4.5 The carnot cycle

 Page 1 / 7
• 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 the weight of the earth in space
As w=mg where m is mass and g is gravitational force... Now if we consider the earth is in gravitational pull of sun we have to use the value of "g" of sun, so we can find the weight of eaeth in sun with reference to sun...
Prince
g is not gravitacional forcé, is acceleration of gravity of earth and is assumed constante. the "sun g" can not be constant and you should use Newton gravity forcé. by the way its not the "weight" the physical quantity that matters, is the mass
Jorge
Yeah got it... Earth and moon have specific value of g... But in case of sun ☀ it is just a huge sphere of gas...
Prince
Thats why it can't have a constant value of g ....
Prince
not true. you must know Newton gravity Law . even a cloud of gas it has mass thats al matters. and the distsnce from the center of mass of the cloud and the center of the mass of the earth
Jorge
please why is the first law of thermodynamics greater than the second
define electric image.obtain expression for electric intensity at any point on earthed conducting infinite plane due to a point charge Q placed at a distance D from it.
explain the lack of symmetry in the field of the parallel capacitor
pls. explain the lack of symmetry in the field of the parallel capacitor
Phoebe
does your app come with video lessons?
What is vector
Vector is a quantity having a direction as well as magnitude
Damilare
tell me about charging and discharging of capacitors
a big and a small metal spheres are connected by a wire, which of this has the maximum electric potential on the surface.
3 capacitors 2nf,3nf,4nf are connected in parallel... what is the equivalent capacitance...and what is the potential difference across each capacitor if the EMF is 500v
equivalent capacitance is 9nf nd pd across each capacitor is 500v
santanu
four effect of heat on substances
why we can find a electric mirror image only in a infinite conducting....why not in finite conducting plate..?
because you can't fit the boundary conditions.
Jorge
what is the dimensions for VISCOUNSITY (U)
Branda
what is thermodynamics
the study of heat an other form of energy.
John
heat is internal kinetic energy of a body but it doesnt mean heat is energy contained in a body because heat means transfer of energy due to difference in temperature...and in thermo-dynamics we study cause, effect, application, laws, hypothesis and so on about above mentioned phenomenon in detail.
ing
It is abranch of physical chemistry which deals with the interconversion of all form of energy
Vishal
what is colamb,s law.?
it is a low studied the force between 2 charges F=q.q`\r.r
Mostafa
what is the formula of del in cylindrical, polar media
prove that the formula for the unknown resistor is Rx=R2 x R3 divided by R3,when Ig=0.