MV² ______ r
Kaka
how lesers can transmit information
griffts bridge derivative
below me
please explain; when a glass rod is rubbed with silk, it becomes positive and the silk becomes negative- yet both attracts dust. does dust have third types of charge that is attracted to both positive and negative
what is a conductor
Timothy
hello
Timothy
below me
why below you
Timothy
no....I said below me ...... nothing below .....ok?
dust particles contains both positive and negative charge particles
Mbutene
corona charge can verify
Stephen
when pressure increases the temperature remain what?
what is frequency
define precision briefly
CT scanners do not detect details smaller than about 0.5 mm. Is this limitation due to the wavelength of x rays? Explain.
hope this helps
what's critical angle
The Critical Angle Derivation So the critical angle is defined as the angle of incidence that provides an angle of refraction of 90-degrees. Make particular note that the critical angle is an angle of incidence value. For the water-air boundary, the critical angle is 48.6-degrees.
okay whatever
Chidalu
pls who can give the definition of relative density?
Temiloluwa
the ratio of the density of a substance to the density of a standard, usually water for a liquid or solid, and air for a gas.
Chidalu
What is momentum
mass ×velocity
Chidalu
it is the product of mass ×velocity of an object
Chidalu
how do I highlight a sentence]p? I select the sentence but get options like copy or web search but no highlight. tks. src
then you can edit your work anyway you want
Wat is the relationship between Instataneous velocity
Instantaneous velocity is defined as the rate of change of position for a time interval which is almost equal to zero
Astronomy
The potential in a region between x= 0 and x = 6.00 m lis V= a+ bx, where a = 10.0 V and b = -7.00 V/m. Determine (a) the potential atx=0, 3.00 m, and 6.00 m and (b) the magnitude and direction of the electric ficld at x =0, 3.00 m, and 6.00 m.