On chapter 8, Entropy, Zemansky and Dittman's book Heat and Thermodynamics has a problem with following details:

A body of constant heat capacity $C_p$ and at a temperature $T_i$ is put in contact a reservoir at a higher temperature $T_f$. The pressure remanis constant while the body comes to equilibrium with the reservoir. Show that the entropy change in the universe is equal to $\Delta S = C_p(x-ln(1+x))$, where $x=-(T_f - T_i)/T_f$. Prove that the entropy change of the universe is postive.

Solution : As the process is isobaric ( pressure kept constant), the first law of thermodynamics gives, $TdS_{system} = C_p dT$

Integrating from $T_i$ to $T_f$, $(S_f - S_i)_{system} = \int_{T_i}^{T_f} \frac{C_p}{T} dT = C_p ln(\frac{T_f}{T_i}) = -C_p ln(\frac{T_i}{T_f}) = -C_p ln(\frac{T_f-T_f+T_i}{T_f}) = -C_p ln(1- \frac{T_f-T_i}{T_f}) = -C_p ln(1+ x) $

But, I am having trouble calculating $\Delta S_{surroundings}$. The change of entropy in surroundings is associated with the change of entropy of the thermal reservoirs. To assure that a reversible isobaric process occurs, the body must not be directly put into thermal contact to the reservoir with temperature $T_f$, as there is finite temperature difference between $T_i$ and $T_f$.

So, it the heating of the body should be done with a series of thermal reservoirs which have infinitesimal temperature difference with the current temperature of the body. But I am not sure how to calculate the entropy change for the infinite number of associated reservoirs here.

1) How to calculate the associated change of entropy here?

The solution given in the book suggests that $\Delta S_{surroundings} = \frac{Q_i}{T_f} - \frac{Q_i}{T_i}$ where $Q_i = C_p T_i$, thereby resulting in $\Delta S_{surroundings} = Q_i(\frac{1}{T_f} - \frac{1}{T_i}) =C_p(\frac{T_i}{T_f} - 1) = C_p x$

So, $\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings} = C_p(x- ln(1+x))$

But, I have confusions regarding this section.

Firstly, why the heat exchanged here has the particular value $Q_i = C_p T_i$?

2) What's actually happening in here?

  • $\begingroup$ @Aaron Stevens didn't get your comment! $\endgroup$ – Partha Sarker Oct 17 '18 at 2:11
  • $\begingroup$ @Aaron Stevens I mean if we need to calculate the entropy change of the surroundings won't we calculate it with how much heat the surroundings has rejected/ absorbed? $\endgroup$ – Partha Sarker Oct 17 '18 at 2:37
  • $\begingroup$ What do you think it mean? $\endgroup$ – Partha Sarker Oct 17 '18 at 3:17
  • $\begingroup$ And what's with that particular value of $Q_i$? $\endgroup$ – Partha Sarker Oct 17 '18 at 3:18
  • $\begingroup$ Do you find anything else wrong here? $\endgroup$ – Partha Sarker Oct 17 '18 at 3:19

When you determine the change in entropy for an irreversible process, the first step is to determine, using the first law of thermodynamics, the final state of the system. This has already been done for you in the problem statement.

The second step is to totally disregard the actual process and to focus, instead, on just the initial and final states. For the system, the initial and final states are $(T_i, P)$ and $(T_f,P)$. For the surroundings, modeled as an ideal infinite reservoir, the change from the initial state to the final state involves an amount of heat $C_p(T_f-T_i)$ that has been transferred out of the reservoir.

To determine the change in entropy for the combined system plus surroundings, we must devise an alternative reversible path between the same initial and final states, and determine the integral of dq/T for that path. To do this, we first entirely separate the system from the surroundings and devise alternate reversible paths for each of them separately.

For the system, rather than putting into contact with the actual reservoir, we put it into contact with an infinite sequence of constant temperature reservoirs at slightly different temperatures. This represents a reversible path for the system between its initial and final states. This results in the entropy change that was correctly determined in the book's solution.

Now for the reservoir. The key feature of an ideal infinite reservoir is that its change in entropy is always equal to the heat transferred to the reservoir $-C_p(T_f-T_i)$ divided by its absolute temperature, in this case $T_f$. This assumes that there is never any entropy generated within the reservoir itself in any process it is involved with (whether reversible or irreversible), but only transfer of entropy at its boundary. So, in this problem, for example, all the entropy generation is assumed to take place within the system. Therefore, the entropy change of the reservoir is just $$\Delta S_{surroundings}=\frac{-C_p(T_f-T_i)}{T_f}$$

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  • $\begingroup$ I get that all. But what's with the quantity $Q_i = C_p T_i$? $\endgroup$ – Partha Sarker Oct 17 '18 at 12:42
  • $\begingroup$ @ParthaSarker Is that something you are using, or does the book actually use that quantity in that part? $\endgroup$ – BioPhysicist Oct 17 '18 at 12:44
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    $\begingroup$ I have no idea. It makes no sense to me. $\endgroup$ – Chet Miller Oct 17 '18 at 12:44
  • $\begingroup$ The book used it. @Aaron Stevens $\endgroup$ – Partha Sarker Oct 17 '18 at 14:24

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