# 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.

#### Questions & Answers

hi guys can you teach me how to solve a logarithm?
how about a conceptual framework can you simplify for me? needed please
Villaflor
Hello what happens when electrone stops its rotation around its nucleus if it possible how
Afzal
I think they are constantly moving
Villaflor
yep what is problem you are stuck into context?
S.M
not possible to fix electron position in space,
S.M
Physics
Beatriz
yes of course Villa flor
David
equations of kinematics for constant acceleration
A bottle full of water weighs 45g when full of mercury,it weighs 360g.if the empty bottle weighs 20g.calculate the relative density of mercury and the density of mercury....pls I need help
well You know the density of water is 1000kg/m^3.And formula for density is density=mass/volume Then we must calculate volume of bottle and mass of mercury: Volume of bottle is (45-20)/1000000=1/40000 mass of mercury is:(360-20)/1000 kg density of mercury:(340/1000):1/50000=(340•40000):1000=13600
Sobirjon
the latter is true
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture...take density of water as 1g/cm3 and density of liquid 1.2g/cm3
Lila
plz hu can explain Heisenberg's uncertainty principle
who can help me with my problem about acceleration?
ok
Nicholas
how to solve this... a car is heading north then smoothly made a westward turn during the travel the speed of the car remains constant at 1.5km/h what is the acceleration of the car? the total travel time of the car as it smoothly changed its direction is 15 minutes
Vann
i think the acceleration is 0 since the car does not change its speed unless there are other conditions
Ben
yes I have to agree, the key phrase is, "the speed of the car remains constant...," all other information is not needed to conclude that acceleration remains at 0 during the entire time
Luis
who can help me with a relative density question
Lila
1cm3 sample of tin lead alloy has mass 8.5g.the relative density of tin is 7.3 and that of lead is 11.3.calculate the percentage by weight of tin in the alloy. assuming that there is no change of volume when the metals formed the alloy
Lila
morning, what will happen to the volume of an ice block when heat is added from -200°c to 0°c... Will it volume increase or decrease?
no
Emmanuel
hi what is physical education?
Kate
BPED..is my course.
Kate
No
Emmanuel
I think it is neither decreases nor increases ,it remains in the same volume because of its crystal structure
Sobirjon
100g of water is mixed with 60g of a liquid of relative density 1.2.assuming no changes in volume occurred,find the average relative density of the mixture. take density of water as 1g/cm3 and density of liquid as 1.2g/cm3
Lila
Sorry what does it means"no changes in volume occured"?
Sobirjon
volume can be the amount of space occupied by an object. But when an object does not change in shape it will still occupy the same space. Thats why the volume will still remain the same
Ben
Most soilds expand when heated but if it changes state at 0C it will have less volume. Ice floats because it is less dense ie a larger mass per unit volume.
Richard
how to calculate velocity
v=d/t
Emeka
his about the speed?
Villaflor
Villaflor
v=d/t
Nisha
hello bro hw is life with you
Mine is good. How about you?
Chase
Hi room of engineers
yes,hi sir
Okwethu
hello
akinmeji
Hello
Mishael
hello
Jerry
hi
Sakhi
hi
H.C
so, what is going on here
akinmeji
u are all wlc just ask your question anybody. can answer
Ajayi
good morning ppl
ABDUL
If someone has not studied Mathematics enough yet, should theu study it first then study Phusics or Study Basics of Physics whilst srudying Math as well?
whether u studied maths or not, it is advisable to start from d basics cuz it is essential to know dem
Nuru
yea you are right
wow, you got this w/o knowing math
Thomas
I guess that's it
Thomas
later people
Thomas
mathematics is everywhere
Anand
thanks but dat doesn't mean it is good without maths @Riaz....... Maths is essential in sciences particularly wen it comes to PHYSICS but PHYSICS must be started from the basic which may also help in ur mathematical ability
Nuru
A hydrometer of mass 0.15kg and uniform cross sectional area of 0.0025m2 displaced in water of density 1000kg/m3.what depth will the hydrometer sink
Lila
16.66 meters?
Darshik
16.71m2
aways
,i have a question of let me give answer
aways
the mass is stretched a distance of 8cm and held what is the potential energy? quick answer
aways
oscillation is a to and fro movement, it can also be referred to as vibration. e.g loaded string, loaded test tube or an hinged door
what property makes the magnet to break?
difference between charge and force
charges: these are groups of ions which can be positive or negative. while force is said to be push or pull of an object. Generally, charge deals with ions while force deals with particles.
akinmeji
define oscillation
Munachukwu
what is up thrust
upward force exerted by water on a body.
Matthew
A hydrometer of mass 0.15kg and of uniform cross sectional area 0.0025m2 displaced in water of density 1000kg/m3.what depth will the hydrometer sink
Lila
archimedes principle
Nicholas
mg=p(area x height)g
Nicholas
solve for height
Nicholas
what is matter
Anything that has mass. And it occupies space. So it has volume.
wenhe
stages of matter.
daniel
States of matter are solid liquid gas and plasma
wenhe
what are Newtown's law
Lila
An object as rest will remain at rest or in motion will remain at motion. Force is mass times acceleration. And a force will have equal and opposite reaction.
wenhe
do we have any thing like plasma as a states of matter
daniel
no
Nicholas
Plasma is a state of matter. It's much later on. The sun for example is not a gas ball. Its plasma.
wenhe
why is plasma a state of matter
Lila
In lower grades you are taught that there are 3 states of matter. But later on, you'll learn that if you eat gas enough. It turns to plasma
wenhe
Heat*
wenhe
yes we have plasma as a state of matter
akinmeji
it is anything that occupie space and has mass
Anand
I never head of that before
daniel
Anything that occupies space and has mass is matter.
wenhe
what is the difference between vapor and gas
Lila
Plasma is best described as an ionized gas because it is made up of positively and negatively charged particles.
akinmeji
Vapour is gas.
wenhe
Aside plasma, we also have Bose- Einstein. this is also another state of matter
akinmeji
was bose Einstein discovered by Albert Einstein
Lila
We have like 6 states or more. But those are newly discovered.
wenhe
OK, thanks at list I have experience about that one now.
daniel
Einstein​ predicted it
wenhe
With another person
wenhe
what are the other states
Lila
Fermionic condensate
wenhe
OK what is Albert Einstein
daniel
And maybe more. I just rmb those. Tbh for school you only need to know 4.
wenhe
Albert Einstein​is the name of a scientist
wenhe
OK
daniel
what is the physical state of water and salt
Lila
can I ask off topic question?
Kristine
Both are matter. So can theoretically exist in all those states. But in room temperature, water is liquid and salt is solid.
wenhe
Yes
wenhe
Kristine
I'm not very good at maths. But if I can answer... Lol
wenhe
The reflector of a radiotelescope is in the shape of a parabola revolve its axis, if the diameter of the reflector is 400ft. above the vertex of the parabola, what should be the depth of reflector?
Kristine
Are there any other factors given?
wenhe
Or just that?
wenhe
matter is merely energy condensed to a slow vibration
Vashist
Cuz with those factors, I can't solve it.
wenhe
none
Kristine
Well. There must be another factor given or something. Like an equation of the parabola. Or else I can't solve it
wenhe
explain why water and salt are compounds
Lila
That's chem. But both water and salt are made of more than 1 element. So they are compounds.
wenhe
Compound meaning 2 or more things added together.
wenhe
salt chemically is combination of sodium n chloride whereas water is oxygen n hydrogen
Friday
deficiency of vitamin E cause ?
Friday
I don't know if this is the equation or formula your looking for; (x-h)²=4c(y-k) (x-h)²=-4c(y-k) (y-k)²=-4c(x-h) (y-k)²=4c(x-h)
Kristine
If the parabola opens downward, the equation will be (x-h)²=-4c(y-k). If the parabola opens upward, (x-h)²=4c(y-k). If the parabola opens to the left it will be -4c and if the parabola opens to the right it will be 4c
Kristine