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In my thermodynamics schoolbook, it is asked to apply the Clausius inequality to a Carnot cycle. According to the book the answer is to evaluate $\oint\frac{\delta Q}{T}$ over the cycle which should yield $$ \oint\frac{\delta Q}{T} = \frac{Q_H}{T_H}+\frac{Q_C}{T_C}=0 $$

  1. $Q_H$ is the heat transferred from the hot reservoir (positive)
  2. $T_H$ is the temperature of the hot reservoir
  3. $Q_C$ is the heat transferred to the cold reservoir (negative)
  4. $T_C$ is the temperature of the cold reservoir

While it is easy enough to show this is true using $\delta Q =TdS$, my schoolbook doesn't introduce entropy until later. Therefore, my question is how to prove this equality without using entropy ?

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  • $\begingroup$ Lay out a closed cycle involving the two adiabats and the two isotherms. You won't be able to design one for which this equation is not satisfied. Try it and see. $\endgroup$ – Chet Miller Jun 14 '20 at 21:54
  • $\begingroup$ Hey there, is there a general way to prove that without using $pV=RT$ ? I know that from $pV=RT$ one can show that for a Carnot cycle $\frac{Q_C}{Q_H}=\frac{T_C}{T_H}$, which then shows that $\frac{Q_H}{T_H}+\frac{Q_C}{T_C}=0$. Otherwise, I don't know how to show $\frac{Q_H}{T_H}+\frac{Q_C}{T_C}=0$ always holds true for those types of cycles. $\endgroup$ – Pacific Jun 14 '20 at 22:43
  • $\begingroup$ What's wrong with using pV=RT? $\endgroup$ – Chet Miller Jun 14 '20 at 23:43
  • $\begingroup$ I thought this was valid for any fluid, not just ideal gases. $\endgroup$ – Pacific Jun 15 '20 at 0:10
  • $\begingroup$ It, of course, should be true for any fluid. But solving the equations for, say, a van der Waals gas might be a little challenging. Maybe you are up to such a challenge. $\endgroup$ – Chet Miller Jun 15 '20 at 0:25
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I only know a long ride for this, sorry :/

You have to imagine two processes iabf and if that have the same work $\Delta W_{iabf}=\Delta W_{if}$: enter image description here (Dashed lines are adiabatic curves and solide lines are isothermes)

The internal energy variation is the same for both processes and $\Delta Q_{ab}=\Delta Q_{if}$.

Now imagine a system S' that provides energy $\Delta Q$ to the cycle C(system S). enter image description here The only way to enter energy is through an isotherm, with temperature $T_0=T_H$. This works as the hot reservoir. The ab isotherm has $T<T_0$, which is your $T_C$. The system S' transfers heat($\Delta Q' $) to S: $$\Delta Q' = T_H \frac{\Delta Q}{T_C}$$

The cd path has the opposite direction. Try drawing one Carnot Cycle above the other and you will see they cancel each other.

Now imagine infine mini carnot cycles inside C, the sum of all cycles cancel each other:

$$ \Sigma \Delta Q' = \oint_C T_C \frac{dQ}{T} = 0 $$ *T varies in each mini cycle.

Moving $T_C$ $$\oint_C \frac{dQ}{T} = 0$$

I believe your book assume reversible processes from the begining. Otherwise, according to the second law:

$$\oint_C \frac{dQ}{T} \leq 0$$

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  • $\begingroup$ Hey there, thanks for answering my question. I'm a little bit confused about $\Delta Q'$ though, is it the heat exchanged in the c-d isothermal ? If so, why does $\Delta Q'$ equal $\frac{T_H}{T_C}\Delta Q$ ? Does that come from $pV=RT$ ? Or can we show that $\Delta Q' = \frac{T_H}{T_C}\Delta Q$ without using $pV=RT$ ? $\endgroup$ – Pacific Jun 14 '20 at 21:17
  • $\begingroup$ cd is an isotherm. The relation in $\Delta Q'$ comes from a reversible cycle between two reservatories, one with temperature $T_H$ and other with $T_C$. Or you can think about the thermal effitiency: $\eta=1- \frac{T_C}{T_H}=1-\frac{Q_C}{Q_H}$. Thus, $Q_H=T_C \frac{Q_C}{T_H}$ which is written with a diffferent notation: $\Delta Q'=\frac{T_H}{T_C} \Delta Q$ . $\endgroup$ – Paula Ferreira Jun 15 '20 at 18:07

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