# 4.5 The carnot cycle

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• Describe the Carnot cycle with the roles of all four processes involved
• Outline the Carnot principle and its implications
• Demonstrate the equivalence of the Carnot principle and the second law of thermodynamics

In the early 1820s, Sadi Carnot (1786−1832), a French engineer, became interested in improving the efficiencies of practical heat engines. In 1824, his studies led him to propose a hypothetical working cycle with the highest possible efficiency between the same two reservoirs, known now as the Carnot cycle    . An engine operating in this cycle is called a Carnot engine    . The Carnot cycle is of special importance for a variety of reasons. At a practical level, this cycle represents a reversible model for the steam power plant and the refrigerator or heat pump. Yet, it is also very important theoretically, for it plays a major role in the development of another important statement of the second law of thermodynamics. Finally, because only two reservoirs are involved in its operation, it can be used along with the second law of thermodynamics to define an absolute temperature scale that is truly independent of any substance used for temperature measurement.

With an ideal gas as the working substance, the steps of the Carnot cycle, as represented by [link] , are as follows.

1. Isothermal expansion. The gas is placed in thermal contact with a heat reservoir at a temperature ${T}_{\text{h}}.$ The gas absorbs heat ${Q}_{\text{h}}$ from the heat reservoir and is allowed to expand isothermally, doing work ${W}_{1}.$ Because the internal energy ${E}_{\text{int}}$ of an ideal gas is a function of the temperature only, the change of the internal energy is zero, that is, $\text{Δ}{E}_{\text{int}}=0$ during this isothermal expansion. With the first law of thermodynamics, $\text{Δ}{E}_{\text{int}}=Q-W,$ we find that the heat absorbed by the gas is
${Q}_{\text{h}}={W}_{1}=nR{T}_{\text{h}}\phantom{\rule{0.2em}{0ex}}\text{ln}\frac{{V}_{N}}{{V}_{M}}.$
2. Adiabatic expansion . The gas is thermally isolated and allowed to expand further, doing work ${W}_{2}.$ Because this expansion is adiabatic, the temperature of the gas falls—in this case, from ${T}_{\text{h}\phantom{\rule{0.2em}{0ex}}}\text{to}\phantom{\rule{0.2em}{0ex}}{T}_{\text{c}}.$ From $p{V}^{\gamma }=\phantom{\rule{0.2em}{0ex}}\text{constant}$ and the equation of state for an ideal gas, $pV=nRT$ , we have
$T{V}^{\text{γ}\phantom{\rule{0.2em}{0ex}}\text{−}\phantom{\rule{0.2em}{0ex}}\text{1}}=\text{constant},$

so that
${T}_{\text{h}}{V}_{N}{}^{\gamma -1}={T}_{\text{c}}{V}_{O}{}^{\gamma -1}.$
3. Isothermal compression . The gas is placed in thermal contact with a cold reservoir at temperature ${T}_{\text{c}}$ and compressed isothermally. During this process, work ${W}_{3}$ is done on the gas and it gives up heat ${Q}_{\text{c}}$ to the cold reservoir. The reasoning used in step 1 now yields
${Q}_{\text{c}}=nR{T}_{\text{c}}\phantom{\rule{0.2em}{0ex}}\text{ln}\frac{{V}_{O}}{{V}_{P}},$

where ${Q}_{\text{c}}$ is the heat dumped to the cold reservoir by the gas.
4. Adiabatic compression . The gas is thermally isolated and returned to its initial state by compression. In this process, work ${W}_{4}$ is done on the gas. Because the compression is adiabatic, the temperature of the gas rises—from ${T}_{\text{c}}\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}{T}_{\text{h}}$ in this particular case. The reasoning of step 2 now gives
${T}_{\text{c}}{V}_{P}{}^{\gamma -1}={T}_{\text{h}}{V}_{M}{}^{\gamma -1}.$

The total work done by the gas in the Carnot cycle is given by
$W={W}_{1}+{W}_{2}-{W}_{3}-{W}_{4}.$

what is motion?
where the solving of questions of this topic?
According to Nernst's distribution law there are about two solvents in which solutes undergo equilibria. But i don't understand how can you know which of two solvents goes bottom and one top? I real want to understand b'coz some books do say why they prefer one to top/bottom.
I need chapter 25 last topic
What is physics?
Abdulaziz
physics is the study of matter and energy in space and time and how they related to each other
Manzoor
interaction of matter and eneegy....
Abdullah
thanks for correcting me bro
Manzoor
What is electrostatics bassically?
study of charge at rest
wamis
A branch in physics that deals with statics electricity
Akona
what is PN junction?
Manzoor
please I don't understand the solution of the first example as in d working
what's the question? Write it here.
SABYASACHI
a cold body of 100°C and a hot body is of 100°F . Transfer heat = ?
you are given two metal spheres mounted on portable insulating support. Find a way to give them equal and opposite charges. you may use a glass rod rubbed with silk but may not touch it to the spheres. Do the spheres have to be of equal size for your method to work?
what is emotion?
Abdulaziz
in the 2nd example, for chapter 8.2 on page 3/3, I don't understand where the value 48uC comes from, I just couldn't get that value in my calculator.
are you talking about the capacitance combination problem
sam
please write the problem or send a snap of th page....I don't have the book in my vicinity.
SABYASACHI
yes, the 2nd example called Network of Capacitors on page 3/3 of section 8.2.
Anita
12 V = (Q1/12uF)+(Q1/6uF). So, Q1 = 12x4 = 48 uC.
sam
ohhhh OK thanks so much!!!!!!!
Anita
hello guys,, I'm asking to know something about, How can i know which solvent goes down and which does up in determination of partion coefficient(Nernst's distribution law). Please Need help because i have seen many contradictions via few of text books even some videos on youtube they don't say
Elia
what is electromagnetic force. do electric and magnetic force happen differently
yes
yes
Pranay
why
Godson
how?
Godson
when electric charge exert force on another electric charge then this force is known as electrostatic force and when a magnet exert force on another magnet then this force is known as magnetic force and when force exerted on magnet due to varying electric field then this electromagnetic force
Ilyas
Yes
Akona
derived the electric potential due to disk of charge
how can we derived potential electric due to the disk
aron
how can you derived electric potential of a disk
aron
how can you derived electric potential due to disk
aron
where is response?
aron
what is difference between heat and temperature?
temperature is the measure of degree of hotness or coldness. on the other hand, heat is the form of energy, which causes temperature. So we can safely say, heat is the reason and temperature is its consequence.
SABYASACHI
Heat is the reason and temperature is the consequences
Angela
how many liquid metals do we have
do we have gasses as metals
Jeffery
who knows should please tell us
yes...gallium & cesium
Idris
Hg is liquid. No metal gasses at standard temp and pressure
Shane
I don't ever understand any of this formulae
which formula
How to determine a temperature scale
what is the formula for absolute error
Nyro