In Wikipedia it is said that the inequality holds only for reversible processes. Why is that?
And also from the statement we arrive at this conclusion:
" In other words, the Clausius statement states that it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir."
How do reach this conclusion? Is the conclusion an interpretation of the in-equation, or arises from intuition ?
Thanks
In Wikipedia it is said that the inequality holds only for reversible processes. Why is that?
First, the Clausius theorem applies to engine cycles, not individual processes. Second, the equality applies to reversible cycles and the inequality applies to irreversible cycles.
" In other words, the Clausius statement states that it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir."
This is not the Clausius theorem. This statement is based on the Clausius' statement the second law or
No refrigeration or heat pump can operate without a net work input.
The Clausius theorem and inequality applies to heat engines, not heat pumps and refrigerators. It is given by
$$\int\frac{dQ}{T}\le 0$$
where $dQ$ is the heat entering the system at any point in the cycle and $T$ is the temperature at the point of entry to the system.
The inequality applies to an irreversible cycle of a real heat engine and says that the entropy given to the environment during the cycle is larger than the entropy transferred to the engine by the environment. Since entropy transfer is always in the form of heat, it means that for an irreversible cycle less heat is available to perform work.
Hope this helps.
The Clausius inequality becomes an equality for reversible processes because reversible processes are defined as those during which the system is always at equilibrium with its surroundings (or at most has infinitesimal deviations therefrom so that the process can actually proceed). I find it is helpful to remind oneself where the Clausius inequality comes from in the first place. A succinct relationship between the reversible and irreversible work done on or by a system can be easily argued, \begin{gather*} \delta w - \delta w_{rev} \geq 0 \end{gather*} In other words, we will always get the most work out of a reversible process. If we then write out the first law in its differential form, \begin{align*} dE &= \delta q + \delta w \\ &= \delta q_{rev} + \delta w_{rev} \end{align*} where in the latter we have noted that the first law holds for any process, reversible or irreversible. Then, we can say, \begin{align*} \delta w - \delta w_{rev} &= \delta q_{rev} - \delta q \\ \implies \delta q_{rev} - \delta q &\geq 0 \\ \implies \delta q_{rev} &\geq \delta q \\ \implies dS &\geq \frac{\delta q}{T} \end{align*} Since the process that involves the most work is a reversible process, we similarly conclude that the process involving the minimal exchange of heat is the reversible process, which gives exactly the Clausius entropy, \begin{gather*} dS = \frac{\delta q_{rev}}{T} \end{gather*} From the Clausius inequality, we thus know that the reversible processes are those that will give the smallest changes in entropy.
To your second question, we can show that heat must exclusively flow from a body with higher temperature to a body with lower temperature using the Clausius inequality. To accomplish this, we simply focus on some system of interest that is at a temperature of $T_{sys}$. The system and all of its surroundings constitute the entire universe, which we can reasonably assume is an isolated system. It is relatively straightforward to argue that in general, \begin{gather*} \delta q_{sys} = -\delta q_{surr} \end{gather*} But we now turn our attention to the entropy, \begin{align*} &dS_{sys} + dS_{surr} = dS \\ \implies &\frac{\delta q_{sys}}{T_{sys}} + \frac{\delta q_{surr}}{T_{surr}} \geq 0 \\ \implies &\frac{\delta q_{sys}}{T_{sys}} - \frac{\delta q_{sys}}{T_{surr}} \geq 0 \\ \implies &\frac{\delta q_{sys}}{T_{sys}} \geq \frac{\delta q_{sys}}{T_{surr}} \\ \implies &\frac{T_{surr}}{T_{sys}} \delta q_{sys} \geq \delta q_{sys} \end{align*} where we have invoked the second law and Clausius inequality such that $dS \geq \frac{\delta q}{T} \geq 0$ for isolated systems. We are almost done. Consider the case where $\delta q_{sys} > 0$. What does this imply about the relative temperatures? In order for this inequality to hold, we must have $T_{surr} > T_{sys}$. In other words, if heat is flowing into the system, then the surroundings must be at a higher temperature than the system. Consider the other case where $\delta q_{sys} < 0$. Now what does this imply about the temperatures? Now, we must flip the inequality which gives $T_{surr} < T_{sys}$. In other words, if the system loses heat, then the system must be at a higher temperature than the surroundings. Finally, if the system does not gain or lose heat, then $T_{sys} = T_{surr}$, and the system is already at thermal equilibrium. Thus, our assertion about heat flowing from hotter to colder regions/objects is entirely correct. It turns out that the reason processes proceed in the direction they do is often nothing more than the machinations of entropy!
